Test problems for operator algebras

Azoff, Edward A.

doi:10.1090/S0002-9947-1995-1321565-4

Test problems for operator algebras
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Trans. Amer. Math. Soc. 347 (1995), 2989-3001 Request permission

Abstract:

Kaplansky’s test problems, originally formulated for abelian groups, concern the relationship between isomorphism and direct sums. They provide a "reality check" for purported structure theories. The present paper answers Kaplansky’s problems in operator algebraic contexts including unitary equivalence of von Neumann algebras and equivalence of representations of (non self-adjoint) matrix algebras. In particular, it is shown that matrix algebras admitting similar ampliations are themselves similar.

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An invitation to

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Lectures on von Neumann algebras