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groupoid C*-convolution algebra (Definition)

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 ${\mathsf{G}}$ as follows: the space of continuous functions with compact support on a groupoid ${\mathsf{G}}$ 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 ${\mathsf{G}}$ 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, $G_{CA}$, 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 $G_{CA}$ 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 $({\mathsf{G}},C)$ with Haar measure $(\nu, \mu)$ in a separable Hilbert space $\H$ induces a *-algebra representation $f \mapsto X_f$ of the associated groupoid algebra $\Pi ({\mathsf{G}}, \nu)$ in $L^2 (U_{{\mathsf{G}}} , \mu, \H )$ with the following properties:

(1) For any $l,m \in \H$ , one has that $\left\vert<X_f(u \mapsto l), (u \mapsto m)>\right\vert\leq \left\Vert f_l\right\Vert \left\Vert l \right\Vert \left\Vert m \right\Vert$ and (2) $M_r (\alpha) X_f = X_{f \alpha \circ r}$, where $M_r: L^\infty (U_{{\mathsf{G}}}, \mu \longrightarrow L[L^2 (U_{{\mathsf{G}}}, \mu, \H ]$, with

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

Conversely, any *- algebra representation with the above two properties induces a groupoid representation, X, as follows: $<X_f , j, k> ~ = ~ \displaystyle{\int} f(x)[X(x)j(d(x)),k(r(x))d \nu (x)].$ (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 $C_c (\mathcal G)$ 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 $(\mathcal G, [\displaystyle{\int} \nu ^u d \tilde{\lambda}(u)] = [\lambda])$ on a separable Hilbert space $\H$ induces a non-degenerate *-representation $f \mapsto X_f$ of the associated groupoid algebra $\Pi (\mathcal G, \nu,\tilde{\lambda})$ 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.

Bibliography

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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Cross-references: Hilbert spaces, groups, quantum groupoids, groupoid representation, separable Hilbert space, Haar measure, *-algebra representations, groupoid representations, concept, Haar system, representations, convolution, groupoid, functions, locally compact groupoid
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This is version 1 of groupoid C*-convolution algebra, born on 2009-01-12.
Object id is 377, canonical name is GroupoidCConvolutionAlgebra.
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Physics Classification: 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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