A Generalization of Hilbert Modules

We show that there is a natural generalization of the notion of a Hilbert C*-module (also called “Hilbert module,” “inner product module,” “rigged module,” and sometimes “Hermitian module” in the literature) to nonselfadjoint operator algebras, and we lay down some foundations for this theory, including direct sums, tensor products, change of rings, and index for subalgebras of operator algebras. These modules in general do not give rise to a Morita equivalence (unlike in the C*-algebra case).