  In the present paper we study some homotopy invariants which can be defined
by means of bundles with fiber a matrix algebra. We also introduce some
generalization of the Brauer group in the topological context and show that any
its element can be represented as a locally trivial bundle with a group of
invertible operators in a Hilbert space as the structure group. Finally, we
discuss its possible applications in the twisted $K$-theory.
