Given an operator ideal , we say that a Banach space X has the approximation property with respect to if T belongs to for every Banach space Y and every T ∈ (Y,X), being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting.
Let α be an isometric automorphism of the algebra of bounded linear operators in (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].
We study Banach spaces X with subspaces Y whose unit ball is densely remotal in X. We show that for several classes of Banach spaces, the unit ball of the space of compact operators is densely remotal in the space of bounded operators. We also show that for several classical Banach spaces, the unit ball is densely remotal in the duals of higher even order. We show that for a separable remotal set E ⊆ X, the set of Bochner integrable functions with values in E is a remotal set in L¹(μ,X).
In this article, we describe some basic properties of the Banach algebra which is constructed from all bounded complex-valued functionals.