A STONE-WEIERSTILASS THEOREM FOR GROUP REPRESENTATIONS

It is well known that if G is a compact group and π a faithful (unitary) representation, then each irreducible representation of G occurs in the tensor product of some number of copies of π and its contragredient. We generalize this result to a separable type I locally compact group G as follows: let π be a faithful unitary representation whose matrix coefficient functions vanish at infinity and satisfy an appropriate integrabillty condition. Then, up to isomorphism, the regular representation of G is contained in the direct sum of all tensor products of finitely many copies of π and its contragredient.


JOE REPKA
i.

INTRODUCTION.
Let G be a separable locally compact group, and n a representation of G ("representation" will always mean a strongly continuous unitary representation on a separable Hilbert space).For every pair of non-negative integers m, n, not both zero, let ( ) where n occurs m times, and its contragredient n occurs n times.
Our starting point is the following easy fact.
THEOREM i.Let G be a compact group and n a faithful representation.
Then every irreducible representation of G occurs in m,n>0 m,n 32.9 namely: THEOREM 2. (Burnside).Let G be a finite group, a faithful representation.Then every irreducible representation of G occurs in the tensor product of some number of copies of Rather than prove Burnside's theorem directly, we obtain it as an obvious special case of the following.
THEOREM 3. Let n be a positive integer.Let G be a compact group, all of whose elements have order dividing n.If is a faithful representation of G, then every irreducible representation of G occurs in the tensor product of some number of copies of PROOF.The argument is the same as that for Theorem I; the only thing that is not obvious is that the algebra spanned by coefficient functions is closed under complex conjugation.But the complex conjugates of the matrix coefficients of (g) are Just the matrix coefficients of (g)-I rearranged (since w is unitary), and the matrix coefficients of ?(g)-I )n-I (g are Just sums of products of matrix coefficients of (g) so we are done.
We generalize Theorem 1 to a non-compact group.
In this section we establish two results needed for the proof of the main theorem.As above, G is a separable locally compact group.
Let be a continuous square-integrable function on G. Let U be the open set where is not zero.We may restrict Haar measure to U and _1", JO REPKA LEMMA .Let A be an algebra of continuous functions on G which vanish at infinity and separate points, and suppose A is closed under complex con- Jugation.Let U be as above.
Then " A {" f f A} is dense in L2(U) and fix e > 0 Then $/ C (U) and by c c < the Stone-Weierstrass Theorem there exists $' A such that II$' $/II= < IIII "e and we have approximated Consequently, I12 I ')I12 2 by a function in " A Since C (U) is dense in L2(U) the proof is C complete.

I
Now let be a representation of G on a Hilbert space H LEMMA 5. Suppose the intersection of L2(G) with the set of coefficient functions of G spans a dense subspace of L2(G) Then (up to isomorphism) the regular representation of G is quasi-contained in G (i.e.contained in a direct sum of copies of ().

PROOF. If U
e H are such that the coefficient function SU,v(g) <G(g),V> is in L2(G) we define a linear map I v from some subspace of H to L2(G) by Iv(w) SW,V The domain of T V will be all vectors W H such that SW,V is square- integrable; it is a G-invariant subspace, containing U It is easy to check that T v is a closed G-map; hence we may apply Schur's Lemma to conclude that __L 2 there are closed G-subspaces V v c H W v c (G) and a surJective unitary By hypothesis we may choose (countably many) coefficient functions whose span is dense in L2(G) The corresponding subspaces W v thus span a dense subspace of L2(G) and, sticking together the maps U 1 and STONE-WEIERSTRASS THEOREM FOR GROUP REPRESENTATIONS 239 applying Schur's Lemma again, we get a unitary G-isomorphism of L2(G) into @ V V c_C @ H a countable direct sum of copies of H V V 3. THE MAIN RESULT.
We shall prove two weaker versions of our main result as a preliminary step.PROPOSITION 6.Let G be a separable locally compact group, and a faithful representation of G. Suppose that a non-trivial coefficient function of n is in LP(G) for some finite p, and that all the coefficient functions vanish at infinity.
Then the regular representation of G is quasi-contained in m,nZ0 m,n PROOF.By Lemma 5, it suffices to find coefficient functions of whose span is dense in L2(G) Suppose is a coefficient function of which is in L p(G) Then some power cn of is in L2(G) Since n is a A n L2(G) is invariant under translations by G we see that it contains a dense subset of L2(g.U) for any g E G But since U is open, any element of L2(G) can be approximated by a finite sum of elements Sinceas desired.