Groupoid methods in wavelet analysis. (English) Zbl 1161.22001

Ionescu, Marius; Muhly, Paul S.

Doran, Robert S. (ed.) et al., Group representations, ergodic theory, and mathematical physics. A tribute to George W. Mackey. AMS special session honoring the memory of George W. Mackey, New Orleans, LA, USA, January 7–8, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4225-6/pbk). Contemporary Mathematics 449, 193-208 (2008).

The authors describe how the Deaconu-Renault groupoids may be used in the study of wavelets and fractals. Palle Jorgensen and his collaborators have showed that much of the analysis of wavelets and fractals that has appeared in recent years may be illuminated in terms of special representations of the Cuntz relations. Some of the most important advances are made by choosing an appropriate representation for these relations. One of the aims of this work is to understand the extent to which the use of the Cuntz relations is intrinsic to the situation under consideration. On the other hand, the Cuntz isometries that arise in the work of Jorgensen et al. may be expressed in terms of representations of the Deaconu-Renault groupoid associated to an appropriate local homeomorphism of a compact Hausdorff space. The second purpose of the paper is to show how the C*-algebra of this groupoid is related to a number of other C*-algebras that one can attach to a local homeomorphism. In particular, the authors show that the C*-algebra may be realized as a Cuntz-Pimsner algebra in two different ways and that, in general, it is a quotient of certain other C*-algebras that one may build from the local homeomorphism.
For the entire collection see [Zbl 1131.22001].

Reviewer: Sh. A. Ayupov (Tashkent)

MSC:

22A22	Topological groupoids (including differentiable and Lie groupoids)
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
28A80	Fractals
46L89	Other “noncommutative” mathematics based on \(C^*\)-algebra theory
46L05	General theory of \(C^*\)-algebras
46L55	Noncommutative dynamical systems
46L40	Automorphisms of selfadjoint operator algebras
58H99	Pseudogroups, differentiable groupoids and general structures on manifolds
37F99	Dynamical systems over complex numbers
32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables

Keywords:

groupoids; wavelets; fractal; C*-algebras; Cuntz relations