On a C*-algebra of matrix-bounded operators

We discuss the C*-algebra generated by bounded operators with matrices (with respect to a fixed basis) such that each row and each column has not more than a fixed number of non-zero elements. This C*-algebra in some aspects behaves like the algebra of all bounded operators (e.g. satisfies the Kuiper theorem), but is not a von Neumann algebra and has more ideals. It can be obtained as the direct limit of the uniform Roe algebras of bounded geometry metrics on $\mathbb N$. We also discuss an application of this C*-algebra to infinite graph theory.