Tracial algebras and an embedding theorem

TL;DR

This paper demonstrates that all countably generated tracial *-algebras can be embedded into ultraproducts of matrix algebras, providing new insights into their structure and relationships with finite von Neumann algebras.

Contribution

It establishes an embedding theorem for tracial *-algebras into ultraproducts of generic matrix algebras, advancing understanding of their structure and approximation properties.

Findings

Positive traces can be approximated by traces on matrix algebras

Countably generated tracial *-algebras embed into ultraproducts of matrix algebras

Finite von Neumann algebras embed into ultraproducts of tracial *-algebras

Abstract

We prove that every positive trace on a countably generated *-algebra can be approximated by positive traces on algebras of generic matrices. This implies that every countably generated tracial *-algebra can be embedded into a metric ultraproduct of generic matrix algebras. As a particular consequence, every finite von Neumann algebra with separable pre-dual can be embedded into an ultraproduct of tracial *-algebras, which as *-algebras embed into a matrix-ring over a commutative algebra.