Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 082, 7 pages      arXiv:1407.6020      https://doi.org/10.3842/SIGMA.2015.082

Equivariant Join and Fusion of Noncommutative Algebras

Ludwik Dąbrowski a, Tom Hadfield b and Piotr M. Hajac c
a) SISSA (Scuola Internazionale Superiore di Studi Avanzati), Via Bonomea 265, 34136 Trieste, Italy
b) G-Research, Whittington House, 19-30 Alfred Place, London WC1E 7EA, UK
c) Institytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, 00-656 Warszawa, Poland

Received June 30, 2015, in final form October 03, 2015; Published online October 13, 2015

Abstract
We translate the concept of the join of topological spaces to the language of $C^*$-algebras, replace the $C^*$-algebra of functions on the interval $[0,1]$ with evaluation maps at $0$ and $1$ by a unital $C^*$-algebra $C$ with appropriate two surjections, and introduce the notion of the fusion of unital $C^*$-algebras. An appropriate modification of this construction yields the fusion comodule algebra of a comodule algebra $P$ with the coacting Hopf algebra $H$. We prove that, if the comodule algebra $P$ is principal, then so is the fusion comodule algebra. When $C=C([0,1])$ and the two surjections are evaluation maps at $0$ and $1$, this result is a noncommutative-algebraic incarnation of the fact that, for a compact Hausdorff principal $G$-bundle $X$, the diagonal action of $G$ on the join $X*G$ is free.

Key words: $C^*$-algebras; Hopf algebras; free actions.

pdf (316 kb)   tex (15 kb)

References

  1. Baum P.F., Dąbrowski L., Hajac P.M., Noncommutative Borsuk-Ulam-type conjectures, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, to appear, arXiv:1502.05756.
  2. Baum P.F., De Commer K., Hajac P.M., Free actions of compact quantum group on unital $C^*$-algebras, arXiv:1304.2812.
  3. Brzeziński T., Hajac P.M., The Chern-Galois character, C. R. Math. Acad. Sci. Paris 338 (2004), 113-116, math.KT/0306436.
  4. Dąbrowski L., Hadfield T., Hajac P.M., Matthes R., Wagner E., Index pairings for pullbacks of ${C}^*$-algebras, in Operator Algebras and Quantum Groups, Banach Center Publ., Vol. 98, Polish Acad. Sci. Inst. Math., Warsaw, 2012, 67-84, math.QA/0702001.
  5. Dąbrowski L., De Commer K., Hajac P.M., Wagner E., Noncommutative bordism of free actions of compact quantum groups on unital $C^*$-algebras, in preparation.
  6. Hajac P.M., Strong connections on quantum principal bundles, Comm. Math. Phys. 182 (1996), 579-617, hep-th/9406129.
  7. Hajac P.M., Krähmer U., Matthes R., Zieliński B., Piecewise principal comodule algebras, J. Noncommut. Geom. 5 (2011), 591-614, arXiv:0707.1344.
  8. Jiang X., Su H., On a simple unital projectionless $C^*$-algebra, Amer. J. Math. 121 (1999), 359-413.
  9. Milnor J., Construction of universal bundles. II, Ann. of Math. 63 (1956), 430-436.
  10. Pflaum M.J., Quantum groups on fibre bundles, Comm. Math. Phys. 166 (1994), 279-315, hep-th/9401085.
  11. Takesaki M., Theory of operator algebras. I, Springer-Verlag, New York - Heidelberg, 1979.
  12. Wassermann S., Exact $C^*$-algebras and related topics, Lecture Notes Series, Vol. 19, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994.
  13. Woronowicz S.L., Compact quantum groups, in Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 845-884.


Previous article  Next article   Contents of Volume 11 (2015)