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 (∗) everyrepresentation 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.
