2026/07/22 by Chunlan Jiang, Minghui Ma, Rui Shi +1
#math.FA #math.OA
Kaplansky's second test problem on similarity asks: if T and S are elements in a unital Banach algebra B and T⊕ T is similar to S⊕ S in \mathbbM2(B), is T similar to S in B? We answer this problem affirmatively if T is an operator with property (J) in a type In von Neumann algebra M, i.e., \T\'\capM 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 B with essentially finite-dimensional commutant, i.e., the relative commutant of T in 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.