  Reproducing kernel Hilbert spaces (RKHSs) are Hilbert spaces of functions
where pointwise evaluation is continuous. There are known examples of RKHSs
that are Banach algebras under pointwise multiplication. These examples are
built from weights on the dual of a locally compact abelian group. In this
paper we define an algebra structure on an RKHS that is equivalent to
subconvolutivity of the weight for known examples (referred to as reproducing
kernel Hilbert algebras, or RKHAs). We show that the class of RKHAs is closed
under the Hilbert space tensor product and the pullback construction on the
category of RKHSs. The subcategory of RKHAs becomes a monoidal category with
the spectrum as a monoidal functor to the category of topological spaces. The
image of this functor is shown to contain all compact subspaces of $\mathbb
R^n$ for $n>0$.
