# PlanetPhysics/Groupoid Representation Theorem

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### Groupoid representation theoremEdit

We shall briefly consider a main result due to Hahn (1978) that relates groupoid and associated groupoid algebra representations:

\begin{theorem} {\rm (source: ^{[1]}.)}~
Any representation of a groupoid **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
with Haar measure </math>(\nu,
\mu)\H 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 , 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))M_r (\alpha)j = \alpha \cdot j

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

### RemarksEdit

Furthermore, according to Seda (1986, on p.116) the continuity of a
Haar system is equivalent to the continuity of the
convolution product for any pair of continuous 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 locally compact groupoids or 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 induce *non-degenerate*
--representations 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 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 Hilbert spaces instead of
single Hilbert spaces. Therefore, groupoid representations appear
as the natural construct for Algebraic Quantum Field Theories (AQFT) in
which nets of local observable operators in Hilbert space fiber
bundles were introduced by Rovelli (1998).

## All SourcesEdit

^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}

## ReferencesEdit

- ↑ Cite error: Invalid
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tag; no text was provided for refs named`Hahn78, Hahn2`

- ↑
R. Gilmore:
*Lie Groups, Lie Algebras and Some of Their Applications.*, Dover Publs., Inc.: Mineola and New York, 2005. - ↑
P. Hahn: Haar measure for measure groupoids.,
*Trans. Amer. Math. Soc*.**242**: 1--33(1978). (Theorem 3.4 on p. 50). - ↑
P. Hahn: The regular representations of measure groupoids.,
*Trans. Amer. Math. Soc*.**242**:34--72(1978). - ↑
R. Heynman and S. Lifschitz. 1958.
*Lie Groups and Lie Algebras*., New York and London: Nelson Press. - ↑ C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008); arXiv:0709.4364v2 [quant--ph]