A recent paper on subfactors of von Neumann factors has stimulated much
research in von Neumann algebras. It was discovered soon after the
appearance of this paper that certain algebras which are used there for
the analysis of subfactors could also be used to define a new polynomial
invariant for links. Recent efforts to understand the fundamental nature
of the new link invariants has led to connections with invariant theory,
statistical mechanics and quantum theory. In turn, the link invariants,
the notion of a quantum group, and the quantum Yang-Baxter equation have
had a great impact on the study of subfactors. Our subject is certain
algebraic and von Neumann algebraic topics closely related to the
original paper. However, in order to promote, in a modest way, the
contact between diverse fields of mathematics, we have tried to make
this work accessible to the broadest audience. Consequently, this book
contains much elementary expository material.