Commutativity properties in Banach∗-algebras

Abstract

1* Introduction* Let Tlf T2, and U be bounded linear operators on a Hubert space where Tx and T2 are normal. The well-known theorem of Fuglede [13] asserts that if TtU = UT, then T?U= UT?. Putnam's generalization [13] states that if TiZ7= UT2 then T?U= UT2*. With this in view Berberian [3] defined an .FT-ring to be a ring with an involution x —> x* such that x*y = yx* whenever x is normal and xy — yx. Likewise a PΓ-ring is one which gives x?y — yxf for all x19 x2 normal and xγy = yx2. The usual examples of Banach *-algebras A [14] are FT and PΓ-algebras since they have faithful ^representations as bounded linear operators on a Hubert space. For our purposes we must demand somewhat more of A. We suppose that A is a semisimple hermitian *-algebra whose maximal commutative *-subalgebras are Shilov algebras and where xx* e W, W a minimal closed two-sided ideal implies that x e W. These requirements may seem special, but are actually satisfied by all J3*-algebras, all iϊ*-algebras and all group algebras of compact groups. Suppose that b Φ 0 in A and ba = ab — 0 for some a Φ 0 in A. We show that there exist c φ 0, h Φ 0, h self-adjoint, with be = cb = 0 and ch = he = 0 provided that either A has two closed two-sided ideals I Φ (0), J Φ (0) with I n J = (0) or A has zero socle. Without such hypotheses the conclusion can fail, as it does for the algebra of all 2x2 matrices over the complex field. 2* Notation and preliminaries* As is customary, a Banach *-algebras A is called hermitian if the spectrum of each self-adjoint element is real. Suppose that A is hermitian and semisimple. Then so is the algebra obtained by adjoining an identity to A. Therefore, the theory expounded in [12] for hermitian Banach *-algebras with an identity applies here to show that A has a faithful *-representation as bounded linear operators on a Hubert space. In particular, if x e A and xx* = 0 then x = 0. Ptak's development [12] involves a pene