# PlanetPhysics/Groupoid C Convolution Algebra

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#### Background and Data for the Definition of a Groupoid --Convolution AlgebraEdit

Jean Renault introduced in ref. ^{[1]} the *--algebra of a \htmladdnormallink{locally compact groupoid* {http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
} as follows: the space of continuous functions with compact support on a groupoid **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
is made into a *-algebra whose multiplication is the *convolution*, and that is also endowed with the smallest --norm which makes its representations continuous, as shown in ref.^{[2]}. Furthermore, for this convolution to be defined, one needs also to have a Haar system
associated to the locally compact groupoids **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
that are then called *measured groupoids* because they are endowed with an associated Haar system which involves the concept of measure, as introduced in ref. ^{[3]} by P. Hahn.

With these concepts one can now sum up the definition (or construction) of the *groupoid -convolution algebra* , or groupoid -algebra, as follows.

agroupoid C*--convolution algebra, , is defined formeasured groupoids

as a **--algebra with "" being defined by convolution so that it has a smallest --norm which makes its representations continuous* .

One can also produce a functorial construction of that has additional interesting properties.

Next we recall a result due to P. Hahn ^{[4]} which shows how groupoid representations relate to
induced *-algebra representations and also how--under certain conditions-- the former can be derived from
the appropriate *-algebra representations.

\begin{theorem}
(source: ref. ^{[4]}). Any representation of a groupoid **Failed to parse (unknown function "\grp"): {\displaystyle (\grp,C)}**
with Haar measure in a separable Hilbert space induces a *-algebra representation 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:

(1) For any , one has that and
\med
(2) , where
\med
**Failed to parse (unknown function "\grp"): {\displaystyle M_r: L^\infty (U_{\grp}, \mu \longrightarrow L[L^2 (U_{\grp}, \mu, \mathbb{H}]}**
, with

.

*Conversely, any *- algebra representation with the above two properties induces a groupoid representation, X, as follows:*
\med
(viz. p. 50 of ref. ^{[4]}).
\end{theorem}

Furthermore, according to Seda (ref. \cite {Seda86,Seda2k8}), 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 "\G"): {\displaystyle C_c (\G)}**
of a groupoid **Failed to parse (unknown function "\G"): {\displaystyle \G}**
is closed with respect to convolution if and only if the fixed Haar system associated with the measured groupoid **Failed to parse (unknown function "\G"): {\displaystyle \G}**
is *continuous* (see ref. ^{[2]}).

Thus, in the case of groupoid algebras of transitive groupoids, it was shown in ^{[2]} that any representation of a measured groupoid **Failed to parse (unknown function "\G"): {\displaystyle (\G, [{\int} \nu ^u d \tilde{\lambda}(u)] = [\lambda])}**
on a separable Hilbert space induces a *non-degenerate* *-representation of the associated groupoid algebra
**Failed to parse (unknown function "\G"): {\displaystyle \Pi (\G, \nu,\tilde{\lambda})}**
with properties formally similar to (1) and (2) above.
Moreover, as in the case of groups, *there is a correspondence between the unitary representations of a groupoid and its associated C*-convolution algebra representations* (p. 182 of ^{[2]}), the latter involving however fiber bundles of Hilbert spaces instead of single Hilbert spaces.

## All SourcesEdit

^{[3]}^{[4]}^{[2]}^{[5]}^{[6]}^{[1]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]}

## ReferencesEdit

- ↑
^{1.0}^{1.1}J. Renault. A groupoid approach to C*-algebras,*Lecture Notes in Math*., 793, Springer, Berlin, (1980). - ↑
^{2.0}^{2.1}^{2.2}^{2.3}^{2.4}M. R. Buneci.*Groupoid Representations*, Ed. Mirton: Timishoara (2003). - ↑
^{3.0}^{3.1}P. Hahn: Haar measure for measure groupoids.,*Trans. Amer. Math. Soc*.**242**: 1--33(1978). - ↑
^{4.0}^{4.1}^{4.2}^{4.3}P. Hahn: The regular representations of measure groupoids.,*Trans. Amer. Math. Soc*.**242**:35--72(1978). Theorem 3.4 on p. 50. - ↑
M.R. Buneci. 2006.,
Groupoid C*-Algebras.,
*Surveys in Mathematics and its Applications*, Volume 1: 71--98. - ↑
M. R. Buneci. Isomorphic groupoid C*-algebras associated with
different Haar systems.,
*New York J. Math.*,**11**(2005):225--245. - ↑
J. Renault. 1997. The Fourier Algebra of a Measured Groupoid and Its Multipliers,
*Journal of Functional Analysis*,**145**, Number 2, April 1997, pp. 455--490. - ↑
A. K. Seda: Haar measures for groupoids, \emph{Proc. Roy. Irish Acad.
Sect. A}
**76**No. 5, 25--36 (1976). - ↑
A. K. Seda: Banach bundles of continuous functions and an integral
representation theorem,
*Trans. Amer. Math. Soc.***270**No.1 : 327-332(1982). - ↑
A. K. Seda: On the Continuity of Haar measures on topological groupoids,
*Proc. Amer Math. Soc.***96**: 115--120 (1986). - ↑
A. K. Seda. 2008.
*Personal communication*, and also Seda (1986, on p.116).