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^{} $\U0001d5a6$ 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) $\U0001d5a6$ 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 groupoids as 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 to induced *-algebra representations and also how–under certain conditions– the former can be derived from the 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 associated groupoid 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 $$ and
(2) ${M}_{r}\mathit{}\mathrm{(}\alpha \mathrm{)}\mathit{}{X}_{f}\mathrm{=}{X}_{f\mathit{}\alpha \mathrm{\circ}r}$, where
${M}_{r}:{L}^{\mathrm{\infty}}({U}_{\U0001d5a6},\mu \u27f6L[{L}^{2}({U}_{\U0001d5a6},\mu ,\mathscr{H}]$, with
${M}_{r}(\alpha )j=\alpha \cdot j$.
Conversely, any *- algebra representation with the above two properties induces a groupoid representation, X, as follows:
$$ (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 ${C}_{c}(\mathcal{G})$ of a groupoid $\mathcal{G}$ is closed with respect to convolution if and only if the fixed Haar system associated with the measured groupoid $\mathcal{G}$ 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 $(\mathcal{G},[{\displaystyle \int {\nu}^{u}\mathit{d}\stackrel{~}{\lambda}(u)}]=[\lambda ])$ on a separable Hilbert space $\mathscr{H}$ induces a non-degenerate *-representation $f\mapsto {X}_{f}$ of the associated groupoid algebra $\mathrm{\Pi}(\mathcal{G},\nu ,\stackrel{~}{\lambda})$ 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.
References
- 1 P. Hahn: Haar measure for measure groupoids., Trans. Amer. Math. Soc. 242: 1–33(1978).
- 2 P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc. 242:35–72(1978). Theorem 3.4 on p. 50.
- 3 M. R. Buneci. Groupoid Representations, Ed. Mirton: Timishoara (2003).
- 4 M.R. Buneci. 2006., http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdfGroupoid C*-Algebras., Surveys in Mathematics and its Applications, Volume 1: 71–98.
- 5 M. R. Buneci. Isomorphic^{} groupoid C*-algebras associated with different Haar systems., New York J. Math., 11 (2005):225–245.
- 6 J. Renault. A groupoid approach to C*-algebras, Lecture Notes in Math., 793, Springer, Berlin, (1980).
- 7 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.
- 8 A. K. Seda: Haar measures for groupoids, Proc. Roy. Irish Acad. Sect. A 76 No. 5, 25–36 (1976).
- 9 A. K. Seda: Banach bundles of continuous functions and an integral representation theorem, Trans. Amer. Math. Soc. 270 No.1 : 327-332(1982).
- 10 A. K. Seda: On the Continuity of Haar measures on topological groupoids, Proc. Amer Math. Soc. 96: 115–120 (1986).
- 11 A. K. Seda. 2008. Personal communication, and also Seda (1986, on p.116).
Title | groupoid C*-convolution algebras |
Canonical name | GroupoidCconvolutionAlgebras |
Date of creation | 2013-03-22 18:13:59 |
Last modified on | 2013-03-22 18:13:59 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 63 |
Author | bci1 (20947) |
Entry type | Topic |
Classification | msc 55Q70 |
Classification | msc 55Q55 |
Classification | msc 55Q52 |
Classification | msc 81R50 |
Classification | msc 22A22 |
Classification | msc 81R10 |
Classification | msc 20L05 |
Classification | msc 18B40 |
Synonym | convolution algebra |
Related topic | GroupCAlgebra |
Related topic | CAlgebra |
Related topic | CAlgebra3 |
Related topic | Convolution |
Related topic | NuclearCAlgebra |
Related topic | QuantumGravityTheories |
Related topic | Algebras2 |
Related topic | C_cG |
Related topic | HomomorphismsOfCAlgebrasAreContinuous |
Related topic | SigmaFiniteBorelMeasureAndRelatedBorelConcepts |
Defines | groupoid C*-convolution algebra |
Defines | groupoid C*-algebra |
Defines | groupoid ${C}^{*}$-algebra |
Defines | Haar systems |
Defines | C*-convolution |
Defines | measured groupoid |