I've been reading Folland's Abstract Harmonic Analysis, and I am currently in the section on the correspondence between unitary representations of a locally compact group $G$ and $*$-representations of $L^1(G, \lambda_G)$, where $\lambda_G$ is the left Haar measure.
If $\pi$ is a unitary representation $G\to U(\mathscr{H})$ on a Hilbert space $\mathscr{H}$, one can construct a $*$-representation $\tilde{\pi}: L^1(G) \to \mathscr{L(H)}$ into the collection of bounded linear operators on $\mathscr{H}$. My question:
Can this correspondence be strengthened into an adjunction of functors? If so, what are the correct categories and functors?
Here are my thoughts. There seems to be a nice functor from some subcategory of topological groups to the category of Banach $*$-algebras, sending a locally compact Hausdorff group $G$ to $L^1(G)$. If one has a nice enough group homomorphism $\phi: G\to H$ ($H$ also a locally compact group), we can define a map $\phi^*: L^1(H) \to L^1(G)$ defined by $\phi^*(f) = f\circ \phi$. This leads to a more specific question:
What kind of conditions on $\phi$ should be placed to make this into a functor?
If $f \in L^1(H,\lambda_H)$, I would like $f\circ \phi \in L^1(G)$. For a characteristic function $f=\chi_A$, $A\subset H$ Borel measurable, $$ \int_G |\chi_A \circ \phi| \,d\lambda_G = \int_G \chi_{\phi^{-1}(A)} \,d\lambda_G = \lambda_G(\phi^{-1}(A)). $$ Therefore, it seems sufficient to require that the homomorphism $\phi$ satisfy a relation of the form $$\lambda_G(\phi^{-1}(A))\leq C \lambda_H(A)\qquad (*)$$ for some constant $C$. This would ensure that $$ \begin{align} \int_G |f\circ \phi| \,d\lambda_G \leq C \int_H |f|\,d\lambda_H \qquad \end{align} $$ by approximating $f$ by simple functions. Hence, $\phi^*$ would be a bounded linear map between $L^1(H)$ and $L^1(G)$. Is the property $(*)$ a well-known property? Any kind of references would be helpful.