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Representations of the direct product of matrix algebras. (English) Zbl 1014.46032

Summary: 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).

MSC:

46L05 General theory of \(C^*\)-algebras
46J10 Banach algebras of continuous functions, function algebras
46L85 Noncommutative topology
03E75 Applications of set theory
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