Vol. 18, No. 2, 1966

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Characterization of group algebras in terms of their translation operators

Frederick Paul Greenleaf

Vol. 18 (1966), No. 2, 243–276

In this paper we will characterize those Banach algebras A which are isometric and isomorphic to the group algebra of some (possibly nonabelian) compact group. The central idea of this characterization is to study the group Gl(A) of translation operators which act on A; here a translation is any linear isometric map of A onto A such that T(xy) = (Tx)y for all x,y A.

We first give a simple characterization of an intermediate class of Banach algebra which includes all group algebras of compact groups and many other closely related algebras. This is the class of QCG algebras: those A isometric and isomorphic to an algebra of the form A = φ(L1(H)) M(H)∕N, where H is a compact group, N a weak closed two-sided ideal in M(H) = C(H), and φ : M(H) M(H)∕N is the canonical homomorphism (M(H)∕N is given the quotient norm). This characterization involves the following axiom on A.

Axiom (CA) If a1 then La(La : x ax) is a strong operator limit of convex sums of translations.

Any QCG algebra has a great number of finite dimensional two-sided ideals; those QCG algebras A which are group algebras are singled out by studying the representations gotten by letting Gl(A) act on these ideals. Examples are given of QCG algebras which are not the group algebra of any compact group.

Mathematical Subject Classification
Primary: 46.80
Secondary: 42.56
Received: 13 November 1964
Published: 1 August 1966
Frederick Paul Greenleaf