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

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.