|
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 ∥a∥≦ 1 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.
|
