Von Neumann algebra

In mathematics, a von Neumann algebra or W*-algebra is a -algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: The ring of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space of square-integrable functions. The algebra of all bounded operators on a Hilbert space is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least . Von Neumann algebras were first studied by in 1929; he and Francis Murray developed the basic theory, under the original name of rings of operators, in a series of papers written in the 1930s and 1940s (; ), reprinted in the collected works of . Introductory accounts of von Neumann algebras are given in the online notes of and and the books by , , and . The three volume work by gives an encyclopedic account of the theory. The book by discusses more advanced topics. There are three common ways to define von Neumann algebras. The first and most common way is to define them as weakly closed -algebras of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by many other common topologies including the strong, ultrastrong or ultraweak operator topologies. The -algebras of bounded operators that are closed in the norm topology are C-algebras, so in particular any von Neumann algebra is a C-algebra.