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

Abstract

Let H be a separable Hilbert space with a fixed orthonormal basis. Let 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∈N) of 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, its group of invertibles is contractible. and it gives rise to an example of a C⁎-algebra with a dense ⁎-subalgebra and a maximal closed ideal with zero intersection.