groupoid C*-convolution algebras

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

Jean Renault introduced in ref. [6] the C*algebraPlanetmathPlanetmath of a locally compact groupoidPlanetmathPlanetmath G as follows: the space of continuous functionsMathworldPlanetmathPlanetmath with compact support on a groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 systemPlanetmathPlanetmath ( associated to the locally compact groupoids ( 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*-convolution algebra, or C*-algebra, as follows.

Definition 0.1.

a groupoid C*–convolution algebra, GCA, 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.

Remark 0.1.

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

Next we recall a result due to P. Hahn [2] which shows how groupoid representationsPlanetmathPlanetmathPlanetmathPlanetmath 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 (G,C) with Haar measure (ν,μ) in a separable Hilbert space H induces a *-algebra representation fXf of the associated groupoid algebra Π(G,ν) in L2(UG,μ,H) with the following properties:

(1) For any l,mH , one has that |<Xf(ul),(um)>|fllm and

(2) Mr(α)Xf=Xfαr, where

Mr:L(U𝖦,μL[L2(U𝖦,μ,], with


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

<Xf,j,k>=f(x)[X(x)j(d(x)),k(r(x))dν(x)]. (viz. p. 50 of ref. [2]).

Furthermore, according to Seda (ref. [10, 11]), the continuity of a Haar system is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the continuity of the convolution productPlanetmathPlanetmath 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 supportMathworldPlanetmathPlanetmathPlanetmath in dealing with convolution products of either locally compact groupoids or quantum groupoidsPlanetmathPlanetmathPlanetmath. Seda’s result also implies that the convolution algebra Cc(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,[νu𝑑λ~(u)]=[λ]) on a separable Hilbert space induces a non-degenerate *-representation fXf of the associated groupoid algebra Π(𝒢,ν,λ~) with properties formally similar to (1) and (2) above. Moreover, as in the case of groups, there is a correspondence between the unitary representationsMathworldPlanetmath of a groupoid and its associated C*-convolution algebra representations (p. 182 of [3]), the latter involving however fiber bundlesMathworldPlanetmath of Hilbert spacesMathworldPlanetmath instead of single Hilbert spaces.


  • 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., C*-Algebras., Surveys in Mathematics and its Applications, Volume 1: 71–98.
  • 5 M. R. Buneci. IsomorphicPlanetmathPlanetmathPlanetmath 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 AnalysisMathworldPlanetmath, 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