**Abstract**. We employ recent advances in the theory of
operator spaces, also known as quantized functional analysis, to provide a
context in which one can compare categories of modules over operator algebras
that are not necessarily self-adjoint. We focus our attention on the category
of Hilbert modules over an operator algebra and on the category of operator
modules over an operator algebra. The module operations are assumed to be
completely bounded - usually, completely contractive. We develop the notion of
a Morita context between two operator algebras $A$ and $B$.
This is a system $(A,B,{}_{A}X_{B},{}_{B}
Y_{A},(\cdot,\cdot),[\cdot,\cdot])$ consisting of the algebras, two
bimodules ${}_{A}X_{B}$ and $_{B}Y_{A}$ and pairings
$(\cdot,\cdot)$ and $[\cdot,\cdot]$ that induce (complete)
isomorphisms between the (balanced) Haagerup tensor products, $X
\otimes_{hB} {} Y$ and $Y \otimes_{hA} {} X$, and the algebras,
$A$ and $B$, respectively. Thus, formally, a Morita context
is the same as that which appears in pure ring theory. The subtleties of the
theory lie in the interplay between the pure algebra and the operator space
geometry. Our analysis leads to viable notions of *projective* operator
modules and *dual* operator modules. We show that two
C$^*$-algebras are Morita equivalent in our sense if and only if they
are $C^{\ast}$-algebraically strong Morita equivalent, and moreover
the equivalence bimodules are the same. The distinctive features of the
non-self-adjoint theory are illuminated through a number of examples drawn from
complex analysis and the theory of incidence algebras over topological partial
orders. Finally, an appendix provides links to the literature that developed
since this Memoir was accepted for publication.

Readership

Graduate students and research mathematicians interested in
operator theory.