groupoid C*-convolution algebras

0.1 Introduction: Background and definition of the groupoid C*–convolution algebra

Jean Renault introduced in ref. [6] the $C^{\mathrm{*}}$–algebra of a locally compact groupoid $\mathrm{G}$ as follows: the space of continuous functions with compact support on a groupoid $𝖦$ 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.[3]. Furthermore, for this convolution to be defined, one needs also to have a Haar system (http://planetmath.org/GroupoidRepresentationsInducedByMeasure)associated to the locally compact groupoids (http://planetmath.org/LocallyCompactGroupoids) $𝖦$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. [1] by P. Hahn.

With these concepts one can now sum up the definition (or construction) of the groupoid $C^{\mathrm{*}}$-convolution algebra, or http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdfgroupoid $C^{*}$-algebra, as follows.

Definition 0.1.

a groupoid C*–convolution algebra, $G_{CA}$, is defined for measured groupoidsas a *–algebra with “$\mathrm{*}$” being defined by convolution so that it has a smallest $C^{\mathrm{*}}$–norm which makes its representations continuous.

Remark 0.1.

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

Next we recall a result due to P. Hahn [2] which shows how groupoid representations relate toinduced *-algebra representations and also how–under certain conditions– the former can be derived fromthe appropriate *-algebra representations.

Theorem 0.1.

(source: ref. [2]). Any representation of a groupoid $\mathrm{(}\mathrm{G}\mathrm{,}C\mathrm{)}$ with Haar measure $\mathrm{(}\nu \mathrm{,}\mu \mathrm{)}$ in a separable Hilbert space $\mathrm{H}$ induces a *-algebra representation $f\mathrm{\mapsto }{X}_f$ of the associatedgroupoid algebra $\mathrm{\Pi }\mathit{}\mathrm{(}\mathrm{G}\mathrm{,}\nu \mathrm{)}$ in $L^{\mathrm{2}}\mathit{}\mathrm{(}{U}_{\mathrm{G}}\mathrm{,}\mu \mathrm{,}\mathrm{H}\mathrm{)}$ with the following properties:

(1) For any $l\mathrm{,}m\mathrm{\in }\mathrm{H}$ , one has that and

(2) $M_r\mathit{}\mathrm{(}\alpha \mathrm{)}\mathit{}{X}_f\mathrm{=}{X}_{f\mathit{}\alpha \mathrm{\circ }r}$, where

$M_r:L^{\mathrm{\infty }}(U_{𝖦},\mu ⟶L[L^2(U_{𝖦},\mu ,\mathscr{H}]$, with

$M_r(\alpha )j=\alpha \cdot j$.

Conversely, any *- algebra representation with the above two properties induces a groupoid representation, X, as follows:

(viz. p. 50 of ref. [2]).

Furthermore, according to Seda (ref. [10, 11]), 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 $C_c(𝒢)$ of a groupoid $𝒢$ is closed with respect to convolution if and only if the fixed Haar system associated with the measured groupoid $𝒢$ is continuous (see ref. [3]).

Thus, in the case of groupoid algebras of transitive groupoids, it was shown in [3] that any representation of a measured groupoid $(𝒢,[{\displaystyle \int {\nu }^u𝑑\tilde{\lambda }(u)}]=[\lambda ])$ on a separable Hilbert space $\mathscr{H}$ induces a non-degenerate *-representation $f\mapsto X_f$ of the associated groupoid algebra$\mathrm{\Pi }(𝒢,\nu ,\tilde{\lambda })$ 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 [3]), the latter involving however fiber bundles of Hilbert spaces instead of single Hilbert spaces.