Von Neumann algebra ($M=M^{\prime \prime }\subseteq B(H)$)

A unital star-subalgebra of the bounded operators on a Hilbert space that is closed in the weak operator topology, equivalently in the strong operator topology. The Von Neumann double commutant theorem makes these conditions equivalent to equality with the double commutant of an operator algebra. It is norm closed and hence a C-star algebra.

Here a Hilbert space module means a Hilbert space with a unital star-representation of a Von Neumann algebra, rather than a Hilbert C-star module with an algebra-valued inner product. A closed invariant subspace is reducing because the algebra is closed under adjoints. Module equivalence means an intertwining unitary. The polar decomposition of a bounded operator converts an intertwiner into a unitary between its initial and final support subspaces.

Two Hilbert space modules are unitarily equivalent if each embeds isometrically as a closed submodule of the other. For isometric intertwiners $u:H_1\to H_2$ and $v:H_2\to H_1$, let $D=H_1\ominus v{H}_2$ and $K={⨁}_{n\ge 0}(vu{)}^{n}D$. These summands are orthogonal. The operator acting as $u$ on $K$ and as $v^{\ast }$ on $K^{\perp }$ is the required unitary onto $H_2$.

A Von Neumann algebra whose center consists exactly of scalar multiples of the identity. A group von Neumann algebra is a factor exactly when every nonidentity conjugacy class is infinite. A central operator has a square-summable coefficient vector constant on those classes; an infinite class must therefore have coefficient zero. Conversely, the sum of left translations over a finite nonidentity class is a nonscalar central operator.

For a discrete group, its group von Neumann algebra is generated on ${\ell }^2(\Gamma )$ by left translations $\lambda (g){\delta }_h={\delta }_{gh}$. The vector ${\delta }_e$ is cyclic and separating, and its vector functional is tracial. Its modular conjugation acts by $J{\delta }_g={\delta }_{g^{-1}}$ with conjugated scalar coefficients. The commutant is the algebra generated by right translations $\rho (g){\delta }_h={\delta }_{h{g}^{-1}}$.