Jean Renault introduced in ref. [6] the
-algebra of a locally compact groupoid
as follows: the space of continuous functions with compact support on a groupoid
is made into a *-algebra whose multiplication is the convolution, and that is also endowed with the smallest
-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
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
-convolution algebra, or groupoid
-algebra, as follows.
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.
(1) For any
, one has that
and
(2)
, where
, with
.
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
for any pair
,
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
of a groupoid
is closed with respect to convolution if and only if the fixed Haar system associated with the measured groupoid
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
on a separable Hilbert space
induces a non-degenerate *-representation
of the associated groupoid algebra
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.
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