PlanetPhysics/Groupoid C Convolution Algebra

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

Background and Data for the Definition of a Groupoid $C^{*}$--Convolution Algebra

Jean Renault introduced in ref. the $C^{*}$--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 $C^{*}$--norm which makes its representations continuous, as shown in ref.. 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. by P. Hahn.

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

a groupoid C*--convolution algebra , $G_{CA}$, is defined for measured groupoids

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

One can also produce a functorial construction of $G_{CA}$ that has additional interesting properties.

Next we recall a result due to P. Hahn 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. ). Any representation of a groupoid Failed to parse (unknown function "\grp"): {\displaystyle (\grp,C)} with Haar measure $(\nu ,\mu )$ in a separable Hilbert space $\mathbb {H}$ induces a *-algebra representation $f\mapsto X_{f}$ 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:

(1) For any $l,m\in \mathbb {H}$ , one has that $\left|<X_{f}(u\mapsto l),(u\mapsto m)>\right|\leq \left\|f_{l}\right\|\left\|l\right\|\left\|m\right\|$ and \med (2) $M_{r}(\alpha )X_{f}=X_{f\alpha \circ r}$, 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

$M_{r}(\alpha )j=\alpha \cdot j$.

Conversely, any *- algebra representation with the above two properties induces a groupoid representation, X, as follows: \med $<X_{f},j,k>~=~{\int }f(x)[X(x)j(d(x)),k(r(x))d\nu (x)].$ (viz. p. 50 of ref. ). \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 $f*g$ for any pair $f$, $g$ 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. ).

Thus, in the case of groupoid algebras of transitive groupoids, it was shown in 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 $\mathbb {H}$ induces a non-degenerate *-representation $f\mapsto X_{f}$ 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 ), the latter involving however fiber bundles of Hilbert spaces instead of single Hilbert spaces.