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On ultraproducts of operator algebras. (English) Zbl 1123.46040

Summary: Some basic questions on ultraproducts of \(C^*\)-algebras and von Neumann algebras are considered, including the relation to \(K\)-theory of \(C^*\)-algebras. More specifically, we prove that under certain conditions, the \(K\)-groups of ultraproducts of \(C^*\)-algebras are isomorphic to the ultraproduct of respective \(K\)-groups of \(C^*\)-algebras. We also show that the ultraproducts of factors of type \(\text{II}_1\) are prime, i.e., not isomorphic to any nontrivial tensor product.

MSC:

46L05 General theory of \(C^*\)-algebras
46L10 General theory of von Neumann algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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