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


SIGMA 10 (2014), 055, 50 pages      arXiv:1304.2284      https://doi.org/10.3842/SIGMA.2014.055
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Ordered $*$-Semigroups and a $C^*$-Correspondence for a Partial Isometry

Berndt Brenken
Department of Mathematics and Statistics, University of Calgary, Calgary, Canada T2N 1N4

Received August 30, 2013, in final form May 22, 2014; Published online May 31, 2014

Abstract
Certain $*$-semigroups are associated with the universal $C^*$-algebra generated by a partial isometry, which is itself the universal $C^*$-algebra of a $*$-semigroup. A fundamental role for a $*$-structure on a semigroup is emphasized, and ordered and matricially ordered $*$-semigroups are introduced, along with their universal $C^*$-algebras. The universal $C^*$-algebra generated by a partial isometry is isomorphic to a relative Cuntz-Pimsner $C^*$-algebra of a $C^*$-correspondence over the $C^*$-algebra of a matricially ordered $*$-semigroup. One may view the $C^*$-algebra of a partial isometry as the crossed product algebra associated with a dynamical system defined by a complete order map modelled by a partial isometry acting on a matricially ordered $*$-semigroup.

Key words: $C^*$-algebras; partial isometry; $*$-semigroup; partial order; matricial order; completely positive maps; $C^*$-correspondence; Schwarz inequality; exact $C^*$-algebra.

pdf (691 kb)   tex (56 kb)

References

  1. Abadie B., Eilers S., Exel R., Morita equivalence for crossed products by Hilbert $C^*$-bimodules, Trans. Amer. Math. Soc. 350 (1998), 3043-3054.
  2. Arveson W., Noncommutative dynamics and $E$-semigroups, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.
  3. Barnes B.A., Duncan J., The Banach algebra $l^{1}(S)$, J. Funct. Anal. 18 (1975), 96-113.
  4. Brenken B., Topological quivers as multiplicity free relations, Math. Scand. 106 (2010), 217-242.
  5. Brenken B., Niu Z., The $C^*$-algebra of a partial isometry, Proc. Amer. Math. Soc. 140 (2012), 199-206.
  6. Brown N.P., Ozawa N., $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, Vol. 88, Amer. Math. Soc., Providence, RI, 2008.
  7. Conway J.B., Duncan J., Paterson A.L.T., Monogenic inverse semigroups and their $C^*$-algebras, Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 13-24.
  8. Dales H.G., Lau A.T.M., Strauss D., Banach algebras on semigroups and on their compactifications, Mem. Amer. Math. Soc. 205 (2010), vi+165 pages.
  9. Duncan J., Paterson A.L.T., $C^*$-algebras of inverse semigroups, Proc. Edinburgh Math. Soc. 28 (1985), 41-58.
  10. Dykema K.J., Shlyakhtenko D., Exactness of Cuntz-Pimsner $C^*$-algebras, Proc. Edinb. Math. Soc. 44 (2001), 425-444, math.OA/9911002.
  11. Fell J.M.G., Doran R.S., Representations of $*$-algebras, locally compact groups, and Banach $*$-algebraic bundles. Vol. 1. Basic representation theory of groups and algebras, Pure and Applied Mathematics, Vol. 125, Academic Press, Inc., Boston, MA, 1988.
  12. Fell J.M.G., Doran R.S., Representations of $*$-algebras, locally compact groups, and Banach $*$-algebraic bundles. Vol. 2. Banach $*$-algebraic bundles, induced representations, and the generalized Mackey analysis, Pure and Applied Mathematics, Vol. 126, Academic Press, Inc., Boston, MA, 1988.
  13. Fowler N.J., Muhly P.S., Raeburn I., Representations of Cuntz-Pimsner algebras, Indiana Univ. Math. J. 52 (2003), 569-605.
  14. Howie J.M., An introduction to semigroup theory, Academic Press, London - New York, 1976.
  15. Katsura T., On $C^*$-algebras associated with $C^*$-correspondences, J. Funct. Anal. 217 (2004), 366-401, math.OA/0309088.
  16. Lance E.C., Hilbert $C^*$-modules. A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, Vol. 210, Cambridge University Press, Cambridge, 1995.
  17. Li X., Semigroup $C^*$-algebras and amenability of semigroups, J. Funct. Anal. 262 (2012), 4302-4340, arXiv:1105.5539.
  18. Muhly P.S., Solel B., Tensor algebras over $C^*$-correspondences: representations, dilations, and $C^*$-envelopes, J. Funct. Anal. 158 (1998), 389-457.
  19. Paulsen V., Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, Vol. 78, Cambridge University Press, Cambridge, 2002.
  20. Pimsner M.V., A class of $C^*$-algebras generalizing both Cuntz-Krieger algebras and crossed products by ${\mathbb Z}$, in Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun., Vol. 12, Amer. Math. Soc., Providence, RI, 1997, 189-212.

Previous article  Next article   Contents of Volume 10 (2014)