Dependencies

Let \(\mathfrak {U}_1\) be a C*-subalgebra of the C*-algebra \(\mathfrak {U}_2\). If \(a\) is an element of \(\mathfrak {U}_1\), then the spectrum \(\sigma _{\mathfrak {U}_1}(a)\) of \(a\) when viewed as an element of \(\mathfrak {U}_1\) is the same as the spectrum \(\sigma _{\mathfrak {U}_2}(a)\) of \(a\) when viewed as an element of \(\mathfrak {U}_2\)*

If \(\mathfrak {U}\) is a *-algebra and there exists a norm on \(\mathfrak {U}\) with the C*-norm property and with respect to which \(\mathfrak {U}\) is closed, then this norm is unique.

Let \(\mathcal{H}\) be a Hilbert space and \(\mathcal{S}\) a *-subalgebra of \(\mathcal{B}(\mathcal{H})\) that contains the identity. Then \(\mathcal{S}\) is strongly dense in \(\mathcal{S}''\).

Let \(V_1\) be a normed space and \(V_2\) a Banach space. Suppose \(W\) is a dense subspace of \(V_1\) and \(T: W \rightarrow V_2\) is a bounded linear map. Then there exists a unique bounded linear map \(\widetilde{T}: V_1 \rightarrow V_2\) such that \(\widetilde{T}|_W = T\). Furthermore, the norm of \(\widetilde{T}\) equals the norm of \(T\).

If \(\mathfrak {U}\) is a unital C*-algebra, then it has a faithful representation. Specifically, its universal representation ¹ is faithful.

A non-zero Abelian C*-algebra \(\mathcal{A}\) is primitive if and only if \(\mathcal{A} = \mathbb {C} \mathbf{1}\).

A smooth manifold \(M\) admits a smooth Lorentz metric if and only if it admits a smooth, nowhere-vanishing global vector field.

Consider an open subset \(\mathbf{O}\) of a smooth, connected, four dimensional Hausdorff manifold \(M\) that is equipped with a Lorentzian metric. \(\mathbf{O}\) is said to be causally convex if for every \(p\) and \(r\) in \(\mathbf{O}\) and \(q\) in \(M\) the existence of a future-directed, time-like curve from \(p\) to \(q\) and a future-directed, time-like curve from \(q\) to \(r\) implies that \(q\) is a member of \(\mathbf{O}\).

Consider a smooth, connected, four dimensional Hausdorff manifold \(M\) that is equipped with a Lorentzian metric. \(M\) is said to be strongly causal at \(p \in M\) if \(p\) has arbitrarily small causally convex neighborhoods. \(M\) is said to be strongly causal if \(M\) is strongly causal at every \(p \in M\).

The following three conditions on a smooth, connected, four dimensional manifold \(M\) with Hausdorff manifold topology are equivalent:

1. \(M\) is strongly causal;

2. the Alexandrov Topology agrees with the manifold topology;

3. the Alexandrov Topology is Hausdorff.

A Lorentzian spacetime is a smooth, connected, four dimensional manifold equipped with a smooth, nowhere-vanishing global vector field \(t^a\) and associated Lorentzian metric. In addition it is equipped with an associated Hausdorff Alexandrov topology.

For Lorentzian spacetime \(M\) the image \(\pi _\omega (a)\) of a self-adjoint member \(a\) of the local algebra \(\mathfrak {U}(\mathbf{B})\) under the GNS *-homomorphism \(\pi _\omega \) of a state \(\omega \) on \(\mathfrak {U}(\mathbf{B})\) is self-adjoint and thus corresponds to an “observable”. Any “observable” corresponding to such a self-adjoint \(\pi _\omega (a)\) is called a local observable.

Alexandrov topology on a spacetime \(M\) is the topology generated by the basis consisting of all sets of the form \(I^+(p) \cap I^-(q)\) for points \(p\) and \(q\) in \(M\).

The \(\iota \)-fold amplification \(\iota \cdot \pi := \bigoplus _{i : \iota } \pi \) of a single representation is the direct sum of \(\iota \) copies of \(\pi \); \(\pi \) embeds as each summand.

The double spacelike complement \(\mathbf{B} \mapsto \mathbf{B}^{\perp \perp }\) is a closure operator on the regions of a Lorentzian spacetime: it is monotone, extensive (\(\mathbf{B} \subseteq \mathbf{B}^{\perp \perp }\)), and idempotent (\(\mathbf{B}^{\perp \perp \perp \perp } = \mathbf{B}^{\perp \perp }\)). Its three defining properties are exactly the order structure of the spacelike complement.

For a spacetime \(M\) and \(p\) in \(M\) the set \(J^+(p) = \{ q \in M : p \prec q\} \) is called the causal future of \(p\). \(J^-(p) = \{ q \in M : q \prec p\} \) is called the causal past of \(p\). The causal future of a set \(S \subset M\) is the union of the causal future of each element of the set

The causal past of \(S\) is defined similarly

A causal trip segment is a curve which is a future-oriented, causal geodesic. (Note a causal geodesic is possibly degenerate.) A causal trip is a curve which is piecewise a future-oriented, causal geodesic: a finite chain of causal trip segments joined at matching endpoints. A causal trip from \(p\) to \(q\) is a causal trip with past endpoint \(p\) and future endpoint \(q\). We write \(p \prec q\) if and only if there exists a causal trip from \(p\) to \(q\).

A region is causally complete if it equals its own double complement, \(\mathbf{B}^{\perp \perp } = \mathbf{B}\) (a fixed point of the causal closure operator). These are the regions on which algebraic QFT is naturally indexed.

For a spacetime \(M\) and \(p\) in \(M\) the set \(I^+(p) = \{ q \in M : p \ll q\} \) is called the chronological future of \(p\). \(I^-(p) = \{ q \in M : q \ll p\} \) is called the chronological past of \(p\). The chronological future of a set \(S \subset M\) is the union of the chronological future of each element of the set

The chronological past of \(S\) is defined similarly

Consider two sets \(\mathbf{O}_1\) and \(\mathbf{O}_2\) in a spacetime. \(\mathbf{O}_1\) and \(\mathbf{O}_2\) are completely spacelike with respect to each other if every \(p_1\) in \(\mathbf{O}_1\) is spacelike related to every \(p_2\) in \(\mathbf{O}_2\).

A covariant quasilocal algebra bundles a Haag-Kastler net (87), a quasilocal algebra of that net (82), and a proof that the embeddings are covariance-compatible for every Lorentz transformation. On such data the quasilocal lift (138) exists for every \(L\) by 140, yielding the covariance action \(L \mapsto \beta _L\) on the quasilocal algebra; the trivial net provides an instance. This is the natural home for the covariance dynamics: the compatibility hypothesis of 140 becomes structural data rather than a side condition.

Given a Haag-Kastler net (87), a covariant family of local states assigns to every region \(\mathbf{B}\) a state \(\omega _{\mathbf{B}}\) on the local algebra \(\mathfrak {U}(\mathbf{B})\) such that, for every Lorentz transformation \(L\) and every \(a \in \mathfrak {U}(\mathbf{B})\), one has \(\omega _{\mathbf{B}}(a) = \omega _{L\cdot \mathbf{B}}(\alpha _L a)\), where \(\alpha _L\) is the covariance isomorphism of 86. The local states are thus intertwined by the Lorentz action.

Given a Haag-Kastler net on a Lorentzian spacetime (176), a covariant family of local states assigns to every region \(\mathbf{B}\) a state \(\omega _{\mathbf{B}}\) on \(\mathfrak {U}(\mathbf{B})\) such that, for every identity-component isometry \(\varphi \) and every \(a \in \mathfrak {U}(\mathbf{B})\), \(\omega _{\mathbf{B}}(a) = \omega _{\varphi (\mathbf{B})}(\alpha _\varphi a)\), where \(\alpha _\varphi \) is the covariance isomorphism of 175.

A curve is an equivalence class of paths equivalent under homeomorphisms of the parameter space. A smooth curve is an equivalence class of smooth paths equivalent under diffeomorphisms of the parameter space.

Let \(\mathcal{A}\) be an algebra represented by the bounded operators \(\pi (\mathcal{A})\) on the Hilbert space \(\mathcal{H}\). A vector \(\Omega \) in \(\mathcal{H}\) is said to be a cyclic vector if the set

is dense in \(\mathcal{H}\).

The direct sum \(\bigoplus _i \pi _i : A \to \mathcal{B}(\ell ^2(\iota , H))\) of a family of \(*\)-representations acts coordinatewise: \((\bigoplus _i \pi _i)(a)\) is the diagonal operator \(x \mapsto (\pi _i(a)\, x_i)_i\) on the \(\ell ^2\)-direct sum. It is a \(*\)-representation, the diagonal being uniformly bounded by \(\Vert a\Vert \) since each \(\pi _i\) is contractive.

An intertwiner \(T : H_1 \to H_2\) between \(\pi _1\) and \(\pi _2\) satisfies \(T(\pi _1(a)\, x) = \pi _2(a)(T x)\); intertwiners are closed under sums, scalars, composition, and the adjoint (the adjoint of a \(\pi _1 \to \pi _2\) intertwiner is a \(\pi _2 \to \pi _1\) intertwiner). Two representations are disjoint when the only intertwiner between them is \(0\). Disjointness is symmetric (take adjoints), and a representation on a nonzero Hilbert space is never disjoint from itself; more generally, unitarily equivalent representations on nonzero spaces are never disjoint, since the implementing unitary is a nonzero intertwiner.

A point \(p\) in a spacetime \(M\) is the endpoint of a path \(\mu \) or its associated curve if it is a member of the image \(\mu (\partial \Sigma )\) of the boundary \(\partial \Sigma \) of the parameter space under \(\mu \). If \(\mu \) is a smooth path and its associated smooth curve is timelike and future-oriented, then an endpoint \(p\) is a past endpoint if it is the image under \(\mu \) of the lesser of the two boundary components of \(\partial \Sigma \). It is a future endpoint if it is the image under \(\mu \) of the greater of the two boundary components of \(\partial \Sigma \).

A state \(\omega \) is an extreme point of the (convex) state space if it is not a nontrivial convex combination of two distinct states: whenever \(\omega = t\, \omega _1 + (1-t)\, \omega _2\) with \(0 {\lt} t {\lt} 1\) and \(\omega _1, \omega _2\) states, then \(\omega _1 = \omega _2\).

Given a one-parameter subgroup \(t \mapsto L_t\) of the inhomogeneous Lorentz group, the covariance-flow automorphism family of \(\mathfrak {U}\) is \(t \mapsto \beta _{L_t}\), the covariance action evaluated along the flow.

A Killing flow fixing a region \(\mathbf{B}\) is a map \(t \mapsto \varphi _t\) from \(\mathbb {R}\) into the stabilizer \(\mathrm{Stab}(\mathbf{B})\). Through the stabilizer automorphism (196) it induces, for each \(t\), an automorphism \(\hat\alpha _{\varphi _t}\) of the single local algebra \(\mathfrak {U}(\mathbf{B})\) - the time evolution of \(\mathfrak {U}(\mathbf{B})\) along the flow.

A future-oriented smooth curve is a smooth curve with a tangent vector that is future-pointing at every point. A past-oriented smooth curve is a smooth curve with a tangent vector that is past-pointing at every point.

For any \(p\) in a spacetime \(M\) a timelike tangent vector \(v \in TM|_p\) is future-pointing if \(g|_p(t,v)\) is negative and past-pointing if \(g|_p(t,v)\) is positive. A null tangent vector \(n \in TM|_p\) is future-pointing if it is the limit of future-pointing timelike tangent vectors and it is past-pointing if it is the limit of past-pointing timelike tangent vectors.

The pure translations of the inhomogeneous Lorentz group: an element is a pair \((\text{linear}, \text{translation})\), and a pure translation \(\mathrm{translationSub}(n) = (\mathrm{id}, n)\) has trivial linear part, so \(n \mapsto (\mathrm{id}, n)\) embeds the additive group of the spacetime carrier (\(\mathrm{translationSub}(0) = 1\), \(\mathrm{translationSub}(n+m) = \mathrm{translationSub}(n)\, \mathrm{translationSub}(m)\)). The one-parameter translation flow in a direction \(n\) is \(\mathrm{translationFlow}(n)(t) = (\mathrm{id}, t \cdot n)\), which is a one-parameter subgroup. A one-parameter subgroup \(\gamma \) is a future-timelike translation when \(\gamma = \mathrm{translationFlow}(n)\) for some future-pointing timelike \(n\) — i.e. \(n\) lies in the forward Minkowski cone at the origin. This wires in the translation subgroup and its causal structure, giving the concrete predicate with which the abstract future-timelike-translation parameter of 149 is discharged.

A state \(\omega \) on the quasilocal algebra \(\mathfrak {U}\) is a ground state for a one-parameter subgroup \(t \mapsto L_t\) of the inhomogeneous Lorentz group (e.g. a translation or boost flow) when it is invariant under the flow and, in a GNS representation reproducing \(\omega \) and implementing the flow by unitaries \(U(t)\) fixing \(\Omega \), the one-parameter unitary group \(t \mapsto U(t)\) has positive energy (147). This is the ground-state (\(\beta \to \infty \), spectrum-condition) counterpart of the covariance-flow KMS state 167: the stationary state whose flow generator, the Hamiltonian for a timelike flow, is positive. Two Stone-free consequences: a ground state is flow-invariant, and its implementing unitary group is strongly continuous. It is the Minkowski analogue of the curved Killing-flow ground state 207.

A state \(\omega \) on \(\mathfrak {U}(\mathbf{B})\) is a ground state for a one-parameter Killing flow \(t \mapsto \varphi _t\) into \(\mathrm{Stab}(\mathbf{B})\) when it is invariant under the flow and, in a GNS representation reproducing \(\omega \) and implementing the flow by unitaries \(U(t)\) fixing \(\Omega \), the one-parameter unitary group \(t \mapsto U(t)\) has positive energy (147). This is the ground-state (\(\beta \to \infty \), spectrum-condition) counterpart of the Killing-flow KMS state 204: it selects the stationary state whose Killing-flow generator, the local Hamiltonian, is positive. Two Stone-free consequences: a ground state is flow-invariant, and its implementing unitary group is strongly continuous. The positive-energy condition is the bounded-generator scaffold; the faithful unbounded form is Stone-gated.

A Haag-Kastler net on Minkowski spacetime is the bundling of the data of 80 together with the properties of 81, 83, 85, and 86. In the Lean formalization this is a single structure whose fields are the assignment \(\mathbf{B} \mapsto \mathfrak {U}(\mathbf{B})\) and proofs that this assignment satisfies the four remaining axioms. Theorems about AQFT take an instance of this structure as a hypothesis and invoke each axiom as a projection.

A Haag-Kastler net in curved spacetime on a Lorentzian spacetime is the bundling of the data of 170 together with the properties of 171, 172, 174, and 175. In the Lean formalization this is a single structure whose fields are the assignment \(\mathbf{B} \mapsto \mathfrak {U}(\mathbf{B})\) and proofs that this assignment satisfies the four remaining axioms. Theorems about AQFT in curved spacetime take an instance of this structure as a hypothesis and invoke each axiom as a projection.

A state \(\omega \) on the quasilocal algebra of a covariant quasilocal algebra (142) is (Poincaré-)invariant if it is a fixed point of the dual covariance action: \(\omega (\beta _L a) = \omega (a)\) for every Lorentz transformation \(L\) and observable \(a\). This is the invariance condition of a vacuum state; further conditions (e.g. the spectrum condition) are imposed separately.

A \(*\)-representation \(\pi : A \to \mathcal{B}(H)\) is irreducible if every bounded operator \(T\) commuting with all \(\pi (a)\) is a scalar multiple of the identity, \(T = c \cdot 1\).

Let \(\mathbf{B}\) be any basis element of the Alexandrov topology on a Lorentzian spacetime \(M\), i.e. any set of the form \(I^+(p) \cap I^-(q)\).

A member \(\varphi \) of the group of isometries of \(M\) connected to the identity acts on \(\mathfrak {U}(\mathbf{B})\) as follows

where \(\varphi (\mathbf{B})\) is the image of the basis element \(\mathbf{B}\) under the isometry \(\varphi \) and \(\alpha _\varphi \) is a unital *-isomorphism generated by \(\varphi \). The map \(\alpha _\varphi \) is such that (1) for the identity isometry \(\mathbf{1}\) it satisfies

(2) for all appropriate \(a\), \(\varphi \), and \(\varphi '\) it satisfies

and (3) for Alexandrov topology basis elements \(\mathbf{B}_\iota \subset \mathbf{B}_\kappa \) and the unital *-monomorphism \(i\) of Axiom 2 (Isotony) \(\alpha _\varphi \) commutes with \(i\). In other words the following diagram

commutes.

Let \(\mathbf{B}_1\) and \(\mathbf{B}_2\) be any two basis elements of the Alexandrov topology on Minkowski spacetime, i.e. any two sets of the form \(I^+(p_1) \cap I^-(q_1)\) and \(I^+(p_2) \cap I^-(q_2)\).

If \(\mathbf{B}_1 \subset \mathbf{B}_2\) then \(\mathfrak {U}(\mathbf{B}_1) \subset \mathfrak {U}(\mathbf{B}_2)\), where inclusion is implemented by

a unital *-monomorphism.

Let \(\mathbf{B}_1\) and \(\mathbf{B}_2\) be any two basis elements of the Alexandrov topology on a Lorentzian spacetime, i.e. any two sets of the form \(I^+(p_1) \cap I^-(q_1)\) and \(I^+(p_2) \cap I^-(q_2)\).

If \(\mathbf{B}_1 \subset \mathbf{B}_2\) then \(\mathfrak {U}(\mathbf{B}_1) \subset \mathfrak {U}(\mathbf{B}_2)\), where inclusion is implemented by

a unital *-monomorphism.

A state \(\omega \) on \(A\) is \((\alpha , \beta )\)-KMS for a one-parameter automorphism group \(\alpha \) (155) at inverse temperature \(\beta \) if for every \(a, b \in A\) the correlation function \(t \mapsto \omega (a\, \alpha _t b)\) extends to a function \(F\) on the closed strip \(0 \le \operatorname {Im} z \le \beta \) that is continuous there, holomorphic on the open strip, bounded, and whose boundary value on \(\operatorname {Im} z = \beta \) is \(t \mapsto \omega (\alpha _t b\, \cdot a)\).

A state \(\omega \) on the quasilocal algebra \(\mathfrak {U}\) is a KMS state for the covariance flow \(L\) at inverse temperature \(\beta \) if it satisfies the KMS condition (156) for the induced one-parameter automorphism group (166).

A state \(\omega \) on \(\mathfrak {U}(\mathbf{B})\) is a KMS state for the Killing flow \(\varphi \) at inverse temperature \(\beta \) if it satisfies the KMS condition (156) for the induced one-parameter automorphism group (203). When \(\beta {\gt} 0\) such a state is automatically invariant under the flow (162), so it induces, by the stabilizer GNS unitary (198), a unitary representation of the flow on its GNS space - the modular/thermal time evolution.

For any basis element \(\mathbf{B}\) of the Alexandrov topology on Minkowski spacetime, i.e. any set of the form \(I^+(p) \cap I^-(q)\), there is a corresponding abstract C*-algebra \(\mathfrak {U}(\mathbf{B})\)

and when \(\mathbf{B}\) is the empty set, we have the distinguished correspondence

where \(\mathbf{1}\) is the multiplicative identity in the abstract C*-algebra \(\mathbb {C} \mathbf{1}\).

For any basis element \(\mathbf{B}\) of the Alexandrov topology on a Lorentzian spacetime, i.e. any set of the form \(I^+(p) \cap I^-(q)\), there is a corresponding abstract C*-algebra \(\mathfrak {U}(\mathbf{B})\)

and when \(\mathbf{B}\) is the empty set, we have the distinguished correspondence

where \(\mathbf{1}\) is the multiplicative identity in the abstract C*-algebra \(\mathbb {C} \mathbf{1}\).

Let \(\mathbf{B}_1\) and \(\mathbf{B}_2\) be any two basis elements of the Alexandrov topology on Minkowski spacetime, i.e. any two sets of the form \(I^+(p_1) \cap I^-(q_1)\) and \(I^+(p_2) \cap I^-(q_2)\).

If \(\mathbf{B_1}\) and \(\mathbf{B_2}\) are completely spacelike, then \(\mathfrak {U}(\mathbf{B_1})\) and \(\mathfrak {U}(\mathbf{B_2})\) commute in the quasilocal algebra \(\mathfrak {U}\), i.e. for any \(a_1\) in \(\mathfrak {U}(\mathbf{B_1})\) and \(a_2\) in \(\mathfrak {U}(\mathbf{B_2})\) it follows that

in the quasilocal algebra \(\mathfrak {U}\).

Let \(\mathbf{B}_1\) and \(\mathbf{B}_2\) be any two basis elements of the Alexandrov topology on a Lorentzian spacetime, i.e. any two sets of the form \(I^+(p_1) \cap I^-(q_1)\) and \(I^+(p_2) \cap I^-(q_2)\).

If \(\mathbf{B_1}\) and \(\mathbf{B_2}\) are completely spacelike, for any Alexandrov topology basis element \(\mathbf{B}\) such that \(\mathbf{B_1}, \mathbf{B_2} \subseteq \mathbf{B}\) the algebras \(\mathfrak {U}(\mathbf{B_1})\) and \(\mathfrak {U}(\mathbf{B_2})\) commute in the C*-algebra \(\mathfrak {U}(\mathbf{B})\), i.e. for any \(a_1\) in \(\mathfrak {U}(\mathbf{B_1})\) and \(a_2\) in \(\mathfrak {U}(\mathbf{B_2})\) we have

in the C*-algebra \(\mathfrak {U}(\mathbf{B})\), where \(i\) is the unital *-monomorphism Axiom 2 (Isotony).

If no such \(\mathbf{B}\) exists, then it simply doesn’t make sense to consider if \(\mathfrak {U}(\mathbf{B_1})\) and \(\mathfrak {U}(\mathbf{B_2})\) commute as they are not in the same algebra.

All “observables” are local observables.

For Lorentzian spacetime \(M\) the image \(\pi _\omega (a)\) of a self-adjoint member \(a\) of the local algebra \(\mathfrak {U}(\mathbf{B})\) under the GNS *-homomorphism \(\pi _\omega \) of a state \(\omega \) on \(\mathfrak {U}(\mathbf{B})\) is self-adjoint and thus corresponds to an “observable”. Any “observable” corresponding to such a self-adjoint \(\pi _\omega (a)\) is called a local observable.

Let \(\pi \) be a \(*\)-representation of the quasilocal algebra on \(H\). The local observable operators of a region \(\mathbf{B}\) are the image \(\pi (\mathfrak {U}(\mathbf{B})) = \{ \pi (\iota _{\mathbf{B}} a) : a \in \mathfrak {U}(\mathbf{B})\} \). The local von Neumann algebra \(R(\mathbf{B})\) is the bicommutant \(\pi (\mathfrak {U}(\mathbf{B}))''\).

The local algebra \(R(\mathbf{B})\) is registered as a genuine VonNeumannAlgebra (Mathlib’s bundled structure), not merely a set of operators. The general fact is that the bicommutant \(S''\) of any self-adjoint set \(S\) of bounded operators is a von Neumann algebra: \(S'\) is a \(*\)-subalgebra (the centralizer of a self-adjoint set is \(*\)-closed), and \(S'' = S''''\) since the triple commutant collapses. The local observable operators \(\pi (\mathfrak {U}(\mathbf{B}))\) are self-adjoint (\(\pi \) and \(\iota \) are \(*\)-homomorphisms), so \(R(\mathbf{B}) = \pi (\mathfrak {U}(\mathbf{B}))''\) is a von Neumann algebra whose underlying set is the bicommutant of 89.

As in the Minkowski case, \(R(\mathbf{B}')\) is registered as a genuine VonNeumannAlgebra: the local observable operators \(\pi (\mathfrak {U}(\mathbf{B}'))\) are self-adjoint (the isotony embedding and \(\pi \) are \(*\)-homomorphisms), so their bicommutant is a von Neumann algebra whose underlying set is the bicommutant of 178. The construction is the same general bicommutant-of-a-self-adjoint-set lemma.

Let \(\pi \) be a \(*\)-representation of a containing basis algebra \(\mathfrak {U}(\mathbf{B})\) on \(H\). For a subregion \(\mathbf{B}' \subseteq \mathbf{B}\), the local observable operators are the image \(\pi (\mathfrak {U}(\mathbf{B}'))\) under the isotony embedding, and the local von Neumann algebra \(R(\mathbf{B}')\) is the bicommutant \(\pi (\mathfrak {U}(\mathbf{B}'))''\).

Let \(\mathbf{B}\) be any basis element of the Alexandrov topology on Minkowski spacetime, i.e. any set of the form \(I^+(p) \cap I^-(q)\).

A member \(L\) of the inhomogeneous Lorentz group connected to the identity acts on \(\mathfrak {U}(\mathbf{B})\) as follows

where \(L\mathbf{B}\) is the image of the region \(\mathbf{B}\) under the transformation \(L\) and \(\alpha _L\) is a unital *-isomorphism generated by \(L\). The map \(\alpha _L\) is such that (1) for the identity element \(\mathbf{1}\) of the Lorentz group it satisfies

(2) for all appropriate \(a\), \(L\), and \(L'\) it satisfies

and (3) for basis elements \(\mathbf{B}_\iota \subset \mathbf{B}_\kappa \) and the unital *-monomorphism \(i\) of Axiom 2 (Isotony) \(\alpha _L\) commutes with \(i\). In other words the following diagram

commutes.

Lorentzian spacetime is a spacetime equipped with a Hausdorff Alexandrov topology.

Minkowski spacetime is standard Minkowski spacetime equipped with Alexandrov topology.

A spacetime \(M\) with time orientation \(t\) satisfies the causality condition — equivalently, has no closed causal curve — when no point causally precedes itself: \(\lnot (p \prec p)\) for every \(p\). A closed causal curve through \(p\) would be a causal trip from \(p\) back to \(p\), i.e. \(p \prec p\), so its absence is exactly this condition. This is the standard causality condition, strictly weaker than strong causality and strictly stronger than the chronology condition (\(\lnot (p \ll p)\) for all \(p\)).

A family \(\alpha : \mathbb {R} \to (A \simeq _{\star \mathrm{a}} A)\) of \(*\)-automorphisms of a C\(^*\)-algebra \(A\) is a one-parameter group if \(\alpha _0 = \mathrm{id}\) and \(\alpha _{s+t} = \alpha _s \circ \alpha _t\). This is the algebraic time evolution; in the curved-spacetime setting it is the automorphism group induced by a Killing flow.

An orientation-reversing reparametrisation flips the time-orientation of the tangent, so future/past orientation is well-defined only on the finer quotient by orientation-preserving reparametrisations (those with positive within-derivative). An oriented smooth curve is an equivalence class of smooth paths under this finer relation.

A path is a continuous map \(\mu :\Sigma \rightarrow M\) from the parameter space–a closed, connected subset \(\Sigma \) of \(\mathbb {R}\) that contains more than a single point–to a spacetime \(M\). A smooth path is a path \(\mu \) that is smooth and has a non-vanishing derivative.

A strongly continuous one-parameter unitary group \(V : \mathbb {R} \to \mathcal{U}(H)\) has positive energy when its generator is a positive operator: there exists a positive bounded operator \(P\) (hence self-adjoint, with non-negative spectrum) such that \(V(t) = e^{i t P}\) for all \(t\). The positivity of \(P\) is the energy-positivity asserted by the spectrum condition. The generator of a physical translation is unbounded, so requiring \(P\) bounded is a genuine restriction; the faithful unbounded form requires Stone’s theorem and the theory of unbounded self-adjoint operators, which Mathlib does not yet provide. This is the bounded-generator scaffold.

A state \(\omega \) is pure if every positive linear functional \(\psi \) dominated by \(\omega \) (that is, \(0 \le \psi (a^*a) \le \omega (a^*a)\) for all \(a\)) is a scalar multiple of \(\omega \). This order-theoretic characterization is the extreme-point notion of purity, phrased to avoid convex-combination and normalization bookkeeping.

Two representations are quasi-equivalent when there is a \(*\)-isomorphism of their generated von Neumann algebras \(\pi _1(A)'' \simeq \pi _2(A)''\) carrying \(\pi _1(a)\) to \(\pi _2(a)\). This is an equivalence relation, and it is coarser than unitary equivalence: unitary equivalence implies quasi-equivalence, because the conjugation \(*\)-isomorphism \(T \mapsto U T U^{-1}\) restricts to a \(*\)-isomorphism of the generated von Neumann algebras.

Consider the set-theoretic union of all \(\mathfrak {U}(\mathbf{B})\). As previously proven, this set-theoretic union is a normed *-algebra. Also, as previously proven, taking its completion one obtains a C*-algebra denoted as \(\mathfrak {U}\). This C*-algebra \(\mathfrak {U}\) is called the quasilocal algebra.

All “observables” are quasilocal observables.

A lift of the fiberwise Lorentz action of \(L\) to a quasilocal algebra \(\mathfrak {U}\) (82) is a \(*\)-automorphism \(\beta _L\) of \(\mathfrak {U}\) intertwining the local embeddings \(\iota _{\mathbf{B}}\) with the covariance isomorphisms: \(\beta _L(\iota _{\mathbf{B}} a) = \iota _{L\cdot \mathbf{B}}(\alpha _L a)\) for every Alexandrov-basis set \(\mathbf{B}\).

The image \(\pi _\omega (a)\) of a self-adjoint member \(a\) of the quasilocal algebra \(\mathfrak {U}\) under a GNS *-homomorphism \(\pi _\omega \) is self-adjoint and thus corresponds to an “observable”. Any “observable” corresponding to such a self-adjoint \(\pi _\omega (a)\) is called a quasilocal observable.

The spacelike complement \(\mathbf{B}^\perp \) of a region \(\mathbf{B}\) is the set of points completely spacelike-separated from all of \(\mathbf{B}\): \(\mathbf{B}^\perp = \{ x \mid \{ x\} \text{ is completely spacelike to } \mathbf{B} \} \). It is the geometric substrate of locality and Haag duality.

Consider two points \(p_1\) and \(p_2\) in a spacetime. The points are said to be spacelike related if \(p_2 \notin J^+(p_1) \cup J^-(p_1)\) or equivalently \(p_1 \notin J^+(p_2) \cup J^-(p_2)\).

A spacetime is a real, four-dimensional, connected, smooth, Hausdorff manifold \(M\) with a globally defined smooth tensor field \(g\) of type \((0,2)\) which is non-degenerate and “Lorentzian”. By Lorentzian we mean that for any \(p \in M\) there is a basis of the tangent space \(TM|_p\) to \(M\) at \(p\) relative to which \(g|_p\) is zero in its non-diagonal entries and on the diagonal takes the form \(\text{diag}(-1,1,1,1)\).

For an identity-component isometry \(\varphi \) fixing the region \(\mathbf{B}\) (so \(\varphi (\mathbf{B}) = \mathbf{B}\)), the covariance isomorphism \(\alpha _\varphi \) of 175 lands back in \(\mathfrak {U}(\mathbf{B})\) and so defines an automorphism \(\hat\alpha _\varphi \) of the single algebra \(\mathfrak {U}(\mathbf{B})\).

Standard Minkowski spacetime is a spacetime in which the underlying real, four-dimensional, connected, smooth, Hausdorff manifold is \(\mathbb {R}^4\) with the Euclidean topology. In addition \(g\) takes the form \(g|_p=\text{diag}(-1,1,1,1)\) for all \(p\) in \(\mathbb {R}^4\) with respect to the standard coordinates on \(\mathbb {R}^4\).

Let \(\mathfrak {U}\) be an abstract C*-algebra. A state is an element \(\omega \) of the dual space \(\mathfrak {U}^*\) that is

• Positive - for any \(a \in \mathfrak {U}\) we have \(0 \le \omega (a^*a)\) and

• Normalized - the operator norm satisfies \(\| \omega \| =1\).

Furthermore, a state \(\omega \) is said to be faithful if for any non-zero \(a\) in \(\mathfrak {U}\), it follows that \(0 {\lt} \omega (a^*a)\).

The strip-Liouville principle at width \(\beta \) is the statement that any function \(F\) continuous and bounded on the closed strip \(0 \le \operatorname {Im} z \le \beta \), holomorphic on the open strip, and with equal boundary values \(F(t) = F(t + i\beta )\) for all real \(t\), is constant along the real axis: \(F(t) = F(0)\). It is the analytic input that turns boundary coincidence into invariance.

A spacetime \(M\) is time-orientable if it admits a smooth, non-vanishing vector field \(t\) that is timelike. Such a smooth, non-vanishing vector field is called a time-orientation.

A timelike smooth curve is a smooth curve with a tangent vector that is timelike at every point along the smooth curve. A causal smooth curve is a smooth curve with a tangent vector that is timelike or null at every point along the smooth curve.

Let \(M\) be a spacetime, \(p\) a point in \(M\), and \(g\) the tensor field of type \((0,2)\) associated to \(M\). Any tangent vector \(v \in TM|_p\) is timelike, spacelike, or null if \(g|_p(v,v)\) is negative, positive, or zero respectively.

A trip segment is a curve which is a future-oriented, timelike geodesic. A trip is a curve which is piecewise a future-oriented, timelike geodesic: a finite chain of trip segments \(p = x_0, x_1, \dots , x_n = q\) joined at matching endpoints. Formally this is the transitive closure of single-segment precedence, which is what makes the relation transitive by concatenation. A trip from \(p\) to \(q\) is a trip with past endpoint \(p\) and future endpoint \(q\). We write \(p \ll q\) if and only if there exists a trip from \(p\) to \(q\).

Two \(*\)-representations \(\pi _1 : A \to \mathcal{B}(H_1)\) and \(\pi _2 : A \to \mathcal{B}(H_2)\) are unitarily equivalent when there is an isometric isomorphism \(U : H_1 \simeq H_2\) of the underlying Hilbert spaces intertwining them: \(U(\pi _1(a)\, x) = \pi _2(a)(U x)\). This is an equivalence relation — reflexive, symmetric, and transitive.

A state \(\omega \) on the quasilocal algebra is a vacuum state (relative to a future-timelike-translation predicate) when it is invariant under the covariance action (144) and, in a GNS representation reproducing \(\omega \) and implementing the action by unitaries \(U(L)\), every future-timelike translation one-parameter subgroup \(\gamma \) has positive energy: \(t \mapsto U(\gamma (t))\) satisfies 147. This packages the two necessary vacuum conditions — invariance and the spectrum condition — with the spectrum condition entering as the positive-energy hypothesis on the implementing unitaries. The future-timelike-translation predicate is a parameter, to be instantiated once the translation subgroup of the Lorentz group and its causal structure are wired in; the positive-energy condition is the bounded-generator scaffold. Constructing/discharging the spectrum condition for a concrete net is the Stone-gated next layer.

The vacuum-state condition of 149 with its future-timelike-translation parameter fixed to the concrete predicate 151: the spectrum condition is imposed on exactly the one-parameter translation subgroups \(t \mapsto (\mathrm{id}, t \cdot n)\) with \(n\) future-pointing timelike. The vacuum definition then depends on no free predicate. Invariance and (for a pure state) the irreducible covariant representation follow as in 150, since the concrete form unfolds to the parameterized one.

Packaging isotony, the assignment \(\mathbf{B} \mapsto R(\mathbf{B})\) is an order-preserving map from the poset of basis regions (ordered by inclusion) to the von Neumann algebras of \(\mathcal{B}(H)\). This realizes the net as a functor on the inclusion poset: containment of regions is sent to containment of algebras. It is the central object of the algebraic approach.

The curved counterpart of the von Neumann net (94). Fixing a containing basis region \(\mathbf{B}\) and a \(*\)-representation \(\pi \) of \(\mathfrak {U}(\mathbf{B})\), the assignment \(\mathbf{B}' \mapsto R(\mathbf{B}')\) is an order-preserving map from the poset of basis subregions of \(\mathbf{B}\) (ordered by inclusion) to the von Neumann algebras of \(\mathcal{B}(H)\) — the local net, restricted to a containing region, as a functor on the inclusion poset.

Unlike the Minkowski case, the coherence of the isotony embeddings (coherent below \(\mathbf{B}\): the \(\mathbf{B}_1 \hookrightarrow \mathbf{B}\) embedding factors through \(\mathbf{B}_2\) for all nested \(\mathbf{B}_1 \subseteq \mathbf{B}_2 \subseteq \mathbf{B}\)) is supplied as a single hypothesis rather than discharged geometrically. This is intrinsic, not a gap: the coherence is a property of the net’s chosen Axiom 3 isotony witnesses, not of the spacetime, so — unlike spacelike-monotonicity — it cannot be discharged over a concrete spacetime. The curved Axiom 3 selects those witnesses by choice (with no built-in composition law), so the coherence is unavailable for free even for the trivial net; it holds for any net whose witnesses form a genuine inclusion family. Given it once, the map’s monotonicity carries no per-edge side condition.

Every basis set \(I^+(p) \cap I^-(q)\) is open in the Alexandrov topology.

On a Lorentzian spacetime with at least two points, the Alexandrov diamonds cover the whole space: every point lies in some diamond \(I^+(p) \cap I^-(q)\). Equivalently, every point has both a chronological past point and a chronological future point (a “no endpoints” condition). This is a genuine consequence of the Hausdorff assumption on the Alexandrov topology, not an extra hypothesis.

On a spacetime equipped with the Alexandrov topology, suppose a point \(x\) lies in no diamond \(I^+(p) \cap I^-(q)\). Then every Alexandrov-open set containing \(x\) is the whole space \(M\): the only open neighbourhood of \(x\) is \(M\) itself.

Instantiating the abstract curved-spacetime interface with the oriented identity component, every isometry \(\varphi \) of the abstract spacetime carries Alexandrov-basis sets to basis sets, \(\varphi \cdot \mathbf{B}\) is again a basis set. This is exactly the well-definedness condition for the Axiom 5 action \(\mathfrak {U}(\mathbf{B}) \to \mathfrak {U}(\varphi (\mathbf{B}))\).

Let \(\mathcal{A}\) be a *-algebra and \(\omega \) a positive element of the dual space \(\mathcal{A}^*\), i.e. \(\omega \) is an element of the dual space \(\mathcal{A}^*\) such that for any \(a \in \mathcal{A}\) one has \(0 \le \omega (a^*a)\). Then

for all \(a\) and \(b\) in \(\mathcal{A}\).

Every tangent vector \(v \in TM|_p\) is exactly one of timelike, null, or spacelike. In particular the three classes are mutually exclusive, and the zero vector is null.

Let \(M\) be a spacetime with time orientation \(t\), equipped with the Alexandrov topology.

• For all points \(p, q \in M\) the set \(I^+(p) \cap I^-(q)\) is open. This is just the basis lemma (61) restated on chronological futures and pasts.

• If every point of \(I^+(p)\) has a chronological-future point, i.e. for all \(x \in I^+(p)\) there exists \(b\) with \(x \ll b\), then the chronological future \(I^+(p)\) is open.

• Dually, if every point of \(I^-(p)\) has a chronological-past point, i.e. for all \(x \in I^-(p)\) there exists \(a\) with \(a \ll x\), then the chronological past \(I^-(p)\) is open.

The per-point hypotheses in the last two items are quantified over the members of the set in question, so they hold vacuously when that set is empty; this is consistent, since the empty set is open. The flat unconditional claim “\(I^+(p)\) is always open” is false for a general spacetime: if \(I^+(p)\) is nonempty and contains a future-endpoint point \(x\) (a point with no \(b\) satisfying \(x \ll b\)), then \(x\) lies in no basis set \(I^+(a) \cap I^-(b)\), since membership there forces \(a \ll x \ll b\) and in particular \(x \ll b\). Hence \(x\) has no basic Alexandrov neighbourhood contained in \(I^+(p)\) and \(I^+(p)\) fails to be open. The hypothesis is exactly the “no future endpoints” condition that removes this obstruction, and dually for pasts.

Every trip is a causal trip, so \(p \ll q\) implies \(p \prec q\). Consequently \(I^+(p) \subseteq J^+(p)\) and \(I^-(p) \subseteq J^-(p)\).

Complete spacelike separation is monotone under shrinking either region; the empty region is completely spacelike to any region; and a union of regions is completely spacelike to \(\mathbf{O}\) if and only if each part is. The same properties hold for the bundled Lorentzian spacetime.

Spacelike relatedness is symmetric, and consequently complete spacelike separation is symmetric in its two regions: \(\mathbf{O}_1, \mathbf{O}_2\) are completely spacelike if and only if \(\mathbf{O}_2, \mathbf{O}_1\) are.

Let \(g\) be a symmetric Lorentzian bilinear form, \(t\) a timelike vector, and write \(t^\perp = \{ u : g(t,u) = 0\} \) for the spacelike complement. Then \(g\) is positive semidefinite on \(t^\perp \) (so the ordinary Cauchy-Schwarz inequality holds there), and consequently for any timelike \(v, w\) with \(g(t,v) {\lt} 0\) and \(g(t,w) {\lt} 0\) one has \(g(v,w) {\lt} 0\). In particular two timelike tangent vectors that are future-pointing with respect to a common time orientation have negative inner product; by time reversal the same holds for two past-pointing timelike vectors (with \(g(t,v) {\gt} 0\) and \(g(t,w) {\gt} 0\)).

For a covariant family of local states, the covariance relation composes along the group: \(\omega _{\mathbf{B}}(a) = \omega _{L'\cdot (L\cdot \mathbf{B})}\big(\alpha _{L'}(\alpha _L a)\big)\), reflecting the multiplicativity of the Lorentz action.

For a covariant family of local states, the covariance relation composes along the isometry group: \(\omega _{\mathbf{B}}(a) = \omega _{\varphi '(\varphi (\mathbf{B}))}\big(\alpha _{\varphi '}(\alpha _\varphi a)\big)\).

The endomorphism algebra of an irreducible representation is \(\mathbb {C}\cdot 1\): every self-intertwiner of an irreducible representation is a scalar multiple of the identity. This is the commutant form of irreducibility read through the intertwiner language, and it identifies each irreducible sector with a simple (one-dimensional-centred) object.

The sum of two timelike future-pointing tangent vectors (with respect to a fixed time orientation) is again timelike and future-pointing. More generally, the sum of any two future-pointing tangent vectors – timelike or null – is future-pointing, so the full future cone, including its null boundary, is convex. Since a vector is past-pointing exactly when its negation is future-pointing, the past cone is convex as well. Downstream this packages as the statement that the future-pointing and past-pointing tangent vectors each form a convex cone: they are closed under positive scaling and, more generally, under positive linear combinations \(a v + b w\) with \(a, b {\gt} 0\).

The set-valued chronological and causal futures and pasts are monotone: if \(S \subseteq T\) then \(I^\pm (S) \subseteq I^\pm (T)\) and \(J^\pm (S) \subseteq J^\pm (T)\).

The future-orientation-preserving isometries (those \(\varphi \) with both \(\varphi \) and \(\varphi ^{-1}\) preserving the orientation) form a subgroup, and intersecting it with the identity component gives the oriented identity component. Every such isometry maps Alexandrov-basis diamonds to diamonds, \(\varphi (I^+(p) \cap I^-(q)) = I^+(\varphi (p)) \cap I^-(\varphi (q))\), both as an image and in pointwise-action form \(\varphi \cdot \mathbf{B}\), and this lifts to the bundled Lorentzian spacetime.

Say an isometry \(\varphi \) preserves the future orientation if its differential sends future-pointing vectors to future-pointing vectors; this property holds for the identity and is closed under composition. Under it, \(\varphi \) carries trips to trips, so \(p \ll q\) implies \(\varphi (p) \ll \varphi (q)\), and the chronological futures and pasts satisfy \(\varphi (I^\pm (p)) = I^\pm (\varphi (p))\).

An isometry \(\varphi \) of a spacetime preserves the metric square of a tangent vector, \(g_{\varphi (x)}(d\varphi _x v, d\varphi _x v) = g_x(v,v)\), and therefore \(d\varphi _x v\) is timelike, null, or spacelike if and only if \(v\) is.

For a KMS state \(\omega \) and any observable \(a\), the correlation function \(F\) of the pair \((1, a)\) has its two boundary values equal - both are \(t \mapsto \omega (\alpha _t a)\). This is the algebraic heart of the invariance argument: it follows directly from the KMS condition with \(a := 1\), since \(\omega (1\cdot \alpha _t a) = \omega (\alpha _t a \cdot 1) = \omega (\alpha _t a)\).

Let \(\omega \) be a state over a unital C*-algebra \(\mathfrak {U}\). Then the set \(\mathcal{N}_1\) defined by

is equivalent to the set \(\mathcal{N}\) defined by

Let \(\omega \) be a state over a unital C*-algebra \(\mathfrak {U}\). Then the set \(\mathcal{N}\) defined by

is a closed, linear subspace of \(\mathfrak {U}\).

On a Lorentzian spacetime, complete spacelike separation of two regions is symmetric, and every basis set \(I^+(p) \cap I^-(q)\) is open in the Alexandrov topology.

Work on standard Minkowski spacetime, with underlying manifold \(\mathbb {R}^4\), and let \(e_0\) denote the unit time vector (the first standard coordinate vector).

1. Every point has a chronological future point and a chronological past point: for all \(x \in \mathbb {R}^4\) there exist \(b, a\) with \(x \ll b\) and \(a \ll x\). Concretely \(b = x + e_0\) lies in the forward cone of \(x\) and \(a = x - e_0\) lies in the backward cone of \(x\), since the connecting vector \(\pm e_0\) is future-, respectively past-pointing timelike.

2. For every point \(p\) the chronological future \(I^+(p)\) and the chronological past \(I^-(p)\) are open in the Euclidean (manifold) topology on \(\mathbb {R}^4\). On standard Minkowski the coordinate cone characterisation identifies \(I^+(p)\) with the forward Minkowski cone of \(p\) and \(I^-(p)\) with the backward Minkowski cone of \(p\), and each such cone is an open subset of \(\mathbb {R}^4\).

3. For every point \(p\) the chronological future \(I^+(p)\) and the chronological past \(I^-(p)\) are open in the Alexandrov topology, unconditionally: the per-point hypothesis of 62 is discharged on standard Minkowski by part (i).

On standard Minkowski spacetime the Alexandrov diamonds have the downward intersection property: for any two diamonds \(B_1 = I^+(p_1) \cap I^-(q_1)\) and \(B_2 = I^+(p_2) \cap I^-(q_2)\) and any \(x \in B_1 \cap B_2\), there is a diamond \(B_3\) with \(x \in B_3 \subseteq B_1 \cap B_2\).

On standard Minkowski spacetime, if \(x \ll q_1\) and \(x \ll q_2\) then there is a point \(b\) with \(x \ll b\), \(b \ll q_1\), and \(b \ll q_2\).

On standard Minkowski spacetime, if \(p_1 \ll x\) and \(p_2 \ll x\) then there is a point \(a\) with \(p_1 \ll a\), \(p_2 \ll a\), and \(a \ll x\).

If \(t \mapsto L_t\) is a one-parameter subgroup (\(L_0 = 1\), \(L_{s+t} = L_s L_t\)), then the induced automorphisms \(t \mapsto \beta _{L_t}\) of \(\mathfrak {U}\) form a one-parameter automorphism group (155), by the coherence of the covariance action (143).

If the flow is a one-parameter subgroup of \(\mathrm{Stab}(\mathbf{B})\) (\(\varphi _0 = 1\) and \(\varphi _{s+t} = \varphi _s\, \varphi _t\)), then the induced family \(t \mapsto \hat\alpha _{\varphi _t}\) is a one-parameter automorphism group of \(\mathfrak {U}(\mathbf{B})\) (155). This is immediate from the group-action laws of the stabilizer action (197).

The parameter space of a path has unique differentials: being a closed, connected subset of \(\mathbb {R}\) with more than one point, it is a non-degenerate interval, hence convex with non-empty interior.

A future-pointing or past-pointing vector is timelike or null. Moreover a timelike vector cannot be both future-pointing and past-pointing with respect to a fixed time orientation.

An isometry \(\varphi \) pushes a smooth path \(\mu \) forward to the smooth path \(\varphi \circ \mu \) on the same parameter space, with tangent vector \(d\varphi (\dot\mu )\). The pushforward preserves the timelike and causal conditions and carries the past and future endpoints of \(\mu \) to those of \(\varphi \circ \mu \).

The covariance action \(L \mapsto \beta _L\) of a covariant quasilocal algebra is a genuine action of the Lorentz group by \(*\)-automorphisms of the quasilocal algebra: \(\beta _{\mathbf{1}} = \mathrm{id}\) and \(\beta _{L'L} = \beta _{L'} \circ \beta _L\).

Any two lifts of the same \(L\) have the same underlying automorphism: they agree on the union of the local images, which is dense in \(\mathfrak {U}\), and \(*\)-automorphisms of a C*-algebra are continuous.

Let \(g\) be a symmetric Lorentzian bilinear form on a real four-dimensional vector space and let \(v, w\) be timelike, that is \(g(v,v) {\lt} 0\) and \(g(w,w) {\lt} 0\). Then the reverse Cauchy-Schwarz inequality holds:

In particular this applies pointwise to the metric \(g|_p\) of any spacetime \(M\) and any two timelike tangent vectors at a point \(p\).

The space of intertwiners between two irreducible representations is at most one-dimensional: any two intertwiners \(S, T\) with \(S \neq 0\) are proportional, \(T = \lambda \cdot S\). Indeed \(S^\ast S = a\cdot 1\) and \(S^\ast T = b \cdot 1\) (commuting with \(\pi _1\), hence scalar), and \(S S^\ast = c \cdot 1\) with \(c \neq 0\) (irreducibility of \(\pi _2\)) makes \(S^\ast \) injective; since \(S^\ast \big(T - (b/a)\, S\big) = 0\), injectivity gives \(T = (b/a)\, S\). So irreducible representations intertwine multiplicity-free.

At the level of the underlying sets, the spacelike complement turns unions into intersections. In binary form, for regions \(\mathbf{B}_1, \mathbf{B}_2\),

and, for an arbitrary indexed family \((\mathbf{B}_i)_{i \in I}\),

A Lorentzian bilinear form is nondegenerate: if \(g(v,w) = 0\) for every \(w\), then \(v = 0\) (this is read off the signature basis, on which the Gram matrix \(\mathrm{diag}(-1,1,1,1)\) is invertible). Consequently \(g\) is positive definite on the spacelike complement: if \(t\) is timelike and \(u \ne 0\) satisfies \(g(t,u) = 0\), then \(g(u,u) {\gt} 0\), i.e. \(u\) is spacelike.

The spacelike complement is antitone (\(\mathbf{B}_1 \subseteq \mathbf{B}_2 \Rightarrow \mathbf{B}_2^\perp \subseteq \mathbf{B}_1^\perp \)), a region is contained in its double complement (\(\mathbf{B} \subseteq \mathbf{B}^{\perp \perp }\)), and the triple complement collapses (\(\mathbf{B}^{\perp \perp \perp } = \mathbf{B}^\perp \)). Moreover \(\mathbf{B}_1 \subseteq \mathbf{B}_2^\perp \) if and only if \(\mathbf{B}_1\) and \(\mathbf{B}_2\) are completely spacelike-separated, so complementation is the Galois connection attached to the spacelike-separation relation.

The assignment \(\varphi \mapsto \hat\alpha _\varphi \) is a monoid action of \(\mathrm{Stab}(\mathbf{B})\) by automorphisms of \(\mathfrak {U}(\mathbf{B})\): \(\hat\alpha _1 = \mathrm{id}\) and \(\hat\alpha _{\varphi '\varphi } = \hat\alpha _{\varphi '}\circ \hat\alpha _\varphi \). These follow from the identity and composition laws of the covariance action (175); the canonical region-casts collapse because all the regions involved equal \(\mathbf{B}\).

Let \(g\) be a symmetric Lorentzian bilinear form on a real four-dimensional vector space and let \(v, w\) be timelike and aligned, that is \(g(v,v) {\lt} 0\), \(g(w,w) {\lt} 0\) and \(g(v,w) \le 0\). Then \(v + w\) is timelike, and the reverse (Lorentzian) triangle inequality holds:

In particular the timelike vectors sharing a time cone (so that \(g(v,w) \le 0\)) form a convex cone, and this applies pointwise to the metric \(g|_p\) of any spacetime.

Locality expressed through the spacelike complement, without attaching any algebra to the unbounded complement: for basis sets \(\mathbf{B}' \subseteq \mathbf{B}^\perp \), the local von Neumann algebra \(R(\mathbf{B}')\) lies in the commutant \(R(\mathbf{B})'\). It is a direct repackaging of bundled microcausality (93) through the Galois bridge \(\mathbf{B}' \subseteq \mathbf{B}^\perp \iff \mathbf{B}', \mathbf{B}\) completely spacelike, staying strictly within the physically local (bounded, diamond) regions. No local algebra is ever attached to the unbounded complement.

The geometric specialisation of additive-free locality to a curved local net over a concrete Lorentzian spacetime \(L\): for basis sets \(\mathbf{B}_1 \subseteq \mathbf{B}_2^\perp \) inside a common containing region \(\mathbf{B}\), the local von Neumann algebra \(R(\mathbf{B}_1)\) lies in the commutant \(R(\mathbf{B}_2)'\). As in Minkowski, the spacelike complement \(\mathbf{B}_2^\perp \) only selects the bounded regions spacelike to \(\mathbf{B}_2\); no algebra is attached to the unbounded complement. The Galois bridge discharges the spacelike hypothesis, and \(L.\mathtt{toAbstract}\) identifies the concrete and abstract spacelike relations.

On a Lorentzian spacetime with at least two points, suppose the diamonds are downward-directed: for any two diamonds \(B_1, B_2\) and any point \(x \in B_1 \cap B_2\) there is a diamond \(B_3\) with \(x \in B_3 \subseteq B_1 \cap B_2\) (the intersection property). Then the diamonds form a genuine topological basis for the Alexandrov topology.

On the complete lattice of causally complete regions the causal complement \(\mathbf{B} \mapsto \mathbf{B}^\perp \) is an order-reversing involution, and therefore satisfies the full De Morgan laws. Explicitly:

• (order-reversing) \(\mathbf{B}_1 \le \mathbf{B}_2 \Rightarrow \mathbf{B}_2^\perp \le \mathbf{B}_1^\perp \);

• (bounds) \(\bot ^\perp = \top \) and \(\top ^\perp = \bot \);

• (binary De Morgan) \((\mathbf{B}_1 \sqcup \mathbf{B}_2)^\perp = \mathbf{B}_1^\perp \sqcap \mathbf{B}_2^\perp \) and \((\mathbf{B}_1 \sqcap \mathbf{B}_2)^\perp = \mathbf{B}_1^\perp \sqcup \mathbf{B}_2^\perp \);

• (infinitary De Morgan) \(\bigl(\bigsqcup _i \mathbf{B}_i\bigr)^\perp = \bigsqcap _i \mathbf{B}_i^\perp \) and \(\bigl(\bigsqcap _i \mathbf{B}_i\bigr)^\perp = \bigsqcup _i \mathbf{B}_i^\perp \).

Assume \(M\) has no closed causal curve. Then chronological precedence is irreflexive, \(\lnot (p \ll p)\) for every \(p\) (since \(p \ll p\) would give \(p \prec p\) because chronological precedence implies causal precedence), and causal precedence is asymmetric and antisymmetric: \(p \prec q\) excludes \(q \prec p\), and \(p \prec q\) together with \(q \prec p\) forces \(p = q\) (indeed both cannot hold, since transitivity would give the forbidden \(p \prec p\)). Thus, under the causality condition, \(\prec \) is a strict partial order on the events of the spacetime.

The causally complete regions form a complete lattice (meets are intersections, joins are causal closures of unions), obtained from the causal closure operator via its Galois insertion. The spacelike complement of any region is causally complete, causally complete regions are closed under intersection, and the causal complement \(\mathbf{B} \mapsto \mathbf{B}^\perp \) is an order-reversing involution on this lattice (\(\mathbf{B}^{\perp \perp } = \mathbf{B}\)). (The full orthocomplement law \(\mathbf{B} \wedge \mathbf{B}^\perp = \bot \) does not hold at this generality, because the trip-based causal relation is irreflexive, so a point is spacelike-separated from itself; what holds is the complete lattice with an order-reversing De Morgan involution.)

A von Neumann algebra and its commutant share the same center: the intersection \(\pi (A)'' \cap (\pi (A)'')' = \pi (A)'' \cap \pi (A)'\) is symmetric under the duality of 129, being simultaneously the center of \(\pi (A)''\) and the center of \(\pi (A)'\). Consequently \(\pi (A)''\) is a factor (trivial center) if and only if its commutant \(\pi (A)'\) is a factor. Dually, at the extreme of triviality, the commutant collapses to the scalars \(\pi (A)' = \mathbb {C}\cdot 1\) if and only if the generated algebra is everything \(\pi (A)'' = \mathcal{B}(H)\); this is the commutant form of the equivalence “irreducible \(\iff \) generates \(\mathcal{B}(H)\)” (128), passing between the two sides through the bicommutant \((\mathbb {C}\cdot 1)' = \mathcal{B}(H)\).

In a cyclic representation reproducing a state \(\omega \), an operator \(T\) commuting with all \(\pi (a)\) is a scalar multiple of the identity if and only if its diagonal coefficient \(a \mapsto \langle \Omega , T\, \pi (a)\Omega \rangle \) is a scalar multiple of \(\omega \). This is the precise operator-theoretic bridge to purity: irreducibility (every commutant element is scalar) is exactly the statement that every commutant coefficient is proportional to \(\omega \).

The commutant \(\pi (A)'\) of a representation is packaged as a von Neumann algebra — the algebra of self-intertwiners, i.e. the intertwiner/gauge algebra of \(\pi \). Its underlying set is the centralizer of \(\pi (A)\) (self-adjoint, and a commutant is always a von Neumann algebra since \(S''' = S'\)), an operator lies in it exactly when it is a self-intertwiner of \(\pi \), and \(\pi \) is irreducible if and only if this algebra is trivial (\(\pi (A)' = \mathbb {C}\cdot 1\)) — the von Neumann form of Schur’s lemma.

A direct sum with two summands carrying nonzero vectors is reducible. Were \(\bigoplus _i \pi _i\) irreducible, every summand projection would be a scalar (its commutant being trivial); but the projection onto one of two nonzero summands is a nontrivial projection, hence not scalar. In particular an amplification \(\iota \cdot \pi \) with at least two copies (and \(\pi \) acting on a nonzero space) is reducible — the multiplicity is visible in the commutant.

Each summand embeds as a subrepresentation: the isometric inclusion \(H_j \hookrightarrow \ell ^2(\iota , H)\) intertwines \(\pi _j\) with \(\bigoplus _i \pi _i\). Moreover the orthogonal projection onto the \(j\)-th summand lies in the commutant of the direct sum, so each summand is a reducing subspace.

The generated von Neumann algebra \(\pi (A)''\) and the commutant von Neumann algebra \(\pi (A)'\) (128) are each other’s commutants. On one side, the commutant of \(\pi (A)''\) is \(\pi (A)'\): this is an instance of the triple-commutant collapse \(S''' = S'\) applied to the self-adjoint image \(\pi (A)\). On the other side, the commutant of \(\pi (A)'\) is \(\pi (A)''\), which is exactly the definition of the bicommutant. Together these are von Neumann’s double-commutant relation for this pair, and they exhibit the bicommutant as idempotent: taking the commutant twice returns \((\pi (A)'')'' = \pi (A)''\), so \(\pi (A)''\) and \(\pi (A)'\) form a mutually dual pair under \(S \mapsto S'\).

Let \(\pi \) be any \(*\)-representation of the quasilocal algebra on a Hilbert space \(H\). If \(\mathbf{B}_1, \mathbf{B}_2\) are completely spacelike-separated basis regions, then for all \(a \in \mathfrak {U}(\mathbf{B}_1)\) and \(b \in \mathfrak {U}(\mathbf{B}_2)\) the operators \(\pi (\iota _{\mathbf{B}_1} a)\) and \(\pi (\iota _{\mathbf{B}_2} b)\) commute. This is the image under \(\pi \) of the local commutativity relation (83); since the GNS representation of any state is such a \(\pi \), spacelike-separated local observables commute on every GNS Hilbert space.

Let \(\pi \) be any \(*\)-representation of a containing basis algebra \(\mathfrak {U}(\mathbf{B})\) on a Hilbert space \(H\). If \(\mathbf{B}_1, \mathbf{B}_2 \subseteq \mathbf{B}\) are completely spacelike-separated basis regions, then the images of the local observables of \(\mathbf{B}_1\) and \(\mathbf{B}_2\) in \(\mathfrak {U}(\mathbf{B})\) commute under \(\pi \) as bounded operators on \(H\). This is the image under \(\pi \) of the curved local commutativity (172); it holds in particular on the GNS Hilbert space of any state on \(\mathfrak {U}(\mathbf{B})\). It is the curved-spacetime counterpart of 88, with the containing local algebra playing the role of the quasilocal algebra.

Let \(\omega \) be a state over a unital C*-algebra \(\mathfrak {U}\). One can then construct a Hilbert space \(\mathcal{H}_\omega \) and *-representation \(\pi _\omega \) of \(\mathfrak {U}\) by bounded operators on \(\mathcal{H}_\omega \) such that

As \(\mathfrak {U}\) is unital, there exists a cyclic vector \(\Omega \) in \(\mathcal{H}_\omega \) for the representation \(\pi _\omega \) such that

The triple \((\mathcal{H}_\omega , \pi _\omega , \Omega )\) is called the GNS triple associated to \((\mathfrak {U}, \omega )\) or the cyclic representation of \((\mathfrak {U}, \omega )\). Furthermore, if \(\omega \) is a faithful state, then the *-representation \(\pi _\omega \) is faithful. In addition the GNS triple associated to \((\mathfrak {U}, \omega )\) is unique up to unitary equivalence.

For a positive functional \(\psi \) dominated by the state \(\omega \) in its cyclic GNS representation, the form values obey \(\Vert \psi (a^* b)\Vert \le \Vert \pi (a)\Omega \Vert \, \Vert \pi (b)\Omega \Vert \). This follows from the Cauchy-Schwarz inequality for \(\psi \), the domination \(\psi \le \omega \), and the reproducing identity \(\omega (x^* x) = \Vert \pi (x)\Omega \Vert ^2\). In particular the form depends only on the GNS vectors \(\pi (a)\Omega , \pi (b)\Omega \), not on the representatives \(a, b\), so it is well-defined on the cyclic subspace.

The von Neumann algebra \(\pi (A)''\) generated by a representation is bundled as a first-class VonNeumannAlgebra (gnsVonNeumannAlgebra), the image \(\pi (A)\) being self-adjoint. For an irreducible representation, this bundled algebra is all of \(\mathcal{B}(H)\): its underlying set is everything, and equivalently it is the greatest von Neumann algebra on \(H\) (every \(S\) satisfies \(S \le \pi (A)''\)). This is the bundled counterpart of 110. Mathlib’s VonNeumannAlgebra carries no lattice \(\top \), so the literal \(\pi (A)'' = \top \) is phrased at the level of the underlying set together with the greatest-element statement under the existing \(\le \).

The bounded form of 106 is the inner product of an operator \(T\) on the GNS space: extending the densely-defined bounded form by the Fréchet-Riesz representation and a norm-controlled dense extension yields \(T\) with \(\langle \pi (a)\Omega , T\, \pi (b)\Omega \rangle = \psi (a^* b)\). From this reproducing identity, \(T\) commutes with every \(\pi (c)\) (an adjoint computation on the dense cyclic vectors), and \(\psi (a) = \langle \Omega , T\, \pi (a)\Omega \rangle \) (taking \(a = 1\) in the first slot).

Let \(\omega \) be a state on \(\mathfrak {U}(\mathbf{B})\) invariant under the stabilizer action, \(\omega (\hat\alpha _\varphi a) = \omega (a)\) for all \(\varphi \in \mathrm{Stab}(\mathbf{B})\). Then the action is implemented on the GNS Hilbert space of \(\omega \) by a unitary representation \(U\) of \(\mathrm{Stab}(\mathbf{B})\): there is a GNS triple \((H, \pi , \Omega )\) and unitaries \(U(\varphi )\) with \(U(\varphi )\, \pi (a)\Omega = \pi (\hat\alpha _\varphi a)\, \Omega \), \(U(\varphi )\Omega = \Omega \), the group laws, and \(U(1) = \mathrm{id}\). This is the curved-spacetime counterpart of 145: in the absence of a quasilocal algebra, the unitary representation is of the stabilizer subgroup acting on the single algebra \(\mathfrak {U}(\mathbf{B})\), rather than of the full isometry group.

If moreover the isometry group carries a topology (the abstract interface supplies none, so it is an added hypothesis) and the matrix coefficients \(\varphi \mapsto \omega \big(a^*\, \hat\alpha _\varphi b\big)\) are continuous on \(\mathrm{Stab}(\mathbf{B})\), then the representation \(U\) is strongly continuous: \(\varphi \mapsto U(\varphi )\psi \) is continuous for every GNS vector \(\psi \). The stabilizer subgroup inherits its topology as a subspace of the isometry group.

For an invariant state \(\omega \), the covariance action is implemented on the GNS Hilbert space by a family of unitaries \(U(L)\) satisfying \(U(L)\, \pi (a)\Omega = \pi (\beta _L a)\, \Omega \) and \(U(L)\Omega = \Omega \). The unitaries are obtained by extending the densely-defined isometry \(\pi (a)\Omega \mapsto \pi (\beta _L a)\Omega \) - isometric because \(\omega \) preserves the GNS inner product \(\langle \pi (a)\Omega , \pi (b)\Omega \rangle = \omega (a^* b)\) - to the whole GNS space.

A state \(\omega \) on the quasilocal algebra that is both invariant under the covariance action and pure yields a GNS representation that is simultaneously covariant - implemented by a unitary representation \(U(L)\) of the inhomogeneous Lorentz group fixing the cyclic vector \(\Omega \), with operator covariance \(U(L)\, \pi (a)\, U(L)^{-1} = \pi (\beta _L a)\) - and irreducible; in particular it generates all of \(\mathcal{B}(H)\) (\(\pi (\mathfrak {U})'' = \mathcal{B}(H)\), by 110). This combines the covariant GNS triple of 145 with purity \(\Rightarrow \) irreducibility (108). It is a necessary precursor to a vacuum representation; the spectrum condition would be the remaining ingredient.

A state \(\omega \) on a local algebra \(\mathfrak {U}(\mathbf{B})\) that is invariant under the stabilizer action and pure yields a GNS representation that is simultaneously covariant - implemented by a unitary representation \(U\) of \(\mathrm{Stab}(\mathbf{B})\) fixing the cyclic vector \(\Omega \), with operator covariance \(U(\varphi )\, \pi (a)\, U(\varphi )^{-1} = \pi (\hat\alpha _\varphi a)\) - and irreducible; in particular it generates all of \(\mathcal{B}(H)\) (\(\pi (\mathfrak {U}(\mathbf{B}))'' = \mathcal{B}(H)\), by 110). This is the curved, per-region analogue of 146 (there is no quasilocal algebra), combining 198 with purity \(\Rightarrow \) irreducibility. It is not a vacuum: curved spacetime admits no global vacuum, and the analogue of the spectrum condition (the Hadamard / microlocal spectrum condition) is a separate requirement not imposed here.

A nonzero intertwiner \(T\) between two irreducible representations rescales to a unitary equivalence: \(T^\ast T\) commutes with \(\pi _1\), hence is a positive scalar \(r \cdot 1\) with \(r {\gt} 0\), so \((\sqrt r)^{-1}\, T\) is a linear isometry; and \(T T^\ast \) is a nonzero scalar (irreducibility of \(\pi _2\)), which makes \(T\) surjective, so the isometry is a unitary. Consequently two irreducible representations are either disjoint or unitarily equivalent — the foundational trichotomy of superselection theory, identifying sectors with unitary-equivalence classes of irreducible representations.

Two irreducible representations of a curved local algebra \(\mathfrak {U}(\mathbf{B})\) are either disjoint or unitarily equivalent. This is the abstract irreducible dichotomy (125) registered per region on the C*-algebra \(\mathfrak {U}(\mathbf{B})\) — the right generality in curved spacetime, where sectors are attached to the local algebras.

The von Neumann algebra \(\pi (A)''\) generated by an irreducible representation has trivial center: an operator lies in the center \(\pi (A)'' \cap (\pi (A)'')'\) if and only if it is a scalar multiple of the identity. Indeed the center is contained in \((\pi (A)'')' = \pi (A)'\) (the triple commutant collapses to the single one), which irreducibility makes the scalars; conversely scalars are central.

A representation is irreducible if and only if the von Neumann algebra \(\pi (A)''\) it generates is all of \(\mathcal{B}(H)\): \(\pi (A)'' = \mathcal{B}(H)\). For the forward direction, irreducibility makes the commutant \(\pi (A)'\) exactly the scalars, and the centralizer of the scalars is everything (every bounded operator commutes with \(c \cdot 1\)), so \(\pi (A)'' = (\text{scalars})' = \mathcal{B}(H)\). The converse uses that \(\mathcal{B}(H)\) is a central \(\mathbb {C}\)-algebra (its center is the scalars): if the bicommutant is everything, every operator commutes with the commutant, so any operator commuting with \(\pi (A)\) is central, hence scalar. This is the density (bicommutant-theorem) form of irreducibility, sharpening the factor statement 109.

A convex combination \(s\, \omega _1 + (1-s)\, \omega _2\) (\(0 \le s \le 1\)) of two \((\alpha , \beta )\)-KMS states is again \((\alpha , \beta )\)-KMS: for each pair \((a, b)\) the analytic interpolant is the convex combination \(s\, F_1 + (1-s)\, F_2\) of the two interpolants, which is continuous, bounded, and holomorphic on the strip, with boundary values that add. Thus the equilibrium states at a fixed temperature form a convex set.

A convex combination \(s\, \omega _1 + (1-s)\, \omega _2\) (\(0 \le s \le 1\)) of two KMS states on \(\mathfrak {U}\) for the same covariance flow \(L\) at the same inverse temperature \(\beta \) is again a KMS state for that flow. This specializes the abstract KMS convexity (157) to the induced one-parameter group \(\mathrm{flowAut}\). Physically, the equilibrium states for a one-parameter symmetry flow form a convex set.

A convex combination \(s\, \omega _1 + (1-s)\, \omega _2\) (\(0 \le s \le 1\)) of two KMS states on \(\mathfrak {U}(\mathbf{B})\) for the same Killing flow \(\varphi \) at the same inverse temperature \(\beta \) is again a KMS state for that flow. This specializes the abstract KMS convexity (157) to the induced one-parameter group \(\mathrm{flowAut}\). Physically, the curved-spacetime thermal equilibrium states for a stationary Killing flow form a convex set.

For \(\beta {\gt} 0\), the analytic completion of a KMS correlation function is unique: any two functions satisfying the KMS analytic data for the same pair \((a, b)\) - continuous and bounded on the strip, holomorphic on the interior, with the prescribed boundary values \(t \mapsto \omega (a\, \alpha _t b)\) and \(t \mapsto \omega (\alpha _t b\, \cdot a)\) - agree on the whole strip. This is the strip uniqueness (163) applied to the two analytic completions, which share both boundary values.

At positive inverse temperature \(\beta {\gt} 0\), every \((\alpha , \beta )\)-KMS state \(\omega \) is \(\alpha \)-invariant: \(\omega (\alpha _t a) = \omega (a)\) for all \(t\) and \(a\). By boundary coincidence (158) the KMS correlation function of \((1, a)\) has equal boundary values \(F(t) = F(t + i\beta ) = \omega (\alpha _t a)\); the strip-Liouville principle for \(\beta {\gt} 0\) (161) forces \(F(t) = F(0)\), so \(\omega (\alpha _t a) = F(t) = F(0) = \omega (\alpha _0 a) = \omega (a)\).

Let \(\omega \) be a KMS state on \(\mathfrak {U}(\mathbf{B})\) for a one-parameter Killing flow \(t \mapsto \varphi _t\) into \(\mathrm{Stab}(\mathbf{B})\), at inverse temperature \(\beta {\gt} 0\), whose matrix coefficients \(t \mapsto \omega (a^* \hat\alpha _{\varphi _t} b)\) are continuous. Then its GNS triple \((H, \pi , \Omega )\) carries a strongly continuous one-parameter unitary group \(U : \mathbb {R} \to \mathcal{U}(H)\) implementing the flow: \(U_t\, \pi (a)\Omega = \pi (\hat\alpha _{\varphi _t} a)\Omega \), \(U_t\Omega = \Omega \), \(U_0 = \mathrm{id}\), \(U_{s+t} = U_s U_t\), and \(t \mapsto U_t\psi \) is continuous for every \(\psi \). This is the curved-spacetime equilibrium (thermal) representation - the analogue of the Minkowski vacuum representation - realized for the Hartle-Hawking and Gibbons-Hawking states. KMS at \(\beta {\gt} 0\) supplies the flow-invariance, which feeds the strongly continuous GNS unitary construction.

On standard Minkowski spacetime the Alexandrov diamonds form a genuine topological basis for the Alexandrov topology, unconditionally.

For a positive linear functional \(\varphi \) on a unital C*-algebra, \(\Vert \varphi \Vert = \mathrm{Re}\, \varphi (1)\). The bound \(\mathrm{Re}\, \varphi (1) \le \Vert \varphi \Vert \) is immediate from \(\Vert 1\Vert = 1\); conversely Cauchy-Schwarz with the first slot equal to \(1\) gives \(\Vert \varphi (b)\Vert ^2 \le \mathrm{Re}\, \varphi (1)\cdot \mathrm{Re}\, \varphi (b^* b) \le \mathrm{Re}\, \varphi (1)\cdot \Vert \varphi \Vert \, \Vert b\Vert ^2\), whence \(\Vert \varphi \Vert ^2 \le \mathrm{Re}\, \varphi (1)\cdot \Vert \varphi \Vert \). In particular every state satisfies \(\omega (1) = 1\).

Every oriented smooth curve has an underlying smooth curve, via a canonical surjection \(\mathrm{OrientedSmoothCurve} \to \mathrm{SmoothCurve}\) that forgets the orientation data. The timelike and causal predicates factor through it, and a future- or past-oriented curve projects to a causal smooth curve.

Future- and past-orientation are well-defined on oriented smooth curves: a smooth path is future-oriented (resp. past-oriented) if and only if its oriented smooth curve is. The positive within-derivative of an orientation-preserving reparametrisation, together with the positive-scaling invariance of pointing vectors (\(c\, v\) is future-pointing iff \(v\) is, for \(c {\gt} 0\)), gives the invariance.

Structural properties of the positive-energy condition, all Stone-free. (i) Trivial subgroup: the constant group \(t \mapsto \mathrm{id}\) has positive energy, with zero generator. (ii) Uniqueness of the generator: if two bounded generators induce the same one-parameter group, \(e^{i t P} = e^{i t Q}\) for all \(t\), then \(P = Q\) — differentiating at \(t = 0\) gives \(i P = i Q\); positivity is not needed. Hence the generator witnessing positive energy is unique. (iii) Unitary invariance: if \(V\) has positive energy, so does its conjugate \(t \mapsto W \circ V(t) \circ W^{-1}\) by a unitary \(W\), with generator \(W P W^{-1}\) (positive, being the unitary conjugate of \(P\)); the energy exponential transports via \(e^{W A W^{-1}} = W e^{A} W^{-1}\). Physically, the spectrum condition does not depend on the choice of unitary frame. (iv) Strong continuity: a positive-energy group is strongly continuous — \(t \mapsto V(t) x\) is continuous for every \(x\), since \(t \mapsto (i t) P\) is continuous and the operator exponential is continuous. This justifies calling it a strongly continuous one-parameter unitary group.

Chronological precedence \(\ll \) and causal precedence \(\prec \) are transitive: if \(p \ll q\) and \(q \ll r\) then \(p \ll r\), and likewise for \(\prec \). Two trips joined at the common point \(q\) concatenate to a single piecewise trip; this is exactly the transitivity of the transitive closure.

For a pure state \(\omega \) there is a cyclic GNS triple \((H, \pi , \Omega )\) reproducing \(\omega \) whose generated von Neumann algebra \(\pi (A)''\) has trivial center, i.e. is a factor. This combines the GNS construction, purity \(\Rightarrow \) irreducibility (108), and 109. It applies verbatim to the quasilocal algebra \(\mathfrak {U}\) and to each local algebra \(\mathfrak {U}(\mathbf{B})\), since both are C*-algebras.

For a pure state \(\omega \) on the local algebra \(\mathfrak {U}(\mathbf{B})\) there is a cyclic GNS triple reproducing \(\omega \) whose generated von Neumann algebra \(\pi (\mathfrak {U}(\mathbf{B}))''\) has trivial center (it is a factor) and is in fact all of \(\mathcal{B}(H)\). These are the abstract factor (112) and density (113) statements registered per region on the curved local algebra \(\mathfrak {U}(\mathbf{B})\), the right generality in curved spacetime where there is no quasilocal algebra.

The density (bicommutant-theorem) sharpening of 112: for a pure state \(\omega \) there is a cyclic GNS triple \((H, \pi , \Omega )\) reproducing \(\omega \) whose generated von Neumann algebra is all of \(\mathcal{B}(H)\), \(\pi (A)'' = \mathcal{B}(H)\). It combines the GNS construction, purity \(\Rightarrow \) irreducibility (108), and the density form 110. As with the factor statement, it applies verbatim to the quasilocal algebra \(\mathfrak {U}\) and to each local algebra \(\mathfrak {U}(\mathbf{B})\).

A state \(\omega \) is pure if and only if it is an extreme point of the state space. If \(\omega \) is pure and \(\omega = t\, \omega _1 + (1-t)\, \omega _2\), then \(t\, \omega _1\) is a positive functional dominated by \(\omega \), hence (by purity) a scalar multiple of \(\omega \); evaluating at \(1\), where every state gives \(1\), pins the scalar to \(t\) and forces \(\omega _1 = \omega = \omega _2\). Conversely, if \(\omega \) is extreme and \(\psi \le \omega \) is dominated, set \(\lambda = \mathrm{Re}\, \psi (1) \in [0,1]\); for \(\lambda \in (0,1)\) the rescaled functionals \(\lambda ^{-1}\psi \) and \((1-\lambda )^{-1}(\omega - \psi )\) are states (normalized via 114) whose convex combination is \(\omega \), so extremality identifies them and forces \(\psi = \lambda \, \omega \); the boundary cases \(\lambda = 0\) and \(\lambda = 1\) give \(\psi = 0\) and \(\psi = \omega \) respectively.

A state \(\omega \) on the local algebra \(\mathfrak {U}(\mathbf{B})\) is pure if and only if it is an extreme point of the state space of \(\mathfrak {U}(\mathbf{B})\). This is the abstract equivalence 116 applied to the C*-algebra \(\mathfrak {U}(\mathbf{B})\).

A state \(\omega \) on the canonical quasilocal algebra \(\mathfrak {U}\) of a Haag-Kastler net is pure if and only if it is an extreme point of the state space of \(\mathfrak {U}\). This is the abstract equivalence 116 applied to the C*-algebra \(\mathfrak {U}\).

A state \(\omega \) is pure if and only if its cyclic GNS representation is irreducible. For the converse direction, given a dominated \(\psi \), its Radon-Nikodym operator \(T\) commutes with \(\pi \), so by irreducibility \(T = c \cdot 1\); the reproducing identity then gives \(\psi (a) = c\, \omega (a)\), so \(\psi \) is a scalar multiple of \(\omega \) and \(\omega \) is pure.

For a state \(\omega \) on the local algebra \(\mathfrak {U}(\mathbf{B})\) there is a GNS triple \((H, \pi , \Omega )\) reproducing \(\omega \) in which \(\omega \) is pure if and only if the representation \(\pi \) is irreducible (its commutant is trivial). This combines the GNS construction with the abstract 108.

For a state \(\omega \) on the quasilocal algebra \(\mathfrak {U}\) there is a GNS triple \((H, \pi , \Omega )\) reproducing \(\omega \) in which \(\omega \) is pure if and only if the representation \(\pi \) is irreducible. This combines the GNS construction with the abstract 108.

If \(\omega \) is pure, then any cyclic representation reproducing \(\omega \) (in particular its GNS representation) is irreducible. Since the commutant is \(*\)-closed, a commuting operator decomposes into self-adjoint real and imaginary parts; for a self-adjoint commuting \(S\), the affine rescaling \(T = (2(\Vert S\Vert +1))^{-1} S + \tfrac 12\) satisfies \(0 \le T \le 1\), so its coefficient functional \(a \mapsto \langle \Omega , T\, \pi (a)\Omega \rangle \) is positive and dominated by \(\omega \). Purity forces it proportional to \(\omega \), whence \(T\) (and so \(S\), and so the original operator) is a scalar by 103.

The GNS representations of two pure states are either disjoint or unitarily equivalent. Indeed the GNS representation of a pure state is irreducible (108), so the irreducible dichotomy (125) applies. Pure states thus fall into superselection sectors: the sector of a pure state is the unitary-equivalence class of its irreducible GNS representation.

Purity is preserved by any \(*\)-automorphism: for \(\Phi : \mathfrak {U} \xrightarrow {\sim } \mathfrak {U}\), the pullback state \(\omega \circ \Phi \) is pure if and only if \(\omega \) is. A dominated positive functional \(\psi \le \omega \circ \Phi \) transports to \(\psi \circ \Phi ^{-1} \le \omega \), which purity sends to a scalar multiple of \(\omega \); transporting back gives \(\psi \) proportional to \(\omega \circ \Phi \). Applied to the covariance automorphism \(\Phi = \beta _L\), this says purity of a state is a Lorentz-covariance-invariant property.

A state \(\omega \) on a local algebra \(\mathfrak {U}(\mathbf{B})\) is pure if and only if its pullback \(\omega \circ \hat\alpha _\varphi \) along the stabilizer automorphism is pure, for every \(\varphi \in \mathrm{Stab}(\mathbf{B})\). This is the curved specialization of 153: purity is invariant under the isometric symmetry that fixes the region.

For a covariance-compatible quasilocal algebra, the fiberwise Lorentz action extends to a \(*\)-automorphism of \(\mathfrak {U}\), so the lift of 138 exists. The intertwiner is defined on the directed union of local images (a dense \(*\)-subalgebra) and extended by uniform continuity; the inverse is supplied by \(L^{-1}\), and the group laws \(\beta _{L'L} = \beta _{L'}\circ \beta _L\) and \(\beta _{1} = \mathrm{id}\) furnish the two-sided inverse.

The covariance-compatibility hypothesis is satisfiable: the trivial net’s quasilocal algebra is covariance-compatible (every \(*\)-automorphism of \(\mathbb {C}\) is the identity), so the quasilocal lift exists unconditionally for the trivial net.

Let \(\Omega \) be a cyclic vector for \(\pi \). If \(T\) commutes with every \(\pi (a)\) and the diagonal coefficient \(a \mapsto \langle \Omega , T\, \pi (a)\Omega \rangle \) equals \(c\) times \(a \mapsto \langle \Omega , \pi (a)\Omega \rangle \), then \(T = c \cdot 1\). The proof is pure Hilbert-space analysis: using the \(*\)-representation property and the commutation relation, every off-diagonal coefficient \(\langle \pi (b)\Omega , (T - c)\, \pi (a)\Omega \rangle \) vanishes, so \((T - c)\) annihilates the dense cyclic orbit and is therefore zero.

Let \(\omega \) be a faithful state and \((\mathcal{H}_\omega , \pi _\omega , \Omega )\) its GNS triple. Then the cyclic vector \(\Omega \) is separating for \(\pi _\omega (\mathfrak {U})\): if \(\pi _\omega (a)\Omega = 0\) then \(a = 0\). Indeed \(\pi _\omega (a)\Omega = 0\) gives \(\omega (a^* a) = \langle \Omega , \pi _\omega (a^* a)\Omega \rangle = \langle \pi _\omega (a)\Omega , \pi _\omega (a)\Omega \rangle = 0\), and faithfulness forces \(a = 0\). This holds in any representation reproducing a faithful state, not only the GNS one.

The timelike and causal predicates are well-defined on smooth curves, independently of the chosen path representative: a smooth path is timelike (resp. causal) if and only if its associated smooth curve is. The invariance rests on the chain rule for the tangent under a smooth reparametrisation \(\varphi \), namely \(\mathrm{tangent}(\mu \circ \varphi )(s) = \varphi '(s)\cdot \mathrm{tangent}(\mu )(\varphi (s))\), together with the scale-invariance of the causal classification: since \(g(c\, v, c\, v) = c^2\, g(v,v)\), the vector \(c\, v\) is timelike/null exactly when \(v\) is, for \(c \neq 0\).

Realizing the state space as the subset \(\mathrm{stateSpace}(A) \subseteq A \to _L \mathbb {C}\) (the latter a real topological vector space via \(\texttt{NormedSpace.complexToReal}\)), it is convex: a real convex combination \(a\, \omega _1 + b\, \omega _2\) of states is again a state, via \(\texttt{State.convexCombo}\). Moreover a state \(\omega \) lies in Mathlib’s \(\mathrm{extremePoints}_{\mathbb {R}}\) of the state space exactly when it is extreme in the sense of 115, connecting the purity characterizations to the Krein-Milman/Choquet API.

The state space, realized inside the weak-* dual \(\mathrm{WeakDual}\, \mathbb {C}\, A\) as the positive functionals with \(\varphi (1) = 1\), is weak-* compact. By Banach-Alaoglu it suffices that it is a weak-* closed subset of the closed unit ball: the positivity conditions \(0 \le \varphi (a^* a)\) and the normalization \(\varphi (1) = 1\) are each weak-* closed (evaluation at a fixed element is weak-* continuous), and a positive functional with \(\varphi (1)=1\) has \(\| \varphi \| = 1\). This is the analytic input for the existence of pure states via Krein-Milman.

If \(\Omega \) is cyclic for the local observables of \(\mathbf{B}_1\), then for a spacelike-separated region \(\mathbf{B}_2\) every \(R \in R(\mathbf{B}_2)\) with \(R\Omega = 0\) is zero, so \(\Omega \) is separating for \(R(\mathbf{B}_2)\). The implication is elementary (\(R\) vanishes on the dense set of vectors \(A\Omega \) for \(A\) a local observable of \(\mathbf{B}_1\), since \(R(\mathbf{B}_2)\) commutes with those observables by microcausality, 91). In Minkowski spacetime the cyclicity hypothesis is exactly the content of the Reeh-Schlieder theorem, which rests on the spectrum condition; here it is taken as an explicit hypothesis, mirroring the curved-spacetime version where no spectrum condition is available.

The separating property phrased on the bundled VonNeumannAlgebra \(R(\mathbf{B}_2)\): with \(\Omega \) cyclic for the local observables of \(\mathbf{B}_1\), every \(R\) in the bundled algebra \(R(\mathbf{B}_2)\) of a spacelike-separated region with \(R\Omega = 0\) is zero. It reduces to 95 through the coercion \(\uparrow R(\mathbf{B}_2) = \pi (\mathfrak {U}(\mathbf{B}_2))''\).

The separating property phrased on the bundled VonNeumannAlgebra \(R(\mathbf{B}_2)\): with \(\Omega \) cyclic for the local observables of \(\mathbf{B}_1\), every \(R\) in the bundled algebra \(R(\mathbf{B}_2)\) of a spacelike-separated subregion with \(R\Omega = 0\) is zero. It reduces to 184 through the coercion \(\uparrow R(\mathbf{B}_2) = \pi (\mathfrak {U}(\mathbf{B}_2))''\).

The abstract mechanism is: if \(\Omega \) is cyclic for a set \(S\) of operators, then any \(R\) commuting with all of \(S\) with \(R\Omega = 0\) is zero (it vanishes on the dense set \(S\Omega \)). Applied to the net: if \(\Omega \) is cyclic for the local observables of \(\mathbf{B}_1\) - the role supplied in Minkowski spacetime by Reeh-Schlieder - then for a spacelike-separated subregion \(\mathbf{B}_2\), every \(R \in R(\mathbf{B}_2)\) with \(R\Omega = 0\) is zero, since \(R(\mathbf{B}_2)\) commutes with the observables of \(\mathbf{B}_1\) by curved microcausality (180). Thus \(\Omega \) is separating for \(R(\mathbf{B}_2)\): a nonzero observable of one region cannot be annihilated by the cyclic vector of a spacelike-separated region, the operator-algebraic form of statistical independence.

At positive width \(\beta {\gt} 0\) the strip-Liouville principle (159) is a theorem. The equal boundary values let \(F\) extend to a bounded \(i\beta \)-periodic entire function by the strip Schwarz reflection (160), which is then constant on \(\mathbb {R}\) by Liouville’s theorem. The hypothesis \(\beta {\gt} 0\) is necessary: at \(\beta = 0\) the open strip is empty and at \(\beta {\lt} 0\) the strip itself is empty, and in both cases the principle is false.

A function \(F\) continuous on the closed strip \(0 \le \operatorname {Im} z \le \beta \), holomorphic on the open strip, bounded, and with equal boundary values \(F(t) = F(t + i\beta )\) on the real axis, admits a bounded entire extension \(H\) agreeing with \(F\) on \(\mathbb {R}\). The extension is the \(i\beta \)-periodic continuation \(H(z) = F\! \left(z - \lfloor \operatorname {Im} z / \beta \rfloor \, i\beta \right)\), which folds every point into the fundamental strip: it is continuous across each gluing line \(\operatorname {Im} z = k\beta \) because the boundary values match by periodicity, holomorphic off the lines as \(F\) composed with a holomorphic shift, and holomorphic on the lines by the horizontal-line removable-singularity theorem (Morera). This is the analytic engine behind the strip-Liouville principle.

Two functions continuous and bounded on the closed strip \(0 \le \operatorname {Im} z \le \beta \) (with \(\beta {\gt} 0\)), holomorphic on the open strip, that agree on both boundary lines \(\operatorname {Im} z = 0\) and \(\operatorname {Im} z = \beta \), agree on the whole strip. Indeed their difference has vanishing boundary values, so its \(i\beta \)-periodic extension (160) is a bounded entire function, constant by Liouville, equal to its value \(0\) at the origin.

Unitarily equivalent representations share their representation-theoretic type: \(\pi _1\) is irreducible if and only if \(\pi _2\) is, and the generated von Neumann algebra \(\pi _1(A)''\) is a factor if and only if \(\pi _2(A)''\) is. The transport is packaged through the cross-space conjugation \(T \mapsto U T U^{-1}\), a multiplicative isomorphism \(\mathcal{B}(H_1) \simeq \mathcal{B}(H_2)\) that carries \(\pi _1(A)\) onto \(\pi _2(A)\), centralizers onto centralizers, and scalars onto scalars; irreducibility (commutant equal to the scalars) and factoriality (trivial center of \(\pi (A)''\)) are then preserved.

The conditions a vacuum state satisfies that need neither the spectrum condition nor Stone’s theorem. First, a vacuum state is invariant (the first conjunct of the definition). Second, a pure vacuum state yields the irreducible covariant GNS representation of 146: a covariant GNS triple with implementing unitaries \(U(L)\) fixing \(\Omega \) and operator covariance \(U(L)\pi (a)U(L)^{-1} = \pi (\beta _L a)\), whose representation is irreducible and generates all of \(\mathcal{B}(H)\). Both follow by projecting to invariance and chaining with purity \(\Rightarrow \) irreducibility; the spectrum-condition content of the vacuum is not used.

Phrased on the bundled VonNeumannAlgebra \(R(\mathbf{B})\) with its order \(\le \) and Mathlib’s commutant: for completely spacelike-separated regions \(R(\mathbf{B}_1) \le R(\mathbf{B}_2)'\), and for \(\mathbf{B}_1 \subseteq \mathbf{B}_2\), \(R(\mathbf{B}_1) \le R(\mathbf{B}_2)\). Both reduce to the set-level statements through the coercion \(\uparrow R(\mathbf{B}) = \pi (\mathfrak {U}(\mathbf{B}))''\).

Phrased on the bundled VonNeumannAlgebra \(R(\mathbf{B}')\) with its order \(\le \) and Mathlib’s commutant: for completely spacelike-separated subregions \(R(\mathbf{B}_1) \le R(\mathbf{B}_2)'\), and for \(\mathbf{B}_1 \subseteq \mathbf{B}_2\) (with the isotony coherence) \(R(\mathbf{B}_1) \le R(\mathbf{B}_2)\). Both reduce to the set-level statements through the coercion \(\uparrow R(\mathbf{B}') = \pi (\mathfrak {U}(\mathbf{B}'))''\).

The geometric-covariance set equality is upgraded to a first-class \(*\)-algebra isomorphism of the bundled local von Neumann algebras, \(R(\mathbf{B}) \cong R(L \cdot \mathbf{B})\). It is the restriction of the conjugation \(*\)-automorphism \(T \mapsto U(L) T U(L)^{-1}\) of \(\mathcal{B}(H)\) (Mathlib’s conjStarAlgEquiv, whose adjoint-is-inverse property makes conjugation by a unitary star-preserving) to \(R(\mathbf{B})\), whose image is exactly \(R(L \cdot \mathbf{B})\). The reusable ingredient is that a \(*\)-automorphism carrying the underlying set of one star-subalgebra onto another restricts to a star-algebra equivalence between them. This makes the unitary equivalence of orbit-related local algebras an explicit transportable object.

For \(g \in \mathrm{Stab}(\mathbf{B})\), the geometric-covariance set equality is upgraded to a first-class \(*\)-algebra isomorphism of the bundled local von Neumann algebras, \(R(\mathbf{B}_1) \cong R(g \cdot \mathbf{B}_1)\), given by restricting the conjugation \(*\)-automorphism \(T \mapsto U(g) T U(g)^{-1}\) of \(\mathcal{B}(H)\) to \(R(\mathbf{B}_1)\). It reuses the same restriction construction as the Minkowski case (100).

If the local von Neumann algebra \(R(\mathbf{B})\) is a factor (its center \(R(\mathbf{B}) \cap R(\mathbf{B})'\) is exactly the scalars), then so is \(R(L \cdot \mathbf{B})\) for every Lorentz transformation \(L\). Geometric covariance (98) exhibits \(R(L \cdot \mathbf{B}) = U(L) R(\mathbf{B}) U(L)^{-1}\); conjugation by a unitary is a multiplicative automorphism that carries the center onto the center and fixes the scalars, so it preserves the factor property. Thus being a factor is constant along the Lorentz orbit of a region.

If the local von Neumann algebra \(R(\mathbf{B}_1)\) of a subregion is a factor, then so is \(R(g \cdot \mathbf{B}_1)\) for every \(g \in \mathrm{Stab}(\mathbf{B})\). Geometric covariance exhibits \(R(g \cdot \mathbf{B}_1) = U(g) R(\mathbf{B}_1) U(g)^{-1}\), and conjugation by a unitary preserves the factor property; so being a factor is constant along the stabilizer orbit of a subregion.

In a covariant representation \(\pi \) of the quasilocal algebra with the operator covariance \(U(L)\, \pi (a)\, U(L)^{-1} = \pi (\beta _L a)\), conjugation by the implementing unitary \(U(L)\) carries the local von Neumann algebra of a region \(\mathbf{B}\) onto that of the Lorentz-transformed region \(L \cdot \mathbf{B}\):

This is the statement that the symmetry group acts geometrically on the net of von Neumann algebras. The proof rests on three ingredients, all already available: operator covariance (the last clause of 146); the fact that \(\beta _L = \) the covariance action sends \(\iota _{\mathbf{B}}(\mathfrak {U}(\mathbf{B}))\) onto \(\iota _{L\cdot \mathbf{B}}(\mathfrak {U}(L\cdot \mathbf{B}))\) (from the action-on-generators identity and surjectivity of the covariance equivalence \(\alpha _L\)); and the algebraic fact that conjugation by a unit is a multiplicative automorphism, which therefore commutes with the bicommutant. The reusable core is that a multiplicative automorphism maps the centralizer of a set onto the centralizer of its image, hence maps bicommutants to bicommutants. In particular \(R(\mathbf{B})\) and \(R(L \cdot \mathbf{B})\) are unitarily equivalent (conjugate by \(U(L)\)), so any unitary-conjugation-invariant property of a local von Neumann algebra — such as being a factor — is constant along the Lorentz orbit of a region.

The curved, stabilizer-subgroup analogue of geometric covariance. There is no quasilocal algebra in curved spacetime, so the only symmetries acting on the containing algebra \(\mathfrak {U}(\mathbf{B})\) are the stabilizer \(\mathrm{Stab}(\mathbf{B}) = \{ g : g \cdot \mathbf{B} = \mathbf{B}\} \). For \(g \in \mathrm{Stab}(\mathbf{B})\), the implementing unitary \(U(g)\) of the stabilizer GNS representation (198) conjugates the local von Neumann algebra of a subregion \(\mathbf{B}_1 \subseteq \mathbf{B}\) onto that of \(g \cdot \mathbf{B}_1\):

Hence \(R(\mathbf{B}_1)\) and \(R(g \cdot \mathbf{B}_1)\) are unitarily equivalent. Unlike Minkowski, the abstract LorentzianSpacetime interface supplies neither basis-set preservation \(M.\mathrm{IsBasisSet}(g \cdot \mathbf{B}_1)\) nor the coherence relating the stabilizer action \(\hat\alpha _g\) to the chosen isotony embeddings (171); both enter as explicit hypotheses, discharged for a net from a concrete geometric spacetime. The proof reuses the shared conjugation core: conjugation by a unit is a multiplicative automorphism, which maps centralizers — hence bicommutants — to those of the image.

For basis regions \(\mathbf{B}_1 \subseteq \mathbf{B}_2\), the local von Neumann algebras are nested: \(R(\mathbf{B}_1) \subseteq R(\mathbf{B}_2)\). The local observables of \(\mathbf{B}_1\) embed into those of \(\mathbf{B}_2\) via the quasilocal isotony coherence, and the double commutant is monotone.

For nested basis subregions \(\mathbf{B}_1 \subseteq \mathbf{B}_2 \subseteq \mathbf{B}\), the local von Neumann algebras are nested: \(R(\mathbf{B}_1) \subseteq R(\mathbf{B}_2)\). Unlike Minkowski, the curved Axiom 3 isotony embeddings are chosen witnesses with no built-in composition law, so the coherence of the embeddings (that the \(\mathbf{B}_1 \hookrightarrow \mathbf{B}\) embedding factors through \(\mathbf{B}_2\)) is taken as an explicit hypothesis; it holds automatically whenever the Axiom 3 witnesses form a genuine inclusion family. Given it, the local observables of \(\mathbf{B}_1\) embed into those of \(\mathbf{B}_2\) and the double commutant is monotone.

For completely spacelike-separated basis regions \(\mathbf{B}_1, \mathbf{B}_2\), the local von Neumann algebras commute: \(R(\mathbf{B}_1) \subseteq R(\mathbf{B}_2)'\). This is the von Neumann form of Einstein causality (88): elementwise commutation \(\pi (\mathfrak {U}(\mathbf{B}_2)) \subseteq \pi (\mathfrak {U}(\mathbf{B}_1))'\), with the antitonicity of the commutant and the identity \(S''' = S'\) collapsing the iterated commutants. A Haag-Kastler net is thus a net of mutually commuting von Neumann algebras for spacelike regions.

For completely spacelike-separated basis subregions \(\mathbf{B}_1, \mathbf{B}_2 \subseteq \mathbf{B}\), the local von Neumann algebras commute: \(R(\mathbf{B}_1) \subseteq R(\mathbf{B}_2)'\). This is the von Neumann form of curved Einstein causality (177), via the antitonicity of the commutant and \(S''' = S'\). It is the curved counterpart of 91.