On ultraproducts of operator algebras

Some basic questions on ultraproducts of C*-algebras and von Neumann algebras, including the relation to K-theory of C*-algebras are considered. More specifically, we prove that under certain conditions, the K-groups of ultraproduct 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 II₁ are prime, i.e. not isomorphic to any non-trivial tensor product.