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Periodic integral transforms and associated noncommutative orbifold projections. (English) Zbl 1349.46083

Summary: We report on recent results on the existence of cubic and hexic integral transforms on self-dual locally compact groups (orders 3 and 6 analogues of the classical Fourier transform) and their application in constructing a canonical continuous section of smooth projections \({\mathcal E}(t)\) of the continuous field of rotation \(C^*\)-algebras \(\{A_t\}_{0<t<1}\) that is invariant under the noncormutative hexic transform automorphism.
This leads to invariant matrix (point) projections of the irrational noncommutative tori \(A_\theta\). We also present a quick method for computing the (quantized) topological invariants of such projections using techniques from classical theta function theory.

MSC:

46L85 Noncommutative topology
46L05 General theory of \(C^*\)-algebras
57R18 Topology and geometry of orbifolds
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