On the Geometry of Projections in Certain Operator Algebras

Abstract

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The general linear group as a complete invariant for $C^*$-algebras

“…The following lemma is a generalisation of Lemma 10 in [8] to simple, unital C˚-algebras. Lemma 1.5.…”

“…The following additional properties of the map θ can be easily checked by adapting the arguments in [1] and [8] to the present situation. Proposition 1.1.…”

Spectral presheaves as quantum state spaces

“…Hence, Wigner's theorem actually says that every automorphism φ : P(H) → P(H) of the projection lattice lifts to a Jordan * -automorphism Φ : B(H) → B(H). The generalization of Wigner's theorem to von Neumann algebras is known as Dye's theorem [43].…”

Spectral Lattices

“…Wigner's theorem states that every orthoisomorphism of the projection lattice P (B(H )), where dim H ≥ 3, is implemented either by a unitary or an antiunitary map. A more general result along this line is the Dye's theorem saying that any orthoisomorphism of a von Neumann projection lattice without Type I 2 direct summand extends to a Jordan isomorphism of the self-adjoint part of the algebra (see [4]). (Let us recall that a Jordan homomorphism is a linear map π : M 1 → M 2 such that π(x 2 ) = π(x) 2 for all x ∈ M 1 .)…”