Let $\mathcal{H}$ be a separable Hilbert space equipped with a strongly continuous unitary representation of a locally compact group $G$. Denote by $\mathcal{L}^{\infty}(H)$ the space of the bounded Borel measurable real valued functions on $\mathcal{H}$ (with respect to the Borel $\sigma$-algebra of the norm topology) equipped with the sup norm.

Then the representation of $G$ on $\mathcal{H}$ induces, via composition, a representation of $G$ on $\mathcal{L}^{\infty}(H)$. Let $f \in \mathcal{L}^{\infty}(H)$ be fixed and set $$ \Psi_{f}(g):=g\cdot f, \quad g\in G, $$ where $\cdot$ stands for the latter representation. Is the map $\Psi_{f}: G \rightarrow \mathcal{L}^{\infty}(H)$ Borel measurable (with respect to the Borel $\sigma$-algebra of the sup norm topology) ?

If not, is the function $\ell\circ \Psi_{f}$, where $\ell$ is any (fixed) continuous linear functional on $\mathcal{L}^{\infty}(H)$, measurable?

Any reference would be greatly appreciated.

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