PlanetPhysics/Groupoid Representation Theorem

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Groupoid representation theorem

We shall briefly consider a main result due to Hahn (1978) that relates [[../QuantumOperatorAlgebra5/|groupoid]] and associated groupoid algebra representations:

\begin{theorem} {\rm (source: .)}~ Any [[../CategoricalGroupRepresentation/|representation]] of a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} with [[../HigherDimensionalQuantumAlgebroid/|Haar measure]] </math>(\nu, \mu)$inaseparableHilbertspace$\H$inducesa*-algebrarepresentation<math>f\mapsto X_f$ of the associated groupoid algebra Failed to parse (unknown function "\grp"): {\displaystyle \Pi \grp, \nu)} in Failed to parse (unknown function "\grp"): {\displaystyle L^2 (U_{\grp} , \mu, \mathbb{H} )} with the following properties:

 \item[(1)] For any $l,m\in H$ , one has that </math>\left|<X_f(u \mapsto l), (u \mapsto m)>\right|\leq \left\|f_l\right\| \left\|l \right\| \left\|m \right\|Failed to parse (unknown function "\item"): {\displaystyle  and  \item[(2)] <math>M_r (\alpha) X_f = X_{f \alpha \circ r}}
, where </math>M_r: L^\infty (U_{\grp}, \mu, \H) \longrightarrow \mathcal L ( L^2 (U_{\grp}, \mu, \H))$,with$M_r (\alpha)j = \alpha \cdot j$.Conversely,any*-algebrarepresentationwiththeabovetwopropertiesinducesa[[../GroupRepresentations/|groupoidrepresentation]],X,asfollows:<center><math>⟨X_f,j,k⟩=\int f(x)[X(x)j(d(x)),k(r(x))d\nu (x)],$

(cf. p. 50 of Hahn, 1978). \end{theorem}

Remarks

Furthermore, according to Seda (1986, on p.116) the continuity of a [[../QuantumOperatorAlgebra5/|Haar system]] is equivalent to the continuity of the convolution product $f*g$ for any pair $f,g$ of continuous [[../Bijective/|functions]] with compact support. One may thus conjecture that similar results could be obtained for functions with locally compact support in dealing with convolution products of either [[../LocallyCompactGroupoid/|locally compact groupoids]] or [[../WeakHopfAlgebra/|quantum groupoids]]. Seda's result also implies that the convolution algebra Failed to parse (unknown function "\grp"): {\displaystyle C_{conv}(\grp)} of a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} is closed with respect to the convolution {*} if and only if the fixed Haar system associated with the measured groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} is continuous (Buneci, 2003).

In the case of groupoid algebras of transitive groupoids, Buneci (2003) showed that representations of a measured groupoid Failed to parse (unknown function "\grp"): {\displaystyle ({\grp, [\int \nu ^u d \tilde{ \lambda}(u)]=[\lambda]})} on a separable Hilbert space $ℍ$ induce non-degenerate $*$--representations $f\mapsto X_f$ of the associated groupoid algebra Failed to parse (unknown function "\grp"): {\displaystyle \Pi (\grp, \nu,\tilde{\lambda})} with properties formally similar to (1) and (2) above. Moreover, as in the case of [[../TrivialGroupoid/|groups]], \textit{there is a correspondence between the unitary representations of a groupoid and its associated C*--convolution algebra representations} (p.182 of Buneci, 2003), the latter involving however fiber bundles of [[../NormInducedByInnerProduct/|Hilbert spaces]] instead of single Hilbert spaces. Therefore, groupoid representations appear as the natural construct for [[../CosmologicalConstant/|Algebraic Quantum Field Theories]] ([[../MetaTheorems/|AQFT]]) in which nets of local [[../QuantumSpinNetworkFunctor2/|observable]] [[../QuantumOperatorAlgebra4/|operators]] in Hilbert space fiber bundles were introduced by Rovelli (1998).