Maximal Tracial Algebras

TL;DR

This paper introduces the concept of maximal tracial algebras, explores their properties in various settings, and connects these ideas to major open problems in operator algebras.

Contribution

It defines maximal tracial algebras, characterizes abelian subalgebras in Banach spaces, and links the concept to the Kadison Similarity and Connes' Embedding Problems.

Findings

Characterization of maximal tracial algebras in dual and multiplier pairs.

Identification of abelian maximal tracial subalgebras in B(X) for rank-one tensors.

Connections established between maximal tracial algebras and major open problems.

Abstract

We introduce and study the notion of maximal tracial algebras. We prove several results in a general setting based on dual pairs and multiplier pairs. In a special case that X is a Banach space we determine the abelian subalgebras of B(X) that are maximal tracial for rank-one tensors. We also make slight connections between our ideas and the Kadison Similarity Problem and also the Connes' Embedding Problem.