Frederic W. Shultz

A Gelfand-Neumark theorem for Jordan algebras

Abstract Let A be a Jordan algebra over the reals which is a Banach space with respect to a norm satisfying the requirements: (i) ∥ a ° b ∥ ≤ ∥ a ∥ ∥ b ∥, (ii) ∥ a2 ∥ = ∥ a ∥2, (iii) ∥ a2 ∥ ≤ ∥ a2 + b2 ∥ for a, bϵA. It is shown that A possesses a unique norm closed Jordan ideal J such that A J has a faithful representation as a Jordan algebra of self-adjoint operators on a complex Hilbert space, while every “irreducible” representation of A not annihilating J is onto the exceptional Jordan algebra M38.

On normed Jordan algebras which are Banach dual spaces

Alfsen, Shultz, and Stormer have defined a class of normed Jordan algebras called JB-algebras, which are closely related to Jordan algebras of self-adjoint operators. We show that the enveloping algebra of a JB-algebra can be identified with its bidual. This is used to show that a JB-algebra is a dual space iff it is monotone complete and admits a separating set of normal states; in this case the predual is unique and consists of all normal linear functionals. Such JB-algebras (“JBW-algebras”) admit a unique decomposition into special and purely exceptional summands. The special part is isomorphic to a weakly closed Jordan algebra of self-adjoint operators. The purely exceptional part is isomorphic to C(X, M38) (the continuous functions from X into M38).

Pure states as a dual object for

We consider the set of pure states of aC*-algebra as a uniform space equipped with transition probabilities and orientation, and show that the pure states with this structure determine theC*-algebra up to *-isomorphism.

C^*

Abstract It is shown that every rational polytope is affinely equivalent to the set of all states of a finite orthomodular lattice, and that every compact convex subset of a locally convex topological vector space is affinely homeomorphic to the set of all states of an orthomodular lattice.

-algebras

We discuss the uniqueness of quantum states compatible with given measurement results for a set of observables. For a given pure state, we consider two different types of uniqueness: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it was known that for a

A characterization of state spaces of orthomodular lattices

d

Uniqueness of quantum states compatible with given measurement results

-dimensional Hilbert space, there exists a set of

Normal state spaces of Jordan and von Neumann algebras

4d\ensuremath{-}5

Events and Observables in Axiomatic Quantum Mechanics

observables that uniquely determines any pure state. We show that for case (2),

Orientations and von Neumann algebras

5d\ensuremath{-}7