Commutative topological algebras on translation-invariant reproducing kernel Hilbert spaces

We study commutative topological algebras naturally associated with translation-invariant reproducing kernel Hilbert spaces whose direct integral decomposition has one-dimensional fibers.

Starting from the bounded algebra of translation-invariant operators, we pass to a common dense domain generated by reproducing kernels and identify the corresponding diagonalizable operators with multiplication by symbols in an intersection of weighted $L^2$-spaces.

On the symbol side this gives a canonical space $\mathcal F_0$ and a maximal multiplicative subalgebra $\mathcal F_M$, which is a complete locally convex $*$-algebra.

Transporting the structure back yields corresponding algebras of operators and integral kernels.

We also discuss when the inclusions $L^\infty(\Omega)=\mathcal F_\infty\subset \mathcal F_M\subset \mathcal F_0$ are strict, and illustrate the results with vertical and radial operators on classical Bergman and Fock spaces.