Dilations of semigroup crossed products as crossed products of dilations

Authors:
Nadia S. Larsen and Xin Li

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1597-1603

MSC (2010):
Primary 46L55

Published electronically:
February 1, 2013

MathSciNet review:
3020847

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Laca constructed a minimal automorphic dilation for every semigroup dynamical system arising from an action of an Ore semigroup by injective endomorphisms of a unital -algebra. Here we show that the semigroup crossed product with its action by inner endomorphisms given by the implementing isometries has as minimal automorphic dilation the group crossed product of the original dilation. Applications include recent examples studied by Cuntz and the second-named author.

**[Cli-Pre]**A. H. Clifford and G. B. Preston,*The algebraic theory of semigroups. Vol. I*, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR**0132791****[Cu1]**Joachim Cuntz,*Simple 𝐶*-algebras generated by isometries*, Comm. Math. Phys.**57**(1977), no. 2, 173–185. MR**0467330****[Cu2]**Joachim Cuntz,*𝐶*-algebras associated with the 𝑎𝑥+𝑏-semigroup over ℕ*, 𝐾-theory and noncommutative geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 201–215. MR**2513338**, 10.4171/060-1/8**[Cu-Li1]**Joachim Cuntz and Xin Li,*The regular 𝐶*-algebra of an integral domain*, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 149–170. MR**2732050****[Cu-Li2]**Joachim Cuntz and Xin Li,*𝐶*-algebras associated with integral domains and crossed products by actions on adele spaces*, J. Noncommut. Geom.**5**(2011), no. 1, 1–37. MR**2746649**, 10.4171/JNCG/68**[La]**Marcelo Laca,*From endomorphisms to automorphisms and back: dilations and full corners*, J. London Math. Soc. (2)**61**(2000), no. 3, 893–904. MR**1766113**, 10.1112/S0024610799008492**[La-Ra1]**Marcelo Laca and Iain Raeburn,*Semigroup crossed products and the Toeplitz algebras of nonabelian groups*, J. Funct. Anal.**139**(1996), no. 2, 415–440. MR**1402771**, 10.1006/jfan.1996.0091**[La-Ra2]**Marcelo Laca and Iain Raeburn,*Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers*, Adv. Math.**225**(2010), no. 2, 643–688. MR**2671177**, 10.1016/j.aim.2010.03.007**[Li]**Xin Li,*Ring 𝐶*-algebras*, Math. Ann.**348**(2010), no. 4, 859–898. MR**2721644**, 10.1007/s00208-010-0502-x**[Mur]**Gerard J. Murphy,*Crossed products of 𝐶*-algebras by endomorphisms*, Integral Equations Operator Theory**24**(1996), no. 3, 298–319. MR**1375977**, 10.1007/BF01204603**[Sta]**P. J. Stacey,*Crossed products of 𝐶*-algebras by ∗-endomorphisms*, J. Austral. Math. Soc. Ser. A**54**(1993), no. 2, 204–212. MR**1200792**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
46L55

Retrieve articles in all journals with MSC (2010): 46L55

Additional Information

**Nadia S. Larsen**

Affiliation:
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N-0316 Oslo, Norway

Email:
nadiasl@math.uio.no

**Xin Li**

Affiliation:
Department of Mathematics, Westfälische Wilhelms-Universität Münster, Einsteinstraße 62, 48149 Münster, Germany

Email:
xinli.math@uni-muenster.de

DOI:
http://dx.doi.org/10.1090/S0002-9939-2013-11475-9

Received by editor(s):
October 6, 2010

Published electronically:
February 1, 2013

Additional Notes:
The first-named author thanks J. Cuntz and S. Echterhoff for their kind hospitality during a sabbatical visit at Westfälische Wilhelms-Universität Münster in October 2009, where this research was initiated

The second-named author thanks the operator algebra group in Oslo for a nice visit at the University of Oslo.

This research was supported by the Research Council of Norway and the Deutsche Forschungsgemeinschaft

Communicated by:
Marius Junge

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.