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groupoid C*-convolution algebras∗
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0.1
Introduction: Background and definition of the groupoid C*–convolution
algebra
Jean Renault introduced in ref. [?] 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.[?]. 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. [?] 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 [?] 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. [?]). Any representation of a groupoid (G, C) with Haar measure (ν, µ) in
a separable Hilbert space H induces a *-algebra representation f 7→ Xf of the associated groupoid algebra
Π(G, ν) in L2 (UG , µ, H) with the following properties:
(1) For any l, m ∈ H , one has that |< Xf (u 7→ l), (u 7→ m) >| ≤ kfl k klk kmk and
(2) Mr (α)Xf = Xf α◦r , where
Mr : L∞ (UG , µ −→ L[L2 (UG , µ, H], with
Mr (α)j = α · j.
Conversely, any *- algebra representation with the above two properties induces a groupoid representation,
X, as follows:
∗ hGroupoidCconvolutionAlgebrasi
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Z
< Xf , j, k > =
f (x)[X(x)j(d(x)), k(r(x))dν(x)]. (viz. p. 50 of ref. [?]).
Furthermore, according to Seda (ref. [?, ?]), 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. [?]).
Thus, in the case of groupoid
Z algebras of transitive groupoids, it was shown in [?] that any representation
of a measured groupoid (G, [ ν u dλ̃(u)] = [λ]) on a separable Hilbert space H induces a non-degenerate
*-representation f 7→ Xf of the associated groupoid algebra Π(G, ν, λ̃) 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.
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).
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