Free Products of Generalized RFD C*-algebras

TL;DR

This paper studies the residual finite-dimensionality of C*-algebras related to infinite cardinals, characterizes this property, and explores how free products preserve it, including new insights for separable cases.

Contribution

It provides characterizations of residual less than k-dimensional property and shows free products preserve this property under certain conditions.

Findings

Free products of R_{<k}D algebras are R_{<k}D.

Sufficient conditions for free products to be R_{<k}D in unital case.

New characterization of RFD via lifting property for separable C*-algebras.

Abstract

If $k$ is an infinite cardinal, we say a C*-algebra $\mathcal{A}$ is residually less than $k$ dimensional, $R_{<k}D,$ if the family of representations of $\mathcal{A}$ on Hilbert spaces of dimension less than $k$ separates the points of $\mathcal{A}.$ We give characterizations of this property, and we show that if $\{\mathcal{A}_{i}:i\in I\} $ is a family of $R_{<k}D$ algebras, then the free product $\underset{i\in I}{\ast}\mathcal{A}_{i}$ is $R_{<k}D$. If each $\mathcal{A}_{i}$ is unital, we give sufficient conditions, depending on the cardinal $k$, for the free product $\underset{i\in I}{\ast_{\mathbb{C}}}\mathcal{A}_{i}$ in the category of unital C*-algebras to be $R_{<k}D$. We also give a new characterization of RFD, in terms of a lifting property, for separable C*-algebras.