0.0.1 Background and Data for the Definition of a Groupoid C∗–Convolution Algebra
Jean Renault introduced in ref. [6] the C∗–algebra of a locally compact groupoid 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 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 groupoid 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 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 (G,C) with Haar measure
(ν,μ) in a separable Hilbert space induces a *-algebra representation f
Xf of the associated
groupoid algebra Π(G,ν) in L2(U
G,μ,) with the following properties:
(1) For any l,m ∈ , one has that 
≤
and
(2) Mr(α)Xf = Xfα∘r, where
Mr : L∞(U
G,μ→L[L2(U
G,μ,], with
Mr(α)j = α ⋅ j.
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 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 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 f
Xf 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 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., Groupoid 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).
