Non-commutative spectral theory for affine function spaces on convex sets

Alfsen, Erik M.; Shultz, Frederic W.

doi:10.1090/memo/0172

Cited by 81 publications

(122 citation statements)

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“…Here A b (K) is the space of all bounded affine functions on K, and A + b (K) its subspace of positive functions. A(K) will stand for the space of continuous affine functions on K [6,12].…”

Section: State Spaces and The Work Of Alfsen Shultz And Hanche-olsenmentioning

confidence: 99%

“…[6, 3.8] or [4]) (alternative definitions are possible [6]). The first axiom of [5] is Axiom AHS1: Every norm-exposed face of K is projective.…”

Section: State Spaces and The Work Of Alfsen Shultz And Hanche-olsenmentioning

confidence: 99%

“…The Jordan product A • B (or, equivalently, A 2 ) is constructed using the non-commutative spectral theory defined by K [6,7]. This product then coincides with the anti-commutator in…”

Section: State Spaces and The Work Of Alfsen Shultz And Hanche-olsenmentioning

confidence: 99%

“…Even if one is happy describing quantum mechanics with superselection rules in terms of C * -algebras, from a physical perspective one should not necessarily regard the above axioms as unique, or as the best ones possible. The notion of a projective face (or, equivalently, a P -projection [6]) is a complicated one (but cf. [11] for a certain simplification in the finite-dimensional case, and [35] for an analogous interpretation in terms of filters in the general case).…”

Section: State Spaces and The Work Of Alfsen Shultz And Hanche-olsenmentioning

confidence: 99%

“…2.17 in [6], it is not difficult to show that the pure state space of a unital JB-algebra (where every pure state is indeed norm-exposed) is a symmetric transition probability space.…”

Section: Transition Probabilities On Pure State Spacesmentioning

confidence: 99%

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Poisson Spaces with a Transition Probability

Landsman

1997

Rev. Math. Phys.

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The common structure of the space of pure states P of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p : P × P → [0, 1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context).In classical mechanics, where p(ρ, σ) = δρσ, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of P as a transition probability space coincide with the symplectic leaves of P as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant).Motivated by E.M. Alfsen, H.

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“…Here A b (K) is the space of all bounded affine functions on K, and A + b (K) its subspace of positive functions. A(K) will stand for the space of continuous affine functions on K [6,12].…”

Section: State Spaces and The Work Of Alfsen Shultz And Hanche-olsenmentioning

confidence: 99%

“…[6, 3.8] or [4]) (alternative definitions are possible [6]). The first axiom of [5] is Axiom AHS1: Every norm-exposed face of K is projective.…”

Section: State Spaces and The Work Of Alfsen Shultz And Hanche-olsenmentioning

confidence: 99%

“…The Jordan product A • B (or, equivalently, A 2 ) is constructed using the non-commutative spectral theory defined by K [6,7]. This product then coincides with the anti-commutator in…”

Section: State Spaces and The Work Of Alfsen Shultz And Hanche-olsenmentioning

confidence: 99%

Section: State Spaces and The Work Of Alfsen Shultz And Hanche-olsenmentioning

confidence: 99%

“…2.17 in [6], it is not difficult to show that the pure state space of a unital JB-algebra (where every pure state is indeed norm-exposed) is a symmetric transition probability space.…”

Section: Transition Probabilities On Pure State Spacesmentioning

confidence: 99%

See 3 more Smart Citations