Subalgebras of the dual of the Fourier algebra of a compact group

Charles F. Dunkl, Donald E. Ramirez
1972 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
The object of this paper is to show that AP(G) and W(G) are algebras for a restricted class of compact groups called groups of bounded representation type. Let G be a compact non-Abelian group; we let G denote the set of equivalence classes of continuous unitary irreducible representations of G. We call G the dual of G. For aeG, let T a be an element of a. Then T a is a homomorphism of G into U(n a ), the group of n a x n a unitary matrices, where n a is the dimension of a. We use TJp)t o
more » ... the matrix entries of TJx), xeG, 1 ^ i,j ^ n a , and T aii to denote the function x i-> TJx)^. Now T«{*y) X. We define the operator norm of A e3S(X) by = 8up{|.4g|: g e Z , \i\ < 1}. The trace of A, TvA, is ^A^Zd ™^™ {Z$-i is i=i any orthonormal basis for X and (•, •) denotes the inner product in X. Let \A\ denote (A *A)i. The value ||^||oo is the spectral radius of \A\; that is, maxlAj: 1 ^ i ^ n}, where A 4 are the eigenvalues of \A\. Let ^ be a set { a :a.eG where is denoted by J §?°°((T). It is a Banach algebra under the norm j|^{|«, = supfll^J^: aeG} and coordinate-wise operations.
doi:10.1017/s0305004100050568 fatcat:majbkic2gvgpraerqoxj56myn4