Homotopy theory of bundles with fiber matrix algebra

In the present paper we consider a special class of locally trivial bundles with fiber a matrix algebra. On the set of such bundles over a finite $CW$-complex we define a relevant equivalence relation. The obtained stable theory gives us a geometric description of the H-space structure ${\text{BSU}}_{\otimes }$ on $\text{BSU}$ related to the tensor product of virtual $\text{SU}$-bundles of virtual dimension 1.