My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure.

Let $G$ be a locally compact group and fix a Haar measure $\mu$. Let $A$ be a $C^*$-algebra and $f \colon G \to A$ be a function. What does it mean for $f$ to be measurable, or integrable and if so how does one define \begin{equation*} \int_G f \, d\mu \, ? \end{equation*}

My guess is that we put the Borel algebra on $A$ and then define the integral using step functions and then simple functions and then using limits like the usual way of defining Lebesgue integration. But is there cleverer way of doing this for example using Gelfand--Naimark Theorem and Riesz representation theorem or something along those lines?

An example I have in mind is the convolution product on $C_c(G,A)$. That is suppose $\alpha \colon G \to \mbox{Aut}(A)$ is a homomorphism such that the map $s \mapsto \alpha_s(a)$ is norm continuous. Then we can define a convolution on $C_c(G,A)$, the space of continuous functions with compact support from $G$ to $A$, by \begin{equation*} f \ast g(s) = \int_G f(t) \alpha_t(g(t^{-1}s)) \, d \mu(t) \end{equation*} This integral presumably takes values in $A$.


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