# $C^{*}$-algebras isomorphically representable on $l^{p}$

## TL;DR

This paper characterizes when $C^{*}$-algebras can be represented on $l^{p}$ spaces for $p
eq 2$, showing such representations are tied to residual finite dimensionality and involve a specific compactness property of homomorphisms.

## Contribution

It establishes a characterization of $C^{*}$-algebras representable on $l^{p}$ spaces via a compactness property and residual finite dimensionality.

## Key findings

- Homomorphisms into $B(l^{p}(J))$ satisfy a compactness property.
- A $C^{*}$-algebra is isomorphic to a subalgebra of $B(l^{p}(J))$ iff it is residually finite dimensional.
- Representation on $l^{p}$ spaces is characterized by residual finite dimensionality.

## Abstract

Let $p\in(1,\infty)\backslash\{2\}$. We show that every homomorphism from a $C^{*}$-algebra $\mathcal{A}$ into $B(l^{p}(J))$ satisfies a compactness property where $J$ is any set. As a consequence, we show that a $C^{*}$-algebra $\mathcal{A}$ is isomorphic to a subalgebra of $B(l^{p}(J))$, for some set $J$, if and only if $\mathcal{A}$ is residually finite dimensional.