Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-xl52z Total loading time: 1.333 Render date: 2021-04-15T07:57:44.794Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

On simplicity of intermediate $C^{\ast }$-algebras

Published online by Cambridge University Press:  06 June 2019

TATTWAMASI AMRUTAM
Affiliation:
Department of Mathematics, 3551 Cullen Blvd, Room 641, Philip Guthrie Hoffman Hall, Houston, TX77204, USA email tamrutam@math.uh.edu, kalantar@math.uh.edu
MEHRDAD KALANTAR
Affiliation:
Department of Mathematics, 3551 Cullen Blvd, Room 641, Philip Guthrie Hoffman Hall, Houston, TX77204, USA email tamrutam@math.uh.edu, kalantar@math.uh.edu
Corresponding

Abstract

We prove simplicity of all intermediate $C^{\ast }$-algebras $C_{r}^{\ast }(\unicode[STIX]{x1D6E4})\subseteq {\mathcal{B}}\subseteq \unicode[STIX]{x1D6E4}\ltimes _{r}C(X)$ in the case of minimal actions of $C^{\ast }$-simple groups $\unicode[STIX]{x1D6E4}$ on compact spaces $X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar [Stationary $C^{\ast }$-dynamical systems. Preprint, 2017, arXiv:1712.10133]. We show that the Powers’ averaging property holds for the reduced crossed product $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$ for any action $\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{A}}$ of a $C^{\ast }$-simple group $\unicode[STIX]{x1D6E4}$ on a unital $C^{\ast }$-algebra ${\mathcal{A}}$, and use it to prove a one-to-one correspondence between stationary states on ${\mathcal{A}}$ and those on $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$.

Type
Original Article
Copyright
© Cambridge University Press, 2019

Access options

Get access to the full version of this content by using one of the access options below.

References

Archbold, R. J. and Spielberg, J. S.. Topologically free actions and ideals in discrete C -dynamical systems. Proc. Edinb. Math. Soc. (2) 37 (1994), 119124.CrossRefGoogle Scholar
Bédos, E.. Discrete groups and simple C -algebras. Math. Proc. Cambridge Philos. Soc. 109(3) (1991), 521537.CrossRefGoogle Scholar
Breuillard, E., Kalantar, M., Kennedy, M. and Ozawa, N.. C -simplicity and the unique trace property for discrete groups. Publ. Math. Inst. Hautes Études Sci. 126 (2017), 3571.CrossRefGoogle Scholar
Brown, N. and Ozawa, N.. C -Algebras and Finite-dimensional Approximations (Graduate Studies in Mathematics, 88). American Mathematical Society, Providence, RI, 2008.CrossRefGoogle Scholar
Bryder, R. S. and Kennedy, M.. Reduced twisted crossed products over C -simple groups. Int. Math. Res. Not. IMRN 6 (2018), 16381655.Google Scholar
de la Harpe, P. and Skandalis, G.. Powers’ property and simple C -algebras. Math. Ann. 273 (1986), 241250.CrossRefGoogle Scholar
Elliott, G. A.. Some simple C -algebras constructed as crossed products with discrete outer automorphism groups. Publ. Res. Inst. Math. Sci. 16(1) (1980), 299311.CrossRefGoogle Scholar
Haagerup, U.. A new look at C -simplicity and the unique trace property of a group. Operator Algebras and Applications (The Abel Symposium 2015) (Abel Sympia, 12). Springer, Cham, 2017, pp. 167176.Google Scholar
Hartman, Y. and Kalantar, M.. Stationary $C^{\ast }$-dynamical systems. Preprint, 2017, arXiv:1712.10133.Google Scholar
Jang, S. Y. and Lee, S. G.. Simplicity of crossed products of C -algebras. Proc. Amer. Math. Soc. 118(3) (1993), 823826.Google Scholar
Kalantar, M. and Kennedy, M.. Boundaries of reduced C -algebras of discrete groups. J. Reine Angew. Math. 727 (2017), 247267.Google Scholar
Kawamura, S. and Tomiyama, J.. Properties of topological dynamical systems and corresponding C -algebras. Tokyo J. Math. 13 (1990), 251257.CrossRefGoogle Scholar
Kennedy, M.. An intrinsic characterization of $C^{\ast }$-simplicity. Ann. Sci. Éc. Norm. Supér., accepted. Preprint, 2015, arXiv:1509.01870.Google Scholar
Sierakowski, A.. The ideal structure of reduced crossed products. Münster J. Math. 3 (2010), 237261.Google Scholar
Suzuki, Y.. Minimal ambient nuclear C -algebras. Adv. Math. 304 (2017), 421433.CrossRefGoogle Scholar
Suzuki, Y.. Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems. Commun. Math. Phys., accepted. Preprint, 2018, arXiv:1805.02077.CrossRefGoogle Scholar
Takesaki, M.. On the cross-norm of the direct product of C -algebras. Tôhoku Math. J. (2) 16 (1964), 111122.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 64 *
View data table for this chart

* Views captured on Cambridge Core between 06th June 2019 - 15th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On simplicity of intermediate $C^{\ast }$-algebras
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On simplicity of intermediate $C^{\ast }$-algebras
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On simplicity of intermediate $C^{\ast }$-algebras
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *