Translation-invariant function algebras on compact groups

Abstract

Let X be a compact group. $(X) denotes the Banach algebra (point multiplication, sup norm) of continuous complexvalued functions on X A is any closed subalgebra of d(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-ad joint. 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.