Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 3-17.

Morita type equivalences and reflexive algebras

Authors: G.K. Eleftherakis
Author institution: Department of Mathematics, University of Athens, Panepistimiopolis, 15784, Greece

Summary: Two unital dual operator algebras $\mathcal{A},\ \mathcal{B}$ are called $\Delta $-equivalent if there exists an equivalence functor $\mathcal{F}: \, _{\mathcal{A}}\mathfrak{M}\rightarrow \, _{\mathcal{B}}\mathfrak{M}$ which `extends" to a $*$-functor implementing an equivalence between the categories $_{\mathcal{A}}\mathfrak{DM}$ and $_{\mathcal{B}}\mathfrak{DM}.$ Here $_{\mathcal{A}}\mathfrak{M}$ denotes the category of normal representations of $\mathcal{A}$ and $_{\mathcal{A}}\mathfrak{DM}$ denotes the category with the same objects as $_{\mathcal{A}}\mathfrak{M}$ and $\Delta (\mathcal{A})$-module maps as morphisms $(\Delta (\mathcal{A})=\mathcal{A}\cap \mathcal{A}^*)$. We prove that any such functor maps completely isometric representations to completely isometric representations, `respects" the lattices of the algebras and maps reflexive algebras to reflexive algebras. We present applications to the class of CSL algebras.

Keywords: Operator algebras, dual operator algebras, Morita equivalence, TRO, reflexive algebras, CSL

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