S. Bhatt and R. Raina studied in Ref. B the behaviour of some fractional operators and Hadamard products on certain analytic functions on the unit disk. More generally, classes of analytic functions on the unit disk constitute a matter of actual intensive research. So, it is desirable to dispose of an adequate theoretic frame which allow relatively simple and expeditious results on this subject. Recently one of the authors considered topics on the structure of Hadamard algebras (cf. Ref. P , Ref. Pe ). In this article our aim is to consider Dirichlet spaces, which constitute well known Hilbert spaces, endowed with an abelian unitary Banach algebra structure induced by a Hadamard type product. The maximal ideal space, complex Hadamard homomorphisms, reproducing kernels, the generating function and spectra of their elements are determined.