Peter M. Alberti

A NOTE ON THE TRANSITION PROBABILITY OVER C*-ALGEBRAS

The algebraic structure of Uhlmanns transition probability between mixed states on unital C*-algebras (see [2]) is analyzed. Several improvements of methods to calculate the transition probability are fixed examples are given (e.g., the case of quasi-local C*-algebras is dealt with) and two more functional characterizations are proved in general (see Theorems 1 and 3).

STOCHASTIC LINEAR MAPS AND TRANSITION PROBABILITY

Some aspects of the transition probability P(ω, ν) between states ω, ν on unital *-algebras are discussed. It is shown that P increases under the action of any stochastic linear map T, i.e., P(Tω, Tν)⩾P(ω, ν). Some properties of P are derived in starting from a recently-proved characterization of the quantity in question.

Existence and Density Theorems for Stochastic Maps on Commutative C*‐algebras

This paper presents theoremes on the structure of stochastic and normalized positive linear maps over commutative C*-algebras. We show how strongly the solution of the n-tupel problem for stochastic maps relates to the fact that stochastic maps of finite rank are weakly dense within stochastic maps in case of a commutative C*-algebra. We give a new proof of the density theorem and derive (besides the solution of the n-tupel problem) results concerning the extremal maps of certain convex subsets which are weakly dense. All stated facts suggest application in Statistical Physics (algebraic approach), especially concerning questions around evolution of classical systems.

Bounds for the C*‐algebraic transition probability yield best lower and upper bounds to the overlap

Bounds are proved for the C*‐algebraic transition probability PA(ω,ν) between the abstract ground state ν with respect to a symmetric subspace N of a unital C* algebra A and a state ω with the restriction ω‖N=σ‖N to N for an arbitrarily given, but fixed state σ. A is assumed to be the unital C*‐algebra generated by N. The results are specified in the case where A is a subalgebra of a vN algebra in standard form and N is dimensionally finite. Under these assumptions, the relationships of the algebraic transition probability to the notion of the (square of the) overlap integral known in quantum physics are clearly established. The general results are used to treat the standard problem of finding upper and lower bounds to the overlap in a quantum mechanical context. The best bounds are found and their properties discussed.

A note on stochastic dynamics in the state space of a commutative C* algebra

In this paper a functional characterization of stochastic evolutions within the state spaces of commutative C* algebras with identity is derived. Consequences concerning the structure of those linear evolution equations (master equations) that give occassion to stochastic evolutions are discussed. In part, these results generalize facts which are well known from the finite‐dimensional classical case. Examples are given and some important particularities of the W* case are developed.

Convexity, unitary invariance and monotonicity under completely positive maps over injective vN-algebras

The action of dynamical maps over the normal state space of a properly infinite, injective vN-algebra is analyzed and shown to be equivalent to convec unitary mixing with respect to some suitably chosen C*-subalgebra. As an application, it is shown that the conditions usually imposed on (convex) relative state functionals (like the relative entropy etc.) necessarily imply their decrease under completely positive maps.