Noncommutative Spectral Theory Research Articles

Tensorial Function Theory: From Berezin Transforms to Taylor’s Taylor Series and Back

Let H ∞(E) be the Hardy algebra of a W*-correspondence E over a W*-algebra M. Then the ultraweakly continuous completely contractive representations of H ∞(E) are parametrized by certain sets $${{\mathcal{AC}}(\sigma)}$$ indexed by NRep(M)—the normal *-representations σ of M. Each set $${{\mathcal{AC}}(\sigma)}$$ has analytic structure, and each element $${F \in H^{\infty}(E)}$$ gives rise to an analytic operator-valued function $${\widehat{F}_{\sigma}}$$ on $${{\mathcal{AC}}(\sigma)}$$ that we call the σ-Berezin transform of F. The sets $${\{{\mathcal{AC}}(\sigma)\}_{\sigma\in\Sigma}}$$ and the family of functions $${\{\widehat{F}_{\sigma}\}_{\sigma\in\Sigma}}$$ exhibit “matricial structure” that was introduced by Joeseph Taylor in his work on noncommutative spectral theory in the early 1970s. Such structure has been exploited more recently in other areas of free analysis and in the theory of linear matrix inequalities. Our objective here is to determine the extent to which the matricial structure characterizes the Berezin transforms.

Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics

Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras.

Spectral resolution in an order-unit space

The operational approach to quantum physics employs an order-unit space in duality with a base-normed space, and in this context, a suitable spectral theory is a prerequisite for the representation of quantum-mechanical observables. An order-unit space is called spectral if it is enriched by a compression base with the comparability and projection cover properties. These notions are explicated in the article. We show that each element in a spectral order-unit space determines and is determined by a spectral resolution and it has a spectrum which is a nonempty closed bounded subset of the real numbers. Our theory is a generalization and a more algebraic version of the well-known non-commutative spectral theory of Alfsen and Shultz.

Quantum Logics and Convex Spaces

An orthomodular σ-lattice with rich set ofstates satisfying the property that every affinefunctional from the set of states into the unit intervalof the reals corresponds to an expectational functional of exactly one real observable (so-calledu-spectral logic) is compared with the noncommutativespectral theory of Alfsen and Shultz. Necessary andsufficient conditions are found under which these two approaches are in correspondence.

Trace inequalities for spaces in spectral duality

Let (A,e) and (V,K) be an order-unit space and a base-norm space in spectral duality, as in noncommutative spectral theory of Alfsen and Shultz. Let t be a norm lower semicontinuous trace on A, and let φ be a nonnegative convex function on ℝ. It is shown

The foundation of quantum theory and noncommutative spectral theory: Part II

The present paper comprises Sects. 5–8 of a work which proposes an axiomatic approach to quantum mechanics in which the concept of a filter is the central primitive concept. Having layed down the foundations in the first part of this work (which appeared in the last issue of this journal and comprises Sects. 0–4), we arrived at a dual pair 〈Y, M〉 consisting of abase norm space Y and anorder unit space M, being in order and norm duality with respect to each other. This is precisely the setting of noncommutative spectral theory, a theory which has been developed during the late nineteen seventies by Alfsen and Shultz. (2,3) In this part we add to the four axioms (Axioms S, DP, R, SP) of Sect. 3 three further axioms (Axioms E, O, L). These axioms are suggested by the work of Alfsen and Shultz and enable us to derive the JB-algebra structure of quantum mechanics (cf. Theorem 8.9).

The foundation of quantum theory and noncommutative spectral theory. Part I

The foundation of quantum theory and noncommutative spectral theory. Part I

On Non-Commutative Spectral Theory and Jordan Algebras

On Non-Commutative Spectral Theory and Jordan Algebras

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

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

On the geometry of noncommutative spectral theory

On the geometry of noncommutative spectral theory