Crossed-product convolution *-algebra C_c(G,A)

Definition

Let $(A,G,\alpha)$ be a $C^*$ -dynamical system, fix a left Haar measure $ds$ , and let $\Delta$ be the modular function of $G$ . The crossed-product convolution $*$ -algebra is the vector space $C_c(G,A)$ of continuous compactly supported functions $G\to A$ , with

(f*g)(s)=\int_G f(t)\alpha_t\!\left(g(t^{-1}s)\right)\,dt

and

f^*(s)=\Delta(s^{-1})\alpha_s\!\left(f(s^{-1})^*\right).

The integrals are $A$ -valued Bochner integrals. These operations make $C_c(G,A)$ an associative involutive algebra, generally without an identity.

Algebraic role

The twisting by $\alpha_t$ in convolution records the interaction between the group and the coefficient algebra. The modular factor in the involution compensates for inversion under a left Haar measure. With these choices, every covariant pair has an integrated form that respects both multiplication and involution Williams, §2.3, equations (2.16)–(2.19).

The full crossed product and reduced crossed product are different $C^*$ -completions of this same dense algebraic core. Consequently, the notation $C_c(G,A)$ alone does not select a $C^*$ -norm.

Standard cases

If $A=\mathbb C$ and the action is trivial, the formulas reduce to the ordinary convolution and involution on $C_c(G)$ . If $G$ is discrete, integration becomes summation, compact support becomes finite support, and

(f*g)(s)=\sum_{t\in G}f(t)\alpha_t\!\left(g(t^{-1}s)\right).

For the trivial action on a general $A$ , the twisting disappears but the coefficients still multiply in $A$ , so the algebra need not be commutative.

Conventions and scope

The formulas depend on choosing a left rather than right Haar measure and on the convention for $\Delta$ . Equivalent conventions move inverses or modular factors between formulas. The displayed pair is one coherent standard convention; mixing it with a different involution formula can destroy the $*$ -algebra identities. For noncompact $G$ , $C_c(G,A)$ is not complete in the crossed-product norms.

References

Dana P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: §2.3, especially equations (2.16)–(2.19) for convolution, involution, and integrated forms.
J. M. G. Fell and R. S. Doran, Representations of $*$ -Algebras, Locally Compact Groups, and Banach $*$ -Algebraic Bundles, Volume 1, Academic Press, 1988. DOI record. Relevant: the convolution algebra of a group action and its representation theory.