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.
Read more about this topic: Group Algebra
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