We shall briefly consider a main result due to Hahn (1978) that relates groupoid and associated groupoid algebra representations:
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(0.1) |
Furthermore, according to Seda (1986, on p.116) 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 the convolution * if and only if the fixed Haar
system associated with the measured groupoid
is
continuous (Buneci, 2003).
In the case of groupoid algebras of transitive groupoids, Buneci
(2003) showed that representations of a measured groupoid
on a
separable Hilbert space
induce non-degenerate
-representations
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 Buneci, 2003), the latter
involving however fiber bundles of Hilbert spaces
instead of
single Hilbert spaces. Therefore, groupoid representations appear
as the natural construct for algebraic quantum field theories
(AQFT) in
which nets of local observable
operators
in Hilbert space fiber
bundles were introduced by Rovelli (1998).
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