Morita equivalence and stable isomorphism via unitary operators
Nikolaos Koutsonikos-Kouloumpis
TL;DR
The paper develops Δ-equivalence for dual operator systems and proves its equivalence with stable isomorphism, extending Morita-type theory to the dual, non-selfadjoint context. It shows that weak TRO-equivalence between dual operator spaces yields stable isomorphisms implemented by unitary operators, with a parallel unitary equivalence in the operator-system setting, and extends these results to the strong TRO-equivalence framework. It also demonstrates that Δ-equivalent dual operator spaces, viewed as bimodules over their adjointable multiplier algebras, admit TRO-equivalent normal CES representations. Collectively, the work unifies stable and Morita-type notions for dual operator spaces and systems and provides explicit unitary constructions realizing stabilized isomorphisms.
Abstract
We define $Δ$-equivalence for dual operator systems and prove that it is an equivalence relation. We show that weak TRO-equivalence of dual operator spaces induces a stable isomorphism between them which is given by multiplication with unitary operators, and in the special case of dual operator systems it is a unitary equivalence. We prove an analogous result for strong TRO-equivalence of operator spaces and for operator systems. Lastly, we show that $Δ$-equivalent dual operator spaces, considered as bimodules over their left and right adjointable multiplier algebras, have TRO-equivalent normal CES representations.