C*–algebras Nearly Contained in Type I Algebras

Erik Christensen; Allan M. Sinclair; Roger R. Smith; Stuart White

doi:10.4153/CJM-2012-001-1

Skip to main content Accessibility help

Home
Journals
Canadian Journal of Mathematics
Volume 65 Issue 1
C*–algebras Nearly Contained in Type I Algebras

Canadian Journal of Mathematics

Article contents

Abstract
References

C*–algebras Nearly Contained in Type I Algebras

Published online by Cambridge University Press: 20 November 2018

Roger R. Smith and

Erik Christensen: Affiliation:
Institute for Mathematiske Fag, University of Copenhagen, Denmark, e-mail: echris@math.ku.dk
Allan M. Sinclair: Affiliation:
School of Mathematics, University of Edinburgh, JCMB, King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, e-mail: a.sinclair@ed.ac.uk
Roger R. Smith: Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A., e-mail: rsmith@math.tamu.edu
Stuart White: Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow Q12 8QW, Scotland, e-mail: stuart.white@glasgow.ac.uk

Article
Metrics

Article contents

Abstract
References

Rights & Permissions

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider near inclusions $A\,{{\subseteq }_{\gamma }}\,B$ of ${{\text{C}}^{*}}$-algebras. We show that if $B$ is a separable type $\text{I}$${{\text{C}}^{*}}$-algebra and $A$ satisfies Kadison's similarity problem, then $A$ is also type $\text{I}$. We then use this to obtain an embedding of $A$ into $B$.

Keywords

46L05 46L45 C*-algebras near inclusions perturbations type I C*-algebras similarity problem

Type: Research Article
Information: Canadian Journal of Mathematics , Volume 65 , Issue 1 , 01 February 2013 , pp. 52 - 65

DOI: https://doi.org/10.4153/CJM-2012-001-1[Opens in a new window]
Copyright: Copyright © Canadian Mathematical Society 2013

References

[A] W. Arveson, , Interpolation problems in nest algebras. J. Functional Analysis 20(1975), no. 3, 208–233. http://dx.doi.org/10.1016/0022-1236(75)90041-5 Google Scholar

[ChC] Choi, M.-D. and Christensen, E., Completely order isomorphic and close be^*-isomorphic. Bull. London Math. Soc. 15(1983), no. 6, 604–610. http://dx.doi.org/10.1112/blms/15.6.604 Google Scholar

[C1] Christensen, E., Perturbations of operator algebras. Invent. Math. 43(1977), no. 1, 1–13. http://dx.doi.org/10.1007/BF01390201 Google Scholar

[C2] Christensen, E., Perturbations of operator algebras. II. Indiana Univ. Math. J. 26(1977), no. 5, 891–904. http://dx.doi.org/10.1512/iumj.1977.26.26072 Google Scholar

[C3] Christensen, E., Near inclusions of C^*-algebras. Acta Math. 144(1980), no. 3-4, 249–265. http://Dx.Doi.Org/10.1007/Bf02392125 Google Scholar

[CSSW] Christensen, E., Sinclair, A., Smith, R. R., and White, S., Perturbations of C^*-algebraic invariants. Geom. Funct. Anal. 20(2010), no. 2, 368–397. http://dx.doi.org/10.1007/s00039-010-0070-y Google Scholar

[CSSWW1] Christensen, E., Sinclair, A. M., Smith, R. R., White, S. A., and W.Winter, , The spatial isomorphism problem for close separable nuclear C^*-algebras. Proc. Natl. Acad. Sci. USA 107(2010), no. 2, 587–591. http://dx.doi.org/10.1073/pnas.0913281107 Google Scholar

[CSSWW2] Christensen, E., Perturbations of nuclear C^*-algebras. Acta Math., to appear. arxiv:0910.4953v1. Google Scholar

[D] Dixmier, J., C^*-algebras. North-Holland Mathematical Library, 15, North Holland, Amsterdam-New York-Oxford, 1977.Google Scholar

[F] Fell, J. M. G., The structure of algebras of operator fields. Acta Math. 106(1961), 233–280. http://dx.doi.org/10.1007/BF02545788 Google Scholar

[HWW] Harmand, P., Werner, D., andWerner, W., M-ideals in Banach spaces and Banach algebras. Lecture Notes in Mathematics, 1547, Springer-Verlag, Berlin, 1993.CrossRef Google Scholar

[HKW] Hirshberg, I., Kirchberg, E., and White, S., Decomposable approximations of nuclear C^*-algebras. CRM Preprint 1028, http://www.crm.es/Publications/11/Pr1028.pdf. Google Scholar

[J1] Johnson, B. E., Perturbations of Banach algebras. Proc. London. Math. Soc. (3) 34(1977), no. 3, 439–458. http://dx.doi.org/10.1112/plms/s3-34.3.439 Google Scholar

[J2] Johnson, B. E., A counterexample in the perturbation theory of C^*-algebras. Canad. Math. Bull. 25(1982), no. 3, 311–316. http://dx.doi.org/10.4153/CMB-1982-043-4 Google Scholar

[J3] Johnson, B. E., Near inclusions for subhomogeneous C^*–algebras. Proc. London Math. Soc. (3) 68(1994), no. 2, 399–422. http://dx.doi.org/10.1112/plms/s3-68.2.399 Google Scholar

[K]R. V. Kadison, , Irreducible operator algebras. Proc. Natl. Acad. Sci. U.S.A. 43(1957), 273–276. http://dx.doi.org/10.1073/pnas.43.3.273 Google Scholar

[KK] Kadison, R. V. and Kastler, D., Perturbations of von Neumann algebras. I. Stability of type. Amer. J. Math. 94(1972), 38–54. http://dx.doi.org/10.2307/2373592 Google Scholar

[KR]Kadison, R. V. and Ringrose, J., Fundamentals of the theory of operator algebras. Vol. II. Advanced theory. Pure and Applied Mathematics, 100, Academic Press, Orlando FL, 1986.Google Scholar

[Ki]Kirchberg, E., The derivation problem and the similarity problem are equivalent. J. Operator Theory 36(1996), no. 1, 59–62.Google Scholar

[Pe] Pedersen, G. K., C^*-algebras and their automorphism groups. London Mathematical Society Monographs, 14, Academic Press, London-New York, 1979.Google Scholar

[Ph]Phillips, J., Perturbations of C^*-algebras. Indiana Univ. Math. J. 23(1973/74), 1167–1176. http://dx.doi.org/10.1512/iumj.1974.23.23093 Google Scholar

[PhR1] Phillips, J. and Raeburn, I., Perturbations of AF-algebras. Canad. J. Math. 31(1979), no. 5, 1012–1016. http://dx.doi.org/10.4153/CJM-1979-093-8 Google Scholar

[PhR2] Phillips, J., Perturbations of C^*–algebras. II. Proc. London Math. Soc. (3) 43(1981), no. 1, 46–72. http://dx.doi.org/10.1112/plms/s3-43.1.46 Google Scholar

[RT]Raeburn, I. and Taylor, J. L., Hochschild cohomology and perturbations of Banach algebras. J. Functional Analysis 25(1977), no. 3, 258–266. http://dx.doi.org/10.1016/0022-1236(77)90072-6.Google Scholar

Access

Reply to: Submit a response

Title * Please enter a title for your response.

Contents * Please enter your response.

Your details

First name * Please enter your first name.

Last name * Please enter your last name.

Please enter a valid email address.

Occupation Please enter your occupation.

Affiliation Please enter any affiliation.

Conflicting interests

Do you have any conflicting interests? * Conflicting interests help

Yes No

More information * Please enter details of the conflict of interest or select 'No'.

Please tick the box to confirm you agree to our Terms of use. *

Please accept terms of use.

Please tick the box to confirm you agree that your name, comment and conflicts of interest (if accepted) will be visible on the website and your comment may be printed in the journal at the Editor’s discretion. *

Please confirm you agree that your details will be displayed.

Article contents

C*–algebras Nearly Contained in Type I Algebras

Abstract

Keywords

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

Conflicting interests