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Let $G$ be a locally compact group and $A$ be a non-unital $C^*$-algebra. )$(A,G,\alpha)$ is a $C^*$-dynamical sysytem. The space of all continous functions from $G$ to $A$ with compact support is denoted by $C_c(G, A)$. Suppose $f,g\in C_c(G,A)$. The product of $f$ and $g$ is defined as following:

$fg(t)=\int f(r)\alpha_r(g(r^{-1}t))dr, \forall t\in G$.

I wonder whether there exist some propositions to determine when the center of $C_c(G,A)$ is 0?

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  • $\begingroup$ Functions of the form $t\to f(t)x$ where $x\in A$ and $f\in C_c(G)$ suggest that $Z(C_c(G,A) = \{0\}$ iff $Z(A)=\{0\}$. $\endgroup$
    – Onur Oktay
    Commented Feb 26, 2022 at 14:06
  • $\begingroup$ But not every $f\in C_c(G,A)$ has the form $t\mapsto f(t)x$. $\endgroup$
    – math112358
    Commented Feb 26, 2022 at 15:34
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    $\begingroup$ Could you clarify what algebra product is placed on $C_c(G,A)$. The pointwise product would not seem to use the structure that $G$ is a group in any way... $\endgroup$
    – Matthew Daws
    Commented Feb 26, 2022 at 16:35
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    $\begingroup$ In the case of $\mathbb{Z}$ with the counting measure, since $C_c(\mathbb{Z},A)$ is spanned by maps with singleton support, you can check that $f \in C_c(\mathbb{Z},A)$ is central with respect to convolution iff $f(n) \cdot \alpha_n(a) = a \cdot \alpha_n(f(n))$ for every $n \in \mathbb{Z}$ and every $a \in A$. I don’t know how much more you can extract without additional information… $\endgroup$
    – Branimir Ćaćić
    Commented Feb 27, 2022 at 17:17
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    $\begingroup$ I think in its current form the question is far too broad and almost asking for people to provide both the hypotheses of a theorem as well as its proof. Consider $G$ discrete ICC and $A$ to be abelian and non-unital, for instance $\endgroup$
    – Yemon Choi
    Commented Feb 27, 2022 at 17:26

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First, $Z(C_c(G,A)) = C_c(G,Z(A))$. Indeed, if there exists an $f\in C_c(G,A)$ and $t\in G$ such that $f(t)\notin Z(A)$, then $f\notin Z(C_c(G,A))$. Thus, $Z(C_c(G,A))\subseteq C_c(G,Z(A))$. $(\supseteq)$ is relatively easier to show.

Second, one can use the functions of the form $t\to \phi(t)x$, where $\phi\in C_c(G)$ and $x\in A$, to show $C_c(G,Z(A)) = \{0\}$ iff $Z(A)=\{0\}$.

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  • $\begingroup$ But if $(A,G,\alpha)$ is a $C^*$-dynamical system, the multiplication of $f,g\in C_c(G,A)$ in terms of the integral. $\endgroup$
    – math112358
    Commented Feb 27, 2022 at 0:41
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    $\begingroup$ @math112358 Please edit the question and write down explicitly which product you consider on $C_c(G,A)$ to remove the ambiguity. $\endgroup$
    – Onur Oktay
    Commented Feb 27, 2022 at 7:46
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    $\begingroup$ I have edited the question. $\endgroup$
    – math112358
    Commented Feb 27, 2022 at 8:17

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