Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-25T01:53:52.449Z Has data issue: false hasContentIssue false
  • English
  • Français

Crossed products by abelian semigroups via transfer operators

Published online by Cambridge University Press:  21 July 2009

NADIA S. LARSEN
NADIA S. LARSEN*
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, N-0316 Oslo, Norway (email: nadiasl@math.uio.no)
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a generalization of Exel’s crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler’s theory of Toeplitz and Cuntz–Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz–Pimsner algebras of product systems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 1147 - 1164
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adji, S.. Invariant ideals of crossed products by semigroups of endomorphisms. Functional Analysis and Global Analysis. Eds. Sunada, T. and Sy, P. W.. Springer, Singapore, 1997, pp. 1–8.Google Scholar
[2]Adji, S., Laca, M., Nilsen, M. and Raeburn, I.. Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups. Proc. Amer. Math. Soc. 122 (1994), 1133–1141.CrossRefGoogle Scholar
[3]Bost, J.-B. and Connes, A.. Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Selecta Math. (N.S.) 1 (1995), 411–457.CrossRefGoogle Scholar
[4]Brownlowe, N.. Exel’s crossed product for non-unital C *-algebras. PhD Thesis, The University of Newcastle, Australia.Google Scholar
[5]Brownlowe, N., Larsen, N. S., Putnam, I. F. and Raeburn, I.. Subquotients of Hecke C *-algebras. Ergod. Th. & Dynam. Sys. 25 (2005), 1503–1520.CrossRefGoogle Scholar
[6]Brownlowe, N. and Raeburn, I.. Exel’s crossed product and relative Cuntz–Pimsner algebras. Math. Proc. Cambridge Philos. Soc. 141 (2006), 497–508.CrossRefGoogle Scholar
[7]Cuntz, J.. The Internal Structure of Simple C *-Algebras (Proceedings of Symposia in Pure Mathematics, 38). American Mathematical Society, Providence, RI, 1982, pp. 85–115.Google Scholar
[8]Exel, R.. A new look at the crossed product of a C *-algebra by an endomorphism. Ergod. Th. & Dynam. Sys. 23 (2003), 1733–1750.CrossRefGoogle Scholar
[9]Exel, R.. A new look at the crossed product of a C *-algebra by a semigroup of endomorphisms. Ergod. Th. & Dynam. Sys. 28 (2008), 749–789.CrossRefGoogle Scholar
[10]Fowler, N.. Discrete product systems of Hilbert bimodules. Pacific J. Math. 204 (2002), 335–375.CrossRefGoogle Scholar
[11]Fowler, N., Muhly, P. and Raeburn, I.. Representations of Cuntz–Pimsner algebras. Indiana Univ. Math. J. 52 (2003), 569–605.CrossRefGoogle Scholar
[12]Fowler, N. and Raeburn, I.. The Toeplitz algebra of a Hilbert bimodule. Indiana Univ. Math. J. 48 (1999), 155–181.CrossRefGoogle Scholar
[13]Laca, M. and Raeburn, I.. Semigroup crossed products and the Toeplitz algebras of nonabelian groups. J. Funct. Anal. 139 (1996), 415–440.CrossRefGoogle Scholar
[14]Laca, M. and Raeburn, I.. A semigroup crossed product arising in number theory. J. Lond. Math. Soc. (2) 59 (1999), 330–344.CrossRefGoogle Scholar
[15]Larsen, N. S.. Non-unital semigroup crossed products. Math. Proc. R. Ir. Acad. 100A (2000), 205–218.Google Scholar
[16]Larsen, N. S. and Raeburn, I.. Faithful representations of crossed products by actions of ℕk. Math. Scand. 89 (2001), 283–296.CrossRefGoogle Scholar
[17]Muhly, P. S. and Solel, B.. Tensor algebras over C *-correspondences (representations, dilations andC *-envelopes). J. Funct. Anal. 158 (1998), 389–457.CrossRefGoogle Scholar
[18]Murphy, G. J.. Crossed products of C *-algebras by endomorphisms. Integral Equations Operator Theory 24 (1996), 298–319.CrossRefGoogle Scholar
[19]Paschke, W. L.. The crossed products of a C *-algebra by an endomorphism. Proc. Amer. Math. Soc. 80 (1980), 113–118.Google Scholar
[20]Pimsner, M. V.. A class of C *-algebras generalising both Cuntz–Krieger algebras and crossed products by ℤ. Fields Inst. Commun. 12 (1997), 189–212.Google Scholar
[21]Raeburn, I. and Williams, D. P.. Morita Equivalence and Continuous-Trace C *-Algebras (Mathematical Surveys and Monographs, 60). American Mathematical Society, Providence, RI, 1998.CrossRefGoogle Scholar
[22]Stacey, P. J.. Crossed products of C *-algebras by *-endomorphisms. J. Aust. Math. Soc. Ser. A 54 (1993), 204–212.CrossRefGoogle Scholar