Vol. 15, No. 3, 1965

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Translation-invariant function algebras on compact groups

Joseph Albert Wolf

Vol. 15 (1965), No. 3, 1093–1099
Abstract

Let X be a compact group. β(X) denotes the Banach algebra (point multiplication, sup norm) of continuous complexvalued functions on X. A is any closed subalgebra of C(X) which is stable under right and left translations and contains the constants. It is shown, by means of the Peter-Weyl Theorem and some multilinear algebra, that the condition () every representation of degree 1 of X has finite image is necessary and sufficient that every possible A be self-adjoint. If X is connected, then () means that X is a projective limit of semisimple Lie groups; if X is a Lie group, then () means that X is semisimple. The Stone-Weierstrass Theorem then gives a quick classification of all possible algebras A on an arbitrary connected semisimple Lie group X.

Mathematical Subject Classification
Primary: 22.65
Milestones
Received: 20 April 1964
Published: 1 September 1965
Authors
Joseph Albert Wolf