On measurability of certain group actions on spaces of bounded measurable functions

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.

• How is Ψ_f(g) an element of L^∞(H)? The definition you wrote down seems to make it an element of Aut(L^∞(H)). Mar 12, 2021 at 19:45
• Each fixed $f$ gives a map from $G$ to $\mathcal{L}^{\infty}(H)$.
– S.Z.
Mar 12, 2021 at 20:06
• What σ-algebra of measurable sets is being used for L^∞(H)? Mar 13, 2021 at 0:10
• So to be more explicit, let $\varphi : G \to U(H)$ be the representation. Then $\Psi_f(g) = f(\varphi(g) \cdot)$, i.e. the function $h \mapsto f(\varphi(g) h)$. Is that right? Mar 13, 2021 at 0:37
• Dimitri Pavlov, I edited the question to clarify. Nade Eldredge, yes, that's correct.
– S.Z.
Mar 13, 2021 at 16:05