In what follows, a ‘$ \* $-representation’ always means a *non-degenerate $ \* $-representation*.




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Let $ (\mathscr{A},G,\alpha) $ be a $ C^{\*} $-dynamical system, and let $ \pi: \mathscr{A} \to B(\mathcal{H}) $ be a faithful $ \* $-representation of $ \mathscr{A} $ on a Hilbert space $ \mathcal{H} $. Define an associated faithful $ \* $-representation $ \tilde{\pi}: \mathscr{A} \to B({L^{2}}(G,\mathcal{H})) $ and a unitary group representation $ \lambda: G \to U({L^{2}}(G,\mathcal{H})) $ as follows.



> 
> For all $ a \in \mathscr{A} $, $ g \in G $, $ x \in G $ and $ \xi \in {L^{2}}(G,\mathcal{H}) $,
>  \begin{align\*}
> [[\tilde{\pi}(a)](\xi)](x) & \stackrel{\text{def}}{=}
> [\pi({\alpha\_{x^{-1}}}(a))](\xi(x)), \\
> [[\lambda(g)](\xi)](x) & \stackrel{\text{def}}{=} \xi(g^{-1} x).
> \end{align\*}
> 
> 
> 


$ \lambda $ is usually called the *left regular representation of $ G $*.


It is well-known that the pair $ (\tilde{\pi},\lambda) $ — called the *left regular representation induced by $ \pi $* — is a covariant representation of $ (\mathscr{A},G,\alpha) $ on $ {L^{2}}(G,\mathcal{H}) $ and that the integrated form $ \tilde{\pi} \rtimes \lambda $ is a faithful $ \* $-representation of the twisted convolution $ \* $-algebra $ ({C\_{c}}(G,\mathscr{A}),\star,^{\*}) $ on $ {L^{2}}(G,\mathcal{H}) $.


It is also well-known that $ \tilde{\pi} \rtimes \lambda $ extends uniquely to a faithful $ \* $-representation of the reduced crossed product $ C^{\*} $-algebra $ \mathscr{A} \rtimes\_{\alpha,\text{r}} G $ on $ {L^{2}}(G,\mathcal{H}) $.



> 
> **My question:** Suppose that we have a faithful $ \* $-representation $ \rho: \mathscr{A} \to B({L^{2}}(G,\mathcal{H})) $ that is not associated to $ \pi $, i.e. is not of the form $ \tilde{\pi} $, for any faithful $ \* $-representation $ \pi: \mathscr{A} \to B(\mathcal{H}) $. Suppose further that the pair $ (\rho,\lambda) $ — which I shall call a *non-induced left regular representation* — is a covariant representation of $ (\mathscr{A},G,\alpha) $ on $ {L^{2}}(G,\mathcal{H}) $. Then is it necessarily true that the integrated form $ \rho \rtimes \lambda $ gives rise to a faithful $ \* $-representation of $ \mathscr{A} \rtimes\_{\alpha,\text{r}} G $ on $ {L^{2}}(G,\mathcal{H}) $?
> 
> 
> 


Thank you very much for your help!



‘Non-Induced’ Left Regular Representations of $ C^{*} $-Dynamical Systems
The answer is no in two ways. Let $G$ be a discrete group and consider the trivial action of $G$ on the reduced group $\mathrm{C}^\*$-algebra $\mathrm{C}^\*\_{\mathrm{r}}G$. The pair of the right regular representation $\rho\colon \mathrm{C}^\*\_{\mathrm{r}}G\to B(\ell\_2G)$ and the left regular representation $\lambda\colon G\to U(\ell\_2G)$ is covariant. The resulting $\*$-homomorphism $\rho\rtimes\lambda$ is continuous on the reduced crossed product (which is nothing but the minimal tensor product $\mathrm{C}^\*\_{\rho}G \otimes\_{\min} \mathrm{C}^\*\_{\lambda}G$) if and only if $G$ is amenable. It is faithful on the algebraic tensor product $\mathrm{C}^\*\_{\rho}G \otimes\_{\mathrm{alg}} \mathrm{C}^\*\_{\lambda}G$ if and only if $G$ is ICC.


