Smallest norms on crossed product $C^*$-algebras

Question

Let $A$ be a commutative $C^*$-algebra with a discrete group $G$ acting on it. The reduced crossed product is the completion of the algebraic crossed product $C_c(G,A)$ in the reduced norm $\Vert \cdot\Vert_r$. Is there a smaller $C^*$-norm $\Vert \cdot\Vert_s$ on the algebraic crossed product, meaning that $$\Vert f\Vert_s\le \Vert f\Vert_r$$ for every $f\in C_c(G,A)$ and the two norms are in fact different?

Answer 1

The answer is yes. Take $A=\mathbb{C}$ and $G=\mathbb{Z}$. Then $C^*(\mathbb{Z})\cong C(\mathbb{T})$ canonically, where $\mathbb{T}$ is the unit circle. The reduced norm corresponds to the canonical sup-norm of functions. Under this identification, the $*$-algebra $C_c(\mathbb{Z},\mathbb{C})$ corrsponds to the polynomial functions in $C(\mathbb{T})$. Let $\mathbb{T}_+$ be the set of those $z\in\mathbb{T}$ with nonnegative real part, i.e., the closed upper half-circle. One may consider $$ \|\cdot\|_0: C^*(\mathbb{Z})\cong C(\mathbb{T}) \to [0,\infty),\quad \|f\|_0=\|f|_{\mathbb{T}_+}\|_\infty. $$ On $C^*(\mathbb{Z})$, this (semi-)norm is definitely smaller than the usual norm. But since any polynomial function is determined by its restriction on some arbitrary open set, it follows that this semi-norm defines a norm on $C_c(\mathbb{Z},\mathbb{C})$ that is smaller than and not equal to the reduced C*-cross norm.

Answer 2

One important feature of the crossed product (be it the reduced or the full one) is that there is a bounded linear mapping $$E:A\rtimes G \to A$$ such that $E(f) = f(1)$, for every $f$ in $C_c(G,A)$. If you require your norm to lead to a completion of $C_c(G,A)$ sharing the above property, than it must be bigger than the reduced norm. This follows easily from Proposition 3.7 in "Amenability for Fell bundles", J. reine angew. Math., 492 (1997), 41-73 (arXiv:funct-an/9604009).

Answer 3

The completion with respect to the minimal and maximal norm for the trivial action give reduced and unreduced C* algebra of groups. They are equal for amenable groups, but usually different. Here is an example of the full C* algebra of the free group in two generators: http://msp.org/pjm/1980/87-1/pjm-v87-n1-p04-s.pdf