Groupoids associated with endomorphisms

Deaconu, Valentin

doi:10.1090/S0002-9947-1995-1233967-5

Groupoids associated with endomorphisms
HTML articles powered by AMS MathViewer

by Valentin Deaconu PDF

Trans. Amer. Math. Soc. 347 (1995), 1779-1786 Request permission

Abstract:

To a compact Hausdorff space which covers itself, we associate an $r$-discrete locally compact Hausdorff groupoid. Its ${{\mathbf {C}}^ * }$-algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the Cuntz algebra ${O_p}$. We consider examples related to coverings of the circle and of a Heisenberg $3$-manifold.

References

Joel Anderson and William Paschke, The rotation algebra, Houston J. Math. 15 (1989), no. 1, 1–26. MR 1002078
V. A. Arzumanian and A. M. Vershik, Star algebras associated with endomorphisms, Operator algebras and group representations, Vol. I (Neptun, 1980) Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984, pp. 17–27. MR 731760
Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867, DOI 10.1007/978-1-4613-9572-0
James R. Brown, Ergodic theory and topological dynamics, Pure and Applied Mathematics, No. 70, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0492177
Alexander Kumjian, Preliminary algebras arising from local homeomorphisms, Math. Scand. 52 (1983), no. 2, 269–278. MR 702958, DOI 10.7146/math.scand.a-12006
Alexander Kumjian, On $C^\ast$-diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008. MR 854149, DOI 10.4153/CJM-1986-048-0

Coordinates in operator algebras

William L. Paschke, The crossed product of a $C^{\ast }$-algebra by an endomorphism, Proc. Amer. Math. Soc. 80 (1980), no. 1, 113–118. MR 574518, DOI 10.1090/S0002-9939-1980-0574518-2
William L. Paschke, $K$-theory for actions of the circle group on $C^{\ast }$-algebras, J. Operator Theory 6 (1981), no. 1, 125–133. MR 637006
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072
Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199. MR 240824, DOI 10.2307/2373276