Stable Properties of Hyperrelexivity

Abstract

Recently, a new equivalence relation between weak* closed operator spaces acting on Hilbert spaces has appeared. Two weak* closed operator spaces U, V are called weak TRO equivalent if there exist ternary rings of operatorsWeak 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 U and V are weak TRO equivalent operator spaces and the space of I × I matrices with entries in U, M w I (U), is hyperreflexive for suitable infinite I, then so is M w I (V). We describe situations where if L 1 , L 2 are isomorphic lattices, then the corresponding algebras Alg(L 1 ), Alg(L 2 ) have the same complete hyperreflexivity constant.2010 Mathematics Subject Classification. 47L05, 47L35, 16D90.

A Morita Characterisation for Algebras and Spaces of Operators on Hilbert Spaces

“…We consider Ω as a subspace of B(P (K)). Then B ′ and Ω are weakly TRO equivalent in the sense of Definition 1.5 and I has cardinality at least dim H. By [11,Theorem 3.6] we obtain…”

“…According to [11,Theorem 4.6] we have that k(M ∞ (A ′ )) = k(M ∞ (P A ′ P )). However, the vector ξ is a seperating vector for the algebra P A ′ P and by [16,Remark 3.5] it holds that k(P A ′ P ) ≤ 48 < ∞.…”