Properties of strictly singular operators have recently become of topical interest because the work of Gowers and Maurey in [GM1] and [GM2] gives (among many other brilliant and surprising results, such as those in [G1] and [G2]) Banach spaces on which every continuous operator is of form λ I + S, where S is strictly singular. So if strictly singular operators had invariant subspaces, such spaces would have the property that all operators on them had invariant subspaces. However, in this paper we...

There has been a considerable search for radical, amenable Banach algebras. Noncommutative examples were finally found by Volker Runde [R]; here we present the first commutative examples. Centrally placed within the construction, the reader may be pleased to notice a reprise of the undergraduate argument that shows that a normed space with totally bounded unit ball is finite-dimensional; we use the same idea (approximate the norm 1 vector x within distance η by a “good” vector ${y}_{1}$; then approximate...

It is a long standing open problem whether there is any infinite-dimensional commutative Banach algebra without nontrivial closed ideals. This is in some sense the Banach algebraists' counterpart to the invariant subspace problem for Banach spaces. We do not here solve this famous problem, but solve a related problem, that of finding (necessarily commutative) infinite-dimensional normed algebras which do not even have nontrivial closed subalgebras. Our examples are incomplete normed algebras rather...

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