Group–measure-space construction

Core idea

For a measure-preserving action of a countable discrete group $G$ on $(X,\mu)$ , put $A=L^\infty(X,\mu)$ and $\alpha_g(f)=f\circ T_{g^{-1}}$ . The group–measure-space algebra is the von Neumann crossed product $A\rtimes_\alpha G$ . Concretely, on $H=\ell^2(G,L^2(X,\mu))$ , set

(\pi(f)\xi)_h=\alpha_{h^{-1}}(f)\xi_h, \qquad (u_g\xi)_h=\xi_{g^{-1}h}.

Then $A\rtimes G=(\pi(A)\cup\{u_g:g\in G\})''$ .

Covariance and trace

The generators satisfy $u_g\pi(f)u_g^*=\pi(\alpha_g(f))$ . Finite sums $\sum_g f_gu_g$ form an ultraweakly dense algebraic core. The formula

\tau\left(\sum_g f_gu_g\right)=\int_X f_e\,d\mu

extends to a faithful normal tracial state.

Why the regular representation matters

The extra $\ell^2(G)$ coordinate retains group coefficients. Using only multiplication and Koopman operators on $L^2(X)$ need not recover this algebra. On a one-point space with trivial $\mathbb Z$ -action, those operators generate $\mathbb C$ , while the crossed product is $L(\mathbb Z)\cong L^\infty(\mathbb S^1)$ .

References

Sorin Popa, Ergodic theory of group actions. §4.1.