A W∗-category is the
categorical counterpart of a von Neumann algebra with an abstract definition
equivalent to a concrete definition in terms of operators between Hilbert spaces. We
develop the elementary theory of W∗-categories including modular theory and the
comparison theory of objects (equivalence and quasiequivalence). We also
characterize certain W∗-categories in terms of the W∗-category of projections in a
von Neumann algebra, self-dual Hermitian modules for a von Neumann algebra
or normal representations of a von Neumann algebra. This leads naturally
to a discussion of the Morita equivalence of von Neumann algebras and of
W∗-categories.
