Subalgebras of B(l1) and the Stone-Cech compactification βN

AbstractLet X be a non-reflexive Banach space and let B(X) denote the Banach algebra of all bounded linear operators on X. Wilansky [9] introduced two classes of subalgebras of B(X), {Ωw} and {Γw}, defined as follows: Ωw = {T ∈ B(X): T∗∗ w ∈ w ⊕ X}, w ∈ X∗∗X and Γw = {T ϵ B(X): T∗∗w ∈ 〈w〉}, w ∈ X∗∗.Brown and Cho [3] have studied the subalgebras Ωw and Γw in the case where X = c and shown that for w ϵ c∗∗c all the algebras Ωw are isomorphic and all the algebras Γw are isomorphic. Brown, Cass and Robinson [2] showed, inter alia, that in the case X = l1, not all the algebras Γw are isomorphic. In this paper the Stone-Čech compactification and the Riesz representation theorem are used to show that when X = l1 not all the algebras Ωw are isomorphic