Canonical map from full to reduced crossed product

Definition

For a $C^*$ -dynamical system $(A,G,\alpha)$ , the canonical map from the full to the reduced crossed product is the surjective $*$ -homomorphism

\Lambda_A:A\rtimes_\alpha G\longrightarrow A\rtimes_{\alpha,r}G

induced by the identity on the common dense convolution algebra $C_c(G,A)$ . It exists because the reduced norm is bounded above by the universal crossed-product norm. Equivalently, $\Lambda_A$ is the quotient associated with a faithful regular covariant representation of $A$ ; its kernel consists precisely of elements annihilated by that regular representation.

Equality of the two completions

The map $\Lambda_A$ is injective exactly when the full and reduced norms agree on $C_c(G,A)$ ; in that case it is an isomorphism. Amenability of the action is a standard sufficient condition for this equality. For the trivial action on $A=\mathbb C$ , $\Lambda_{\mathbb C}$ becomes the regular quotient

C^*(G)\longrightarrow C_r^*(G),

and it is injective exactly when $G$ is amenable Williams, discussion of regular representations and amenability.

Functorial role

The quotient compares two universal procedures applied to the same algebraic data. Every regular covariant representation factors through $A\rtimes_{\alpha,r}G$ , whereas every covariant representation factors through the full crossed product. Consequently, any property that passes to quotients passes from the full crossed product to the reduced one, but the converse generally requires additional hypotheses.

On the coefficient algebra, the canonical embeddings are compatible with $\Lambda_A$ . For non-discrete $G$ , those embeddings may land naturally in multiplier algebras, so compatibility should not be mistaken for the assertion that $A$ is literally a subalgebra of both crossed products.

Conventions and scope

References

Dana P. Williams, Crossed Products of C-Algebras*, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: Chapter 2 on full and reduced crossed products and Chapter 7 on amenability.