Nonstandard Hulls of C*-Algebras and Their Applications

Stefano Baratella

doi:10.3390/math9202598

Abstract

For the sake of providing insight into the use of nonstandard techniques à la A. Robinson and into Luxemburg’s nonstandard hull construction, we first present nonstandard proofs of some known results about C*-algebras. Then we introduce extensions of the nonstandard hull construction to noncommutative probability spaces and noncommutative stochastic processes. In the framework of internal noncommutative probability spaces, we investigate properties like freeness and convergence in distribution and their preservation by the nonstandard hull construction. We obtain a nonstandard characterization of the freeness property. Eventually we provide a nonstandard characterization of the property of equivalence for a suitable class of noncommutative stochastic processes and we study the behaviour of the latter property with respect to the nonstandard hull construction.

Highlights

In this work we apply nonstandard techniques à la Abraham Robinson to C∗-algebras, C∗-probability spaces and noncommutative stochastic processes.Our starting point is the nonstandard hull construction due to Luxemburg [1]
After some preliminary results about states, we show that the property of freeness of a family of subalgebras is preserved by forming the nonstandard hull of a C∗-probability space
We introduce the nonstandard notion of almost freeness and we show that it coincides with freeness on standard families of subalgebras of a standard C∗-probability space, obtaining a nonstandard characterization of the ordinary freeness property

Summary

Introduction

In this work we apply nonstandard techniques à la Abraham Robinson to C∗-algebras, C∗-probability spaces ( known as noncommutative probability spaces) and noncommutative stochastic processes. After recalling the notion of stochastic process over a C∗-algebra given in [9], we extend the nonstandard hull construction to an internal noncommutative stochastic process. In this setting we deal with the notion of equivalence. We just recall that a nonstandard universe allows to properly extend each infinite mathematical object X under consideration of an object ∗X, in a way that X and ∗X satisfy the same properties which are definable by means of bounded quantifier formulas in the first order language of set theory This property is referred to as the Transfer Principle. The nonstandard proofs that we present below show how to apply the nonstandard techniques in combination with the nonstandard hull construction

Infinite Dimensional Nonstandard Hulls Fail to Be von Neumann Algebras

Nonstandard Hulls of Internal Function Spaces

Noncommutative Loeb Theory

Nonstandard Noncommutative Probability

Nonstandard Fock Spaces