I have a question about a definition from the book of Gert Pedersen, $C^*$-algebras and their automorphism groups.
In the book setup, we have a locally compact group $G$ and a $C^*$-algebra $A$ and we have a range of maps $f \colon G \to A^*$, where $A^*$ denotes the continuous dual of $A^*$.
He then argues that a certain net of maps as above converges in 'the topology of weak$^*$ convergence uniformly on compacta of $G$'.
I am a bit confused regarding the exact definition here, does he mean for all fixed $a \in A$, the map $t \mapsto f(t)(a)$ is uniformly convergent when restricted to compacts of $G$? He does not define the topology nor does he give a neighborhood basis, but it seems that this is what he shows.
Regards,
Based on your description it looks like Pedersen is placing a topology on the collection of such maps $f:G\to A$ such that convergence is characterized as follows: We say a net $(f_\gamma)_{\gamma\in\Gamma}$ converges to $f$ if for all $\varepsilon>0$, all compact sets $K\subset G$, and all $a\in A$, there exists a $\gamma\in\Gamma$ such that if $\gamma\leq\gamma'$ then $|(f_{\gamma'}g)(a)-(fg)(a)|<\varepsilon$ for all $g\in K$.
Thus, this topology would be generated by the seminorms $\rho_{K,a}$, where $K\subset G$ is compact, $a\in A$, and $$\rho_{K,a}(f)=\sup_{g\in K}|(fg)(a)|.$$