The Ulam--Kaplansky problem in operator algebras

Abstract

The Ulam--Kaplansky problem up to similarity in operator algebras asks whether $A\oplus A\sim B\oplus B$ implies $A\sim B$. We construct block-diagonal compact operators for which this implication fails. For von Neumann algebras with separable predual, we prove that this cancellation property is equivalent to subhomogeneity. Moreover, we obtain a cancellation law for finite ampliations in unital Banach algebras when the commutant of one element is finite dimensional modulo its Jacobson radical.

arXiv comment21 pages