  Harmonic Hilbert spaces on locally compact abelian groups are reproducing
kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier
transform of weighted $L^2$ spaces on the dual group. It is known that for
suitably chosen subadditive weights, every such space is a Banach algebra with
respect to pointwise multiplication of functions. In this paper, we study RKHSs
associated with subconvolutive functions on the dual group. Sufficient
conditions are established for these spaces to be symmetric Banach
$^*$-algebras with respect to pointwise multiplication and complex conjugation
of functions (here referred to as RKHAs). In addition, we study aspects of the
spectra and state spaces of RKHAs. Sufficient conditions are established for an
RKHA on a compact abelian group $G$ to have the same spectrum as the
$C^*$-algebra of continuous functions on $G$. We also consider one-parameter
families of RKHSs associated with semigroups of self-adjoint Markov operators
on $L^2(G)$, and show that in this setting subconvolutivity is a necessary and
sufficient condition for these spaces to have RKHA structure. Finally, we
establish embedding relationships between RKHAs and a class of Fourier--Wermer
algebras that includes spaces of dominating mixed smoothness used in
high-dimensional function approximation.
