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STABLE PROPERTIES OF HYPERRELEXIVITY

Part of:Modules, bimodules and idealsLinear spaces and algebras of operators

Published online by Cambridge University Press:  21 July 2015

G. K. ELEFTHERAKIS
G. K. ELEFTHERAKIS*
Affiliation:
Department of Mathematics, Faculty of Sciences, University of Patras, 26500 Patras, Greece e-mail: gelefth@math.upatras.gr
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Abstract

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Recently, a new equivalence relation between weak* closed operator spaces acting on Hilbert spaces has appeared. Two weak* closed operator spaces ${\mathcal{U}}$, ${\mathcal{V}}$ are called weak TRO equivalent if there exist ternary rings of operators ${\mathcal{M}}$i, i = 1, 2 such that ${\mathcal{U}}=[{\mathcal{M}}_2{\mathcal{V}}{\mathcal{M}}_1^*]^{-w^*}, {\mathcal{V}}=[{\mathcal{M}}_2^*{\mathcal{U}}{\mathcal{M}}_1]^{-w^*}.$ Weak TRO equivalent spaces are stably isomorphic, and conversely, stably isomorphic dual operator spaces have normal completely isometric representations with weak TRO equivalent images. In this paper, we prove that if ${\mathcal{U}}$ and ${\mathcal{V}}$ are weak TRO equivalent operator spaces and the space of I × I matrices with entries in ${\mathcal{U}}$, MIw(${\mathcal{U}}$), is hyperreflexive for suitable infinite I, then so is MIw(${\mathcal{V}}$). We describe situations where if ${\mathcal{L}}$1, ${\mathcal{L}}$2 are isomorphic lattices, then the corresponding algebras Alg($\mathcal{L}$1), Alg($\mathcal{L}$2) have the same complete hyperreflexivity constant.

MSC classification

Primary:47L05: Linear spaces of operators
Secondary:47L35: Nest algebras, CSL algebras16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 1 , January 2016, pp. 205 - 218
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

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