Representations of the direct product of matrix algebras
Suppose $B$ is a unital algebra which is an algebraic product of full matrix algebras over an index set $X$. A bijection is set up between the equivalence classes of irreducible representations of $B$ as operators on a Banach space and the $\sigma $-complete ultrafilters on $X$ (Theorem 2.6). Therefore, if $X$ has less than measurable cardinality (e.g. accessible), the equivalence classes of the irreducible representations of $B$ are labeled by points of $X$, and all representations of $B$ are described (Theorem 3.3).