Group Algebra - Von Neumann Algebras Associated To Groups

Von Neumann Algebras Associated To Groups

The group von Neumann algebra W*(G) of G is the enveloping von Neumann algebra of C*(G).

For a discrete group G, we can consider the Hilbert space l2(G) for which G is an orthonormal basis. Since G operates on l2(G) by permuting the basis vectors, we can identify the complex group ring CG with a subalgebra of the algebra of bounded operators on l2(G). The weak closure of this subalgebra, NG, is a von Neumann algebra.

The center of NG can be described in terms of those elements of G whose conjugacy class is finite. In particular, if the identity element of G is the only group element with that property (that is, G has the infinite conjugacy class property), the center of NG consists only of complex multiples of the identity.

NG is isomorphic to the hyperfinite type II₁ factor if and only if G is countable, amenable, and has the infinite conjugacy class property.