Kaplansky's second test problem in operator algebras

Abstract

Kaplansky's second test problem on similarity asks: if $T$ and $S$ are elements in a unital Banach algebra $\mathcal{B}$ and $T\oplus T$ is similar to $S\oplus S$ in ${\mathbb{M}}_2(\mathcal{B})$, is $T$ similar to $S$ in $\mathcal{B}$? We answer this problem affirmatively if $T$ is an operator with property $(J)$ in a type ${\mathrm{I}}_n$ von Neumann algebra $\mathcal{M}$, i.e., $\{T{\}}^{\prime }\cap \mathcal{M}$ contains a bounded maximal abelian family of idempotents. Moreover, the condition of property $(J)$ can be removed for $1⩽n⩽3$. A similar result is proved if $T$ is an element in a unital Banach algebra $\mathcal{B}$ with essentially finite-dimensional commutant, i.e., the relative commutant of $T$ in $\mathcal{B}$ is finite-dimensional modulo its Jacobson radical. Finally, we point out that one of our main results can be applied to the implementation of local unitary (LU) equivalence of quantum states.

54 pages