For a Banach D-bimodule M over an abelian unital C -algebraD, we define E 1 (M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual M # . Equip E 1 (M) with the weak*-topology. We develop general properties of E 1 (M). It is properly viewed as a coordinate system for M when M C, where C is a unital C -algebra containing D as a regular MASA with the extension property; moreover, E 1 (C) coincides with Kumjian's twist in the context of C -diagonals. We identify the C -envelope of a subalgebra A of a C -diagonal when D A C. For triangular subal- gebras, each containing the MASA, a bounded isomorphism induces an al- gebraic isomorphism of the coordinate systems which can be shown to be continuous in certain cases. For subalgebras, each containing the MASA, a bounded isomorphism that maps one MASA to the other MASA induces an isomorphism of the coordinate systems. We show that the weak operator clo- sure of the image of a triangular algebra in an appropriate representation is a CSL algebra and that a bounded isomorphism of triangular algebras extends to an isomorphism of these CSL algebras. We prove that for triangular alge- bras in our context, any bounded isomorphism is completely bounded. Our methods simplify and extend various known results; for example, isometric isomorphisms of the triangular algebras extend to isometric isomorphisms of the C -envelopes, and the conditional expectation E : C ! D is multiplicative when restricted to a triangular subalgebra. Also, we use our methods to prove that the inductive limit of C -diagonals with regular connecting maps is again a C -diagonal.

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Theory of Operator Algebras II

Coordinate systems and bounded isomorphisms

A physical and mathematical framework for the analysis of probabilities in quantum theory is proposed and developed. One purpose is to surmount the problem, crucial to any reconciliation between quantum theory and space-time physics, of requiring instantaneous “wave-packet collapse” across the entire universe. The physical starting point is the idea of an observer as an entity, localized in space-time, for whom any physical system can be described at any moment, by a set of (not necessarily pure) quantum states compatible with his observations of the system at that moment. The mathematical starting point is the theory of local algebras from axiomatic relativistic quantum field theory. A function defining thea priori probability of mistaking one local state for another is analysed. This function is shown to possess a broad range of appropriate properties and to be uniquely defined by a selection of them. Through a general model for observations, it is argued that the probabilities defined here are as compatible with experiment as the probabilities of conventional interpretations of quantum mechanics but are more likely to be compatible, not only with modern developments in mathematical physics, but also with a complete and consistent theory of measurement.

A priori probability and localized observers

We prove that if A and B are bounded self-adjoint operators such that A − B belongs to the trace class, then |A|−|B| belongs to the principal ideal L1,1 in the algebra L(H) of all bounded operators on an infinite-dimensional Hilbert space generated by an operator whose sequence of eigenvalues is {1, 1 , 1 ,...}. Moreover, µ(j;|A|−|B|) ≤ const(1+j) 1 kA− Bk1. We also obtain a semifinite version of this result, as well as the corresponding commutator estimates.

Weak type estimates for the absolute value mapping

The sufficiency and weak sufficiency in * -algebras are discussed. Some properties are studied concerning the relative entropy and the sufficiency for invariant states and KMS states in W* and C*-dynami- cal systems. Introduction. The concept of sufficiency is very important in mathematical statistics. The abstract measure theoretic investigation of sufficient statistics was initiated by Halmos and Savage (13). Kullback and Leibler (19) gave the characterization of sufficiency in terms of the information (i.e., the classical relative entropy). Umegaki (33,34) studied the sufficiency and the relative entropy in the noncommutative case of semi-finite von Neumann algebras. Araki (4,5) extended the relative entropy to the case for normal positive linear functionals of general von Neumann algebras and showed its several properties. Furthermore Uhlmann (32) showed the general WYDL concavity using a quadratic inteφolation theory and defined the relative entropy of positive linear functionals of arbitrary *-algebras. In the previous paper (14), we discussed the sufficiency and the relative entropy in von Neumann algebras and gave the characterizations of invariant states and KMS states with respect to the modular automor- phism group of a faithful normal state. In this paper, we further develop the sufficiency and the relative entropy in * -algebras. In §1, we introduce besides the sufficiency another notion of weak sufficiency and establish the relation between them. In §2, we deal with the weak sufficiency of positive linear maps between ^alge- bras. In §3, we mention the Araki's and Uhlmann's relative entropies which are equal in the von Neumann algebra case. We further give a formula of relative entropy for states of C*-algebras. In §4, we establish some properties of invariant states and KMS states in WΓ*-dynamical systems and C*-dynamical systems through the relative entropy and the sufficiency. The theorems there improve or extend the results obtained in (14). Finally we give an application to the Gibbs states of quantum lattice systems.

/pdf/sufficiency-and-relative-entropy-in-algebras-with-4utn3pu8rb.pdf

Theory of Operator Algebras II

Citations

Coordinate systems and bounded isomorphisms

A priori probability and localized observers

Weak type estimates for the absolute value mapping

Sufficiency and relative entropy in *-algebras with applications in quantum systems

A bicommutant theorem for completely bounded module homomorphisms

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Theory of Operator Algebras II

Citations

Coordinate systems and bounded isomorphisms

A priori probability and localized observers

Weak type estimates for the absolute value mapping

Sufficiency and relative entropy in *-algebras with applications in quantum systems

A bicommutant theorem for completely bounded module homomorphisms

Related Papers (5)

Fundamentals of the Theory of Operator Algebras

Operator algebras and quantum statistical mechanics

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Completely Bounded Maps and Operator Algebras

C[]-algebras and W[]-algebras