Let $ (A,G,\alpha) $ be a $ C^{\*} $-dynamical system, i.e., $ A $ is a $ C^{\*} $-algebra, $ G $ is a locally compact Hausdorff group and $ \alpha $ is a strongly continuous action of $ G $ on $ A $ by $ \* $-automorphisms. Equip $ {C\_{c}}(G,A) $, the linear space of continuous $ A $-valued functions on $ G $ with compact support, with an associative multiplication $ \star\_{\alpha} $ and an involution $ ^{\*\_{\alpha}} $ by
\begin{align}
\forall f,g \in {C\_{c}}(G,A), ~ \forall x \in G: \quad
(f \star\_{\alpha} g)(x) & \stackrel{\text{df}}{=}
\int\_{G} f(y) ~ {\alpha\_{y}}(g(y^{-1} x)) ~ \mathrm{d}{y}, \\
{f^{\*\_{\alpha}}}(x) & \stackrel{\text{df}}{=}
{\alpha\_{x}}(f(x^{-1})^{\*}) \cdot \Delta(x^{-1}),
\end{align}
where $ \Delta $ denotes the modular function of $ G $.


Let $ \pi $ be a faithful $ \* $-representation of $ A $ on the Hilbert space $ \mathcal{H} $. From this, fashion a $ \* $-representation $ \tilde{\pi} $ of $ A $ on $ {L^{2}}(G,\mathcal{H}) \cong {L^{2}}(G) \otimes \mathcal{H} $ by
$$
\forall a \in A, ~ \forall \xi \in {L^{2}}(G,\mathcal{H}), ~ \forall x \in G: \quad
[[\tilde{\pi}(a)](\xi)](x) \stackrel{\text{df}}{=} [\pi({\alpha\_{x^{-1}}}(a))](\xi(x)).
$$
Define also a unitary representation $ \lambda $ of $ G $ on $ {L^{2}}(G,\mathcal{H}) $ by
$$
\forall x,y \in G, ~ \forall \xi \in {L^{2}}(G,\mathcal{H}): \quad
(\lambda\_{x} \xi)(y) \stackrel{\text{df}}{=} \xi(x^{-1} y).
$$
Then the integrated form $ \tilde{\pi} \rtimes\_{\alpha} \lambda $ defines a $ \* $-representation of $ ({C\_{c}}(G,A),\star\_{\alpha},^{\*\_{\alpha}}) $ on $ {L^{2}}(G,\mathcal{H}) $:
$$
\forall f \in {C\_{c}}(G,A), ~ \forall \xi \in {L^{2}}(G,\mathcal{H}), ~ \forall x \in G: \\
 [[(\tilde{\pi} \rtimes\_{\alpha} \lambda)(f)](\xi)](x) \stackrel{\text{df}}{=}
 \int\_{G} [[\tilde{\pi}(f(y))](\lambda\_{y} \xi)](x) ~ \mathrm{d}{y}
= \int\_{G} [\pi({\alpha\_{x^{-1}}}(f(y)))](\xi(y^{-1} x)) ~ \mathrm{d}{y}.
$$
Finally, the $ C^{\*} $-algebraic reduced crossed product $ A \rtimes\_{\alpha,\text{r}} G $ is taken to be the completion of $ {C\_{c}}(G,A) $ under the $ C^{\*} $-norm $ \| \cdot \|\_{\*} $ defined by
$$
\forall f \in {C\_{c}}(G,A): \quad
\| f \|\_{\*} \stackrel{\text{df}}{=}
\| (\tilde{\pi} \rtimes\_{\alpha} \lambda)(f) \|\_{\mathscr{B}({L^{2}}(G,\mathcal{H}))}.
$$




---


I have also seen an unconventional definition of $ A \rtimes\_{\alpha,\text{r}} G $. For each $ f \in {C\_{c}}(G,A) $, define a function $ f^{\Delta} \in {C\_{c}}(G,A) $ by $ f^{\Delta} \stackrel{\text{df}}{=} f \sqrt{\Delta} $. Define a $ \* $-representation $ \rho $ of $ ({C\_{c}}(G,A),\star\_{\alpha},^{\*\_{\alpha}}) $ on $ {L^{2}}(G,\mathcal{H}) $ by
$$
\forall f \in {C\_{c}}(G,A), ~ \forall \xi \in {L^{2}}(G,\mathcal{H}), ~ \forall x \in G: \\
[[\rho(f)](\xi)](x) \stackrel{\text{df}}{=}
\int\_{G} [\pi({\alpha\_{x}}({f^{\Delta}}(x^{-1} y)))](\xi(y)) ~ \mathrm{d}{y}.
$$
Then the $ C^{\*} $-algebraic reduced crossed product $ A \rtimes\_{\alpha,\text{r}} G $ is taken to be the completion of $ {C\_{c}}(G,A) $ under the $ C^{\*} $-norm $ \| \cdot \|\_{\*\*} $ defined by
$$
\forall f \in {C\_{c}}(G,A): \quad
\| f \|\_{\*\*} \stackrel{\text{df}}{=}
\| \rho(f) \|\_{\mathscr{B}({L^{2}}(G,\mathcal{H}))}.
$$



> 
> **Question.** What is the exact equation that relates $ \tilde{\pi} \rtimes\_{\alpha} \lambda $ to $ \rho $?
> 
> 
> 


Thanks for your help!



An unconventional definition of the $ C^{*} $-algebraic reduced crossed product
It seems that I have answered my own question. For the benefit of anyone who might have an interest in this sort of thing, I have decided to post my answer.


My idea is to find a unitary mapping
$$
U: {L^{2}}(G,\mathcal{H}) \to {L^{2}}(G,\mathcal{H})
$$
that intertwines $ (\tilde{\pi} \rtimes\_{\alpha} \lambda)(f) $ and $ \rho(f) $, i.e., $ U $ satisfies the following commutative diagram:
\begin{equation}
\require{AMScd}
\begin{CD}
{L^{2}}(G,\mathcal{H}) @>{(\tilde{\pi} \rtimes\_{\alpha} \lambda)(f)}>>
{L^{2}}(G,\mathcal{H}) \\
@V{U}VV @VV{U}V \\
{L^{2}}(G,\mathcal{H}) @>>{\rho(f)}> {L^{2}}(G,\mathcal{H}).
\end{CD}
\end{equation}
Naïvely, one can try to define $ U $ by
$$
\forall \xi \in {L^{2}}(G,\mathcal{H}), ~ \forall x \in G: \quad
(U \xi)(x) \stackrel{\text{df}}{=} \xi(x^{-1}),
$$
but this is incorrect because then $ U $ is not isometric in the case that $ G $ is not unimodular. Therefore, one has to modify this flawed definition using the modular function $ \Delta $ of $ G $ so that $ U $ is indeed isometric. The theory of integration on locally compact Hausdorff groups then yields the following correct definition of $ U $:
$$
\forall \xi \in {L^{2}}(G,\mathcal{H}), ~ \forall x \in G: \quad
(U \xi)(x) \stackrel{\text{df}}{=} \sqrt{\Delta(x^{-1})} \cdot \xi(x^{-1}).
$$
Straightforward computations show that $ U = U^{-1} = U^{\*} $. Hence,
$$
\rho(f) = U \circ (\tilde{\pi} \rtimes\_{\alpha} \lambda)(f) \circ U,
$$
i.e., $ \rho(f) $ and $ (\tilde{\pi} \rtimes\_{\alpha} \lambda)(f) $ are unitarily equivalent.


