Geodesic
Derivations and Spectral Triples on Quantum Domains I: Quantum Disk
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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We study unbounded invariant and covariant derivations on the quantum disk. In particular we answer the question whether such derivations come from operators with compact parametrices and thus can be used to define spectral triples.
Keywords: invariant and covariant derivations; spectral triple; quantum disk. 
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     title = {Derivations and {Spectral} {Triples} on {Quantum} {Domains} {I:} {Quantum} {Disk}},
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}
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Slawomir Klimek; Matt McBride; Sumedha Rathnayake; Kaoru Sakai; Honglin Wang. Derivations and Spectral Triples on Quantum Domains I: Quantum Disk. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a74/

[1] Barría J., Halmos P. R., “Asymptotic Toeplitz operators”, Trans. Amer. Math. Soc., 273 (1982), 621–630 | DOI | MR | Zbl

[2] Battisti U., Seiler J., “Boundary value problems with Atiyah–Patodi–Singer type conditions and spectral triples”, J. Noncommut. Geom. (to appear) , arXiv: 1503.02897 | MR

[3] Booß-Bavnbek B., Wojciechowski K. P., Elliptic boundary problems for Dirac operators, Mathematics: Theory Applications, Birkhäuser Boston, Inc., Boston, MA, 1993 | DOI | MR | Zbl

[4] Bratteli O., Derivations, dissipations and group actions on $C^*$-algebras, Lecture Notes in Math., 1229, Springer-Verlag, Berlin, 1986 | DOI | MR

[5] Bratteli O., Elliott G. A., Jorgensen P. E. T., “Decomposition of unbounded derivations into invariant and approximately inner parts”, J. Reine Angew. Math., 346 (1984), 166–193 | DOI | MR | Zbl

[6] Brown A., Halmos P. R., Shields A. L., “Cesàro operators”, Acta Sci. Math. (Szeged), 26 (1965), 125–137 | MR | Zbl

[7] Carey A. L., Klimek S., Wojciechowski K. P., “A Dirac type operator on the non-commutative disk”, Lett. Math. Phys., 93 (2010), 107–125 | DOI | MR | Zbl

[8] Coburn L. A., “The $C^{\ast} $-algebra generated by an isometry”, Bull. Amer. Math. Soc., 73 (1967), 722–726 | DOI | MR | Zbl

[9] Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994 | MR | Zbl

[10] Connes A., “Cyclic cohomology, quantum group symmetries and the local index formula for ${\rm SU}_q(2)$”, J. Inst. Math. Jussieu, 3 (2004), 17–68, arXiv: math.QA/0209142 | DOI | MR | Zbl

[11] D'Andrea F., Da̧browski L., “Local index formula on the equatorial Podleś sphere”, Lett. Math. Phys., 75 (2006), 235–254, arXiv: math.QA/0507337 | DOI | MR | Zbl

[12] Forsyth I., Goffeng M., Mesland B., Rennie A., Boundaries, spectral triples and $K$-homology, arXiv: 1607.07143

[13] Hadfield T., “The noncommutative geometry of the discrete Heisenberg group”, Houston J. Math., 29 (2003), 453–481 | MR | Zbl

[14] Jorgensen P., “Approximately inner derivations, decompositions and vector fields of simple $C^*$-algebras”, Mappings of Operator Algebras (Philadelphia, PA, 1988), Progr. Math., 84, Birkhäuser Boston, Boston, MA, 1990, 15–113 | DOI | MR

[15] Kadison R. V., Ringrose J. R., Fundamentals of the theory of operator algebras, v. II, Pure and Applied Mathematics, 100, Advanced theory, Academic Press, Inc., Orlando, FL, 1986 | MR | Zbl

[16] Klimek S., Lesniewski A., “Quantum Riemann surfaces. I. The unit disc”, Comm. Math. Phys., 146 (1992), 103–122 | DOI | MR | Zbl

[17] Klimek S., McBride M., “D-bar operators on quantum domains”, Math. Phys. Anal. Geom., 13 (2010), 357–390, arXiv: 1001.2216 | DOI | MR | Zbl

[18] Klimek S., McBride M., Rathnayake S., Derivations and spectral triples on quantum domains II: Quantum annulus, in preparation

[19] Sakai S., Operator algebras in dynamical systems, Encyclopedia of Mathematics and its Applications, 41, Cambridge University Press, Cambridge, 1991 | DOI | MR | Zbl

[20] Schechter M., “Basic theory of Fredholm operators”, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 261–280 | MR | Zbl

[21] Stacey P. J., “Crossed products of $C^\ast$-algebras by $\ast$-endomorphisms”, J. Austral. Math. Soc. Ser. A, 54 (1993), 204–212 | DOI | MR | Zbl