On the C*-algebra of matrix-finite bounded operators

TL;DR

This paper investigates a specific C*-algebra generated by matrix-finite bounded operators on a separable Hilbert space, revealing its structural properties and distinctions from other well-known operator algebras.

Contribution

It introduces and analyzes the C*-algebra formed by limits of matrix-finite operators, highlighting its unique properties and differences from classical operator algebras.

Findings

Not an AW*-algebra

Contains a proper closed ideal larger than compact operators

Group of invertibles is contractible

Abstract

Let $H$ be a separable Hilbert space with a fixed orthonormal basis. Let $\mathbb B^{(k)}(H)$ denote the set of operators, whose matrices have no more than $k$ non-zero entries in each line and in each column. The closure of the union (over $k\in\mathbb N$) of $\mathbb B^{(k)}(H)$ is a C*-algebra. We study some properties of this C*-algebra. We show that this C*-algebra is not an AW*-algebra, has a proper closed ideal greater than compact operators, and its group of invertibles is contractible.