Skip to main content
MathOverflow

Return to Question

Notice removed Authoritative reference needed by CommunityBot
Bounty Ended with no winning answer by CommunityBot
Notice added Authoritative reference needed by S.Z.
Bounty Started worth 50 reputation by S.Z.
added 148 characters in body
Source Link
S.Z.
S.Z.
  • 505
  • 3
  • 12

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.

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) ?

Any reference would be greatly appreciated.

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.

added 124 characters in body
Source Link
S.Z.
S.Z.
  • 505
  • 3
  • 12

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 $$ \Psi_{f}(g):=g\cdot f, \quad g\in G. $$$$ \Psi_{f}(g):=g\cdot f, \quad g\in G, $$ Iswhere $\cdot$ 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) ?

Any reference would be greatly appreciated.

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 $$ \Psi_{f}(g):=g\cdot f, \quad g\in G. $$ Is the map $\Psi_{f}: G \rightarrow \mathcal{L}^{\infty}(H)$ Borel measurable?

Any reference would be greatly appreciated.

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) ?

Any reference would be greatly appreciated.

added 10 characters in body
Source Link
S.Z.
S.Z.
  • 505
  • 3
  • 12

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 $\mathcal{L}^{\infty}(H)$. GivenLet $f \in \mathcal{L}^{\infty}(H)$, be fixed and set $$ \Psi_{f}(g):=g\cdot f, \quad g\in G. $$ Is the map $\Psi_{f}: G \rightarrow \mathcal{L}^{\infty}(H)$ Borel measurable?

Any reference would be greatly appreciated.

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 $\mathcal{L}^{\infty}(H)$. Given $f \in \mathcal{L}^{\infty}(H)$, set $$ \Psi_{f}(g):=g\cdot f, \quad g\in G. $$ Is the map $\Psi_{f}: G \rightarrow \mathcal{L}^{\infty}(H)$ Borel measurable?

Any reference would be greatly appreciated.

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 $$ \Psi_{f}(g):=g\cdot f, \quad g\in G. $$ Is the map $\Psi_{f}: G \rightarrow \mathcal{L}^{\infty}(H)$ Borel measurable?

Any reference would be greatly appreciated.

Source Link
S.Z.
S.Z.
  • 505
  • 3
  • 12