to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative
Geometry The theory of von Neumann algebras was initiated in a series of
papers by Murray and von Neumann in the 1930's and 1940's. A von Neumann
algebra is a self-adjoint unital subalgebra M of the algebra of bounded
operators of a Hilbert space which is closed in the weak operator
topology. According to von Neumann's bicommutant theorem, M is closed in
the weak operator topology if and only if it is equal to the commutant
of its commutant. A factor is a von Neumann algebra with trivial centre
and the work of Murray and von Neumann contained a reduction of all von
Neumann algebras to factors and a classification of factors into types
I, IT and III. C* -algebras are self-adjoint operator algebras on
Hilbert space which are closed in the norm topology. Their study was
begun in the work of Gelfand and Naimark who showed that such algebras
can be characterized abstractly as involutive Banach algebras,
satisfying an algebraic relation connecting the norm and the involution.
They also obtained the fundamental result that a commutative unital C*
-algebra is isomorphic to the algebra of complex valued continuous
functions on a compact space - its spectrum. Since then the subject of
operator algebras has evolved into a huge mathematical endeavour
interacting with almost every branch of mathematics and several areas of
theoretical physics.
