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Group von Neumann Algebra

The group von Neumann algebra L(G) of a discrete group G is the von Neumann algebra generated on l^2(G) by its left regular group representation. Here l^2(G) is the Hilbert space of square-summable functions on G, and the representing operators satisfy

L_gxi(h)=xi(g^(-1)h).

Equivalently, L(G) is the bicommutant of the represented complex group algebra. Its canonical trace is tau(T)=<Tdelta_e,delta_e>, where delta_e is 1 at the identity element of G and 0 elsewhere (de la Harpe 2012). It satisfies tau(1)=1 and tau(L_g)=0 for g!=e.

For example, if G is the trivial group, then L(G)=C. Matrices over L(G) carry the trace tau_n(T)=sum_(i=1)^(n)tau(T_(ii)), with tau_n(1)=n. This normalization is used for the group-ring determinant conjecture.

See also

Bifinite Correspondence, Fuglede-Kadison Determinant, Group Algebra, Group Representation, Group-Ring Determinant Conjecture, L2-Betti Number, Twisted Group von Neumann Algebra, von Neumann Algebra, von Neumann Dimension

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References

de la Harpe, P. "The Fuglede-Kadison Determinant, Theme and Variations." 7 Jul 2012. https://arxiv.org/abs/1107.1059v2.

Cite this as:

Weisstein, Eric W. "Group von Neumann Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GroupvonNeumannAlgebra.html

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