Stable properties of hyperreflexivity

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 operators M_i, i=1,2 such that U=[ M_2 V M_1^*]^{-w^*}, V=[ M_2^* U 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 cl U and V are weak TRO equivalent operator spaces and the space of I x I matrices with entries in U, M_I^w( U), is hyperreflexive for suitable infinite I, then so is M_I^w( V). We describe situations where if L1, L are isomorphic lattices, then the corresponding algebras Alg{L1}, Alg{L2} have the same complete hyperreflexivity constant.