The integrated form of a unitary representation

Assume the Axiom of Choice. Let $G$ be an LCH group with a fixed left Haar measure, let $(\pi ,H)$ be a strongly continuous unitary representation of $G$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Hilbert space), and let $f\in L^1(G)$ with its norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$ (Complex Haar L^p spaces and compactly supported functions). The integrated form of $\pi$ at $f$ is the operator $\pi (f)\in \mathcal{B}(H)$ (A bounded linear operator between normed spaces) characterised by the weak integral identity $⟨\pi (f)\xi ,\eta ⟩={\int }_{G}f(g)⟨\pi (g)\xi ,\eta ⟩dg(\xi ,\eta \in H)\mathrm{.}$ The integral is a Haar integral over the fixed measure, and the right-hand side is the pairing convention of the Hilbert space, linear in the first argument and conjugate-linear in the second.

• Well-definedness (existence). Fix $\xi \in H$. Since $\pi$ is unitary, $\mathrm{\mid }⟨\pi (g)\xi ,\eta ⟩\mathrm{\mid }\le \mathrm{\parallel }\xi \mathrm{\parallel }\mathrm{\parallel }\eta \mathrm{\parallel }$ for all $g\in G$ and $\eta \in H$, so $g\mapsto f(g)⟨\pi (g)\xi ,\eta ⟩$ is measurable with $\mathrm{\mid }f(g)⟨\pi (g)\xi ,\eta ⟩\mathrm{\mid }\le \mathrm{\mid }f(g)\mathrm{\mid }\mathrm{\parallel }\xi \mathrm{\parallel }\mathrm{\parallel }\eta \mathrm{\parallel }$; this majorant lies in $L^1(G)$ when $\xi$ and $\eta$ are fixed. The assignment $\eta \mapsto {\int }_{G}f(g)⟨\pi (g)\xi ,\eta ⟩dg$ is therefore a well-defined conjugate-linear functional, bounded by $\mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }\xi \mathrm{\parallel }\mathrm{\parallel }\eta \mathrm{\parallel }$; by the Hilbert Riesz representation theorem (Riesz representation for Hilbert spaces) there is a unique vector, written $\pi (f)\xi$, with $⟨\pi (f)\xi ,\eta ⟩={\int }_{G}f(g)⟨\pi (g)\xi ,\eta ⟩dg$ for every $\eta \in H$, and $\mathrm{\parallel }\pi (f)\xi \mathrm{\parallel }\le \mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }\xi \mathrm{\parallel }$.
• Linearity. For scalars $a,b$ and $\xi ,{\xi }^{\mathrm{\prime }}\in H$ the defining functionals satisfy the identity for $a\xi +b{\xi }^{\mathrm{\prime }}$ by linearity of the integral and of the inner product in the first argument, so $\pi (f)(a\xi +b{\xi }^{\mathrm{\prime }})=a\pi (f)\xi +b\pi (f){\xi }^{\mathrm{\prime }}$; thus $\xi \mapsto \pi (f)\xi$ is a linear map $H\to H$ with $\mathrm{\parallel }\pi (f)\mathrm{\parallel }\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$. Likewise, for scalars $a,b$ and $f,h\in L^1(G)$ the integral identity gives $\pi (af+bh)=a\pi (f)+b\pi (h)$, because both sides have the same pairing with every $\eta \in H$.
• No further hypotheses. The construction applies to every strongly continuous unitary representation of every LCH group: no irreducibility, separability, unimodularity or compactness is assumed. The modular function enters this page only through the involution of $L^1(G)$ (Involution on L1 of a locally compact group), not through the definition of $\pi (f)$.
• Use of choice. The Axiom of Choice is declared as a standing hypothesis of the completion chain of this page. In this definition it is needed only through the Hilbert Riesz representation step, which is established under Countable Choice (Riesz representation for Hilbert spaces) and therefore under AC (The Axiom of Choice).