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Twisted index theory on good orbifolds. I: Noncommutative Bloch theory. (English) Zbl 0959.58035
The authors identify the range of the canonical trace applied to the $$K$$-theory of a class of twisted group $$C^{*}$$ algebras. The $$C^{*}$$ algebras in this class arise from a Fuchsian group and a multiplier with trivial Dixmier-Douady invariant. One step in the reasoning is a characterization of the $$K$$-theory of the twisted group $$C^{*}$$ algebras that, in the presence of trivial Dixmier-Douady invariant, establishes that this $$K$$-theory is isomorphic to the $$K$$-theory of the associated untwisted group $$C^{*}$$ algebra. A result of G. G. Kasparov [Sov. Math. Dokl. 29, 256-260 (1984; Zbl 0584.22004); translation from Dokl. Akad. Nauk SSSR 275, 541-545 (1984)] implies that this $$K$$-theory is generated by invariant elliptic operators. The authors use a twisted $$L^{2}$$ index theorem for orbifolds to provide a formula for the trace of such elements expressed via characteristic classes. With the help of an integrality result based on the orbifold index theorem of T. Kawasaki [Nagoya Math. J. 84, 135-157 (1981; Zbl 0465.58026)], the authors evaluate the characteristic class formula to reach their conclusion about the range of the trace.
The authors use their characterization of the trace’s range to classify up to isomorphism the twisted group $$C^{*}$$ algebras they study. When the multiplier takes values in roots of unity, the authors show that the reduced twisted group $$C^{*}$$ algebra has positive Kadison constant and finitely many unitary equivalence classes of projections. The authors combine their results with a theorem of J. Brüning and T. Sunada [Astérisque 210, 65-74 (1992; Zbl 0788.58008)] to show that certain self-adjoint elliptic operators have only finitely many spectral gaps in each half-line $$(-\infty, \lambda ]$$. The elliptic operators studied live on the hyperbolic plane and are invariant under the action determined by the Fuchsian group and the multiplier, which must have vanishing Dixmier-Douady invariant and take values in roots of unity.
The authors extend many of their results from Fuchsian groups to discrete torsion-free cocompact groups of isometries of four-dimensional real and complex hyperbolic space. They indicate that a later paper will address twisted higher index theory.

##### MSC:
 58J22 Exotic index theories on manifolds 46L80 $$K$$-theory and operator algebras (including cyclic theory) 19K56 Index theory 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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