Hyperreflexivity and operator ideals

Suppose (B,β) is an operator ideal, and A is a linear space of operators between Banach spaces X and Y . Modifying the classical notion of hyperreflexivity, we say that A is called B-hyperreflexive if there exists a constant C such that, for any T ∈ B(X,Y) with α = supβ(qT i) <∞ (the supremum runs over all isometric embeddings i into X, and all quotient maps of Y , satisfying qAi = 0), there exists a ∈ A, for which β(T − a) Cα. In this paper, we give examples of B-hyperreflexive spaces, as well as of spaces failing this property. In the last section, we apply SE -hyperreflexivity of operator algebras (SE is a regular symmetrically normed operator ideal) to constructing operator spaces with prescribed families of completely bounded maps. © 2007 Elsevier Inc. All rights reserved.