On constructing multiplications on Banach spaces

Abstract

This paper deals with the general problem of determining what kinds of multiplications can be put on a Banach space to make it into a Banach algebra. It has been shown that for a unital Banach algebra, the identity must be a vertex and a point of local uniform convexity [I ; 2, pp. 33-381. This implies that many Banach spaces (e.g., Hilbert space) cannot be made into unital Banach algebras. Grabiner has shown that any separable Banach space can be made into a radical Banach algebra of power series [6] or a semisimple Banach algebra of sequences with termwise multiplication [5]. In this paper we point out that many of the problems in the theory of strictly cyclic operator algebras as studied by Embry [4], Herrero [7], and Lambert [8] are closely related to the problem of putting multiplications on Banach spaces. In this paper we prove that any Banach space with a weak-* separable dual can be made into a radical Banach algebra of power series of any finite or countable number of indeterminates, commutative or not; a radical Banach algebra of strictly lower (or strictly upper) triangular matrices; or a radical Banach algebra of Dirichlet series. We prove that given a Banach space B with a weak* separable dual and a complemented subspace with an unconditional basis, and given an incidence algebra I(P) on a countable locally finite partially-ordered set P (as defined by Doubilet, Rota and Stanley in [3, pp. 269-271]), we can give B a separately continuous multiplication so that it is algebraically isomorphic to a dense subalgebra of I(P). As a corollary of this theorem, we get that any Banach space which satisfies the conditions of the theorem can be made into a Banach algebra of lower (or upper) triangular matrices. In this paper B denotes an infinite-dimensional real or complex Banach