Canonical conditional expectation on a reduced crossed product

Definition

Let a discrete group $G$ act by automorphisms $\alpha$ on a $C^*$ -algebra $A$ . The canonical conditional expectation on the reduced crossed product is the map

E_A:A\rtimes_{\alpha,r}G\longrightarrow A,\qquad E_A\!\left(\sum_{s\in F}a_su_s\right)=a_e

on finite Fourier sums, extended continuously. It is a contractive positive $A$ -bimodule projection fixing $A$ , hence a conditional expectation. Moreover, it is faithful: $E_A(x^*x)=0$ implies $x=0$ . The discreteness hypothesis ensures that $A$ is the degree- $e$ coefficient algebra inside the crossed product.

Fourier coefficients

For $x\in A\rtimes_{\alpha,r}G$ , its $s$ -th Fourier coefficient is

\widehat{x}(s)=E_A(xu_s^*).

These coefficients recover the expected values on finite sums and are compatible with left and right multiplication by $A$ . Faithfulness makes $E_A$ particularly useful: positivity questions can often be tested after forming $x^*x$ and extracting its identity coefficient. The construction is canonical, so it does not depend on a choice of faithful representation used to realize the reduced crossed product Brown–Ozawa, §4.1.

Example

For the trivial action on $A=\mathbb C$ , the map is the canonical trace on the reduced group $C^*$ -algebra:

E_{\mathbb C}\!\left(\sum_{s\in F}c_s\lambda_s\right)=c_e.

For a nontrivial action it is usually not tracial, because the coefficient algebra need not commute with the implementing unitaries. It remains an $A$ -bimodule map, which is the relevant replacement for scalar traciality.

Beyond discrete groups

References

Nathanial P. Brown and Narutaka Ozawa, C-Algebras and Finite-Dimensional Approximations*, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. DOI record. Relevant: §4.1 on reduced crossed products by discrete groups and the canonical expectation.
Dana P. Williams, Crossed Products of C-Algebras*, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: Chapter 2 on reduced crossed-product constructions.