Let G be a locally compact
group, C∞(G) the Banach algebra of C-valued continuous functions on G vanishing
at infinity, and let 𝒬 be a translation-invariant dense ∗-subalgebra. We assume that
𝒬 has its own norm, such that it is a Banach G-algebra with involution. Then the
twisted convolution algebra ℒ = L1(G,𝒬) is simple and symmetric and there exists
— up to unitary equivalence — exactly one irreducible ∗-representation
λ, mapping ℒ into the compact operators of L2(G). Thus for hermitian
f ∈ℒ one has the canonical spectral decomposition λ(f) = ΣjαjEj with
{αj} =Specλ(f) =Specℒ(f), Ej finite-dimensional projections in L2(G). It turns
out that Ej= λ(ej) for idempotent ej∈ℒ, hence every hermitian f ∈ℒ defines
uniquely a Fourier series Σαjej in ℒ. Different convergence properties of such
expansions are studied.
The main result states that for “radial functions” f the eigenfunctions ej span a
maximal commutative subalgebras of ℒ and that there exists a summation method
for these f, generalizing the Fejer kernel for periodic functions. More precisely: There
exists a bounded approximate identity for ℒ, consisting of finite linear combinations
of the ej. Applications are given to algebras L1(N) for nilpotent Lie groups N, in
particular all such N are determined, on which a compact abelian group K acts
such that the subalgebra LK1(N) of radial (i.e. K-invariant) functions is
commutative.
