Mathematics for infinite dimensional objects is becoming more and more
important today both in theory and application. Rings of operators,
renamed von Neumann algebras by J. Dixmier, were first introduced by J.
von Neumann fifty years ago, 1929, in [254] with his grand aim of
giving a sound founda- tion to mathematical sciences of infinite nature.
J. von Neumann and his collaborator F. J. Murray laid down the
foundation for this new field of mathematics, operator algebras, in a
series of papers, [240], [241], [242], [257] and [259], during
the period of the 1930s and early in the 1940s. In the introduction to
this series of investigations, they stated Their solution 1 {to the
problems of understanding rings of operators) seems to be essential for
the further advance of abstract operator theory in Hilbert space under
several aspects. First, the formal calculus with operator-rings leads to
them. Second, our attempts to generalize the theory of unitary
group-representations essentially beyond their classical frame have
always been blocked by the unsolved questions connected with these
problems. Third, various aspects of the quantum mechanical formalism
suggest strongly the elucidation of this subject. Fourth, the knowledge
obtained in these investigations gives an approach to a class of
abstract algebras without a finite basis, which seems to differ
essentially from all types hitherto investigated. Since then there has
appeared a large volume of literature, and a great deal of progress has
been achieved by many mathematicians.
