Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of C$^*$-algebras

TL;DR

This paper introduces a new topological orbit dimension for elements in unital C*-algebras, demonstrating that the topological free entropy dimension of nuclear C*-algebras is at most 1 and is additive in full free products.

Contribution

It defines a new topological orbit dimension and proves key properties of free entropy dimension in nuclear and free product C*-algebras.

Findings

Topological free entropy dimension of nuclear C*-algebras is ≤ 1.

Additivity of topological free entropy dimension in full free products.

Full free products of unital MF algebras are also MF algebras.

Abstract

In the paper, we introduce a new concept of topological orbit dimension of $n$-tuples of elements in a unital C$^*$ algebra. Using this concept, we conclude that the Voiculescu's topological free entropy dimension of any family of self-adjoint generators of a nuclear C$^*$ algebra is less than or equal to 1. We also show that the topological free entropy dimension is additive in the full free products of unital C$^*$ algebras. In the appendix, we show that unital full free product of Blackadar and Kirchberg's unital MF algebras is also MF algebra.