von Neumann algebra (original) (raw)

Definition

Since the weak and strong operator topology are weaker than the norm topology, it follows that every von Neumann algebra is a norm closed *-subalgebra of B⁢(H). Thus, von Neumann algebras are a particular class of C*-algebras (http://planetmath.org/CAlgebra) and the results and tools from the C* theory are also applicable in the setting of von Neumann algebras. Nevertheless, the philosophy behind von Neumann algebras is quite different from that of C*-algebras and the tools and techniques for each theory turn out to be different as well.

Examples:

1. 1. B⁢(H) is itself a von Neumann algebra.
2. 1. L∞⁢(ℝ) (http://planetmath.org/LinftyXDmu) as subalgebra of B⁢(L2⁢(ℝ)) is a von Neumann algebra.