Representations of normed algebras with minimal left ideals

Abstract

The theory of *-representations of Banach *-algebras on Hilbert space is one of the most useful and most successful parts of the theory of Banach algebras. However, there are only scattered results concerning the representations of general Banach algebras on Banach spaces. It may be that a comprehensive representation theory is impossible. Nevertheless, for some special algebras interesting and worthwhile results can be proved. This is true for (Y), the algebra of all bounded operators on a Banach space Y, and for (Y), the subalgebra of (Y) consisting of operators with finite dimensional range. The representations of (Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of (Y) (and also of some more general algebras of operators). In both these papers, (Y), which is the socle of the algebras under consideration, plays an important role in the theory. This suggests the possibility that a more general representation theory can be formulated in the case of a normed algebra with a nontrivial socle. This we attempt to do in this paper.

Periodicity of functionals and representations of normed algebras on reflexive spaces

“…The same applies to the mapping ^-><^, ft} for fixed rj, and together these amount precisely to the continuity requirements of (1). Thus (2) implies (1).…”

“…There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra 2(E) of all continuous linear operators on E where E -I" with 1 <p # 2<oo, for Berkson and Porta proved in (2) that if E, F are taken from the spaces l p with 1 <p < oo and E # F then the only continuous homomorphism from £(£) into £(F) is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space-for example the algebra of finiterank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).…”