Stable isomorphism of dual operator spaces

TL;DR

This paper characterizes stable isomorphism of dual operator spaces through completely isometric representations and ternary rings of operators, linking it to dual operator algebra isomorphisms and providing examples from CSL algebra theory.

Contribution

It establishes a new criterion for stable isomorphism of dual operator spaces using completely isometric normal representations and ternary rings of operators.

Findings

Stable isomorphism characterized via representations and ternary rings

Dual operator algebras are stably isomorphic iff they are isomorphic

Provides examples motivated by CSL algebra theory

Abstract

We prove that two dual operator spaces $X$ and $Y$ are stably isomorphic if and only if there exist completely isometric normal representations $\phi$ and $\psi$ of $X$ and $Y$, respectively, and ternary rings of operators $M_1, M_2$ such that $\phi (X)= [M_2^*\psi (Y)M_1]^{-w^*}$ and $\psi (Y)=[M_2\phi (X)M_1^*].$ We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. We provide examples motivated by CSL algebra theory.