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 σ-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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm169-2-4,
author = {Daniele Guido and Lars Tuset},
title = {Representations of the direct product of matrix algebras},
journal = {Fundamenta Mathematicae},
volume = {167},
year = {2001},
pages = {145-160},
zbl = {1014.46032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm169-2-4}
}
Daniele Guido; Lars Tuset. Representations of the direct product of matrix algebras. Fundamenta Mathematicae, Tome 167 (2001) pp. 145-160. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm169-2-4/