Representation theorems for complemented algebras
Freda E. Alexander · Transactions of the American Mathematical Society · 1970
Introduction.In this paper we obtain Hubert space representations for as large a class of complemented algebras as possible.In [2] the same problem was considered for complemented £*-algebras.It was shown then that if A is a topologically simple £*-algebra, then a sufficient (subject to a dimension restriction) and a necessary condition that a complementor p be expressible in terms of a Hubert space representation of A (or, equivalently, in the form Rp = (£,)# for some involution # in A) was that p be continuous.Throughout the present paper the same dimension restriction (that the algebra has no minimal left ideals of dimension less than three) will be imposed.In §5 a counterexample shows that it cannot be removed.The definition of continuity in [2] is not applicable to a general complemented algebra, but in §2 we give an alternative definition and show that this is an extension of the previous definition.In §3 we consider the case when A is a primitive Banach algebra.We obtain a faithful, continuous, strictly dense Hubert space representation for A when endowed with a continuous complementor.Under this we identify A with a left ideal of £(//) that is closed under a norm that majorises the operator norm.We then show that this representation characterizes primitive Banach algebras with continuous complementers.In §4 we use the results of §3 to obtain a faithful, continuous Hubert space representation for any semisimple Banach algebra A with a continuous complementor.Conversely, we show that, if any complemented algebra admits a representation of this form, then the complementor is continuous.We deduce that, if A is £*, then a necessary and sufficient condition that its complementor be expressible in the form £" = (£,)# is that it be continuous.This extends the result of [2].In §5 we apply the results of §4 to show that the condition C2 in the definition of a complementor cannot be relaxed.1. Preliminaries.Throughout the paper A will denote a semisimple complex Banach algebra whose norm is || || ; {/A : A e A} is the set of all minimal closed two-sided ideals of A ; RA is the set of all closed right ideals of A and MA the set of all minimal right ideals of A.Following [8] we say that A is a right complemented algebra if there is a mapping p: R -> R" of Ra onto itself that has the following properties:Cx: RnRp = iO) iReRA);