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., $\{T\}'\cap\mathcal{M}$ contains a bounded maximal abelian family of idempotents. Moreover, the condition of property $(J)$ can be removed for $1\leqslant n\leqslant 3$. 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.