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Twisted Group von Neumann Algebra

The twisted group von Neumann algebra L_mu(G) of a discrete group G and a normalized unit-circle-valued 2-cocycle mu is generated on l^2(G) by the operators

u_gdelta_h=mu(g,h)delta_(gh).

Here delta_h is the standard basis vector, and the cocycle identity is mu(g,h)mu(gh,k)=mu(h,k)mu(g,hk). The generators satisfy u_gu_h=mu(g,h)u_(gh). The canonical trace has tau(u_g)=0 for g!=e and tau(1)=1. For an ICC group, this algebra is a von Neumann factor. The trivial cocycle gives the ordinary group von Neumann algebra.

For a type II_1 factor M, its amplification M^t is a corner pM_n(M)p with (Tr_n tensor tau)(p)=t>0, using unnormalized matrix trace.

See also

Bifinite Correspondence, Factor Fundamental Group, Group von Neumann Algebra, ICC Group, Lattice in a Locally Compact Group, Projective Unitary Representation

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References

Anantharaman, C. and Popa, S. An Introduction to II_1 Factors. https://www.math.ucla.edu/~popa/Books/IIun.pdf.Donvil, M. and Vaes, S. "W^*-Superrigidity for Cocycle Twisted Group von Neumann Algebras." Invent. Math. 240, 193-260, 2025. https://doi.org/10.1007/s00222-025-01320-5.

Cite this as:

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

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