Skip to main content Accesibility Help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
Cancel
×
×
Home
  • Home
Article

Elements of $C^{\ast }$ -algebras Attaining their Norm in a Finite-dimensional Representation

  • Kristin Courtney (a1) and Tatiana Shulman (a2)
Abstract

We characterize the class of RFD $C^{\ast }$ -algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^{\ast }$ -algebra is finite-dimensional, which is equivalent to the $C^{\ast }$ -algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^{\ast }$ -algebras whose norms in finite-dimensional representations fit certain prescribed properties.

Copyright
© Canadian Mathematical Society 2017 
Footnotes
Hide All

The research of author T. S. was supported by the Polish National Science Centre grant under the contract number DEC-2012/06/A/ST1/00256 and by the Eric Nordgren Research Fellowship Fund at the University of New Hampshire.

Footnotes
References
Hide All
[1] Akemann, C. A. and Pedersen, G. K., Ideal perturbation of elements in C ∗-algebras . Math. Scand. 41(1977), 117–139. https://doi.org/10.7146/math.scand.a-11707.
[2] Archbold, R. J., On residually finite-dimensional C ∗-algebras . Proc. Amer. Math. Soc. 123(1995), no. 9, 2935–2937. https://doi.org/10.2307/2160599.
[3] Bekka, B., Operator superrigidity for SL n (ℤ), n⩾3 . Invent. Math. 169(2007), no. 2, 401–425. https://doi.org/10.1007/s00222-007-0050-5.
[4] Blackadar, B., Shape theory for C ∗-algebras . Math. Scand. 56(1985), 249–275. https://doi.org/10.7146/math.scand.a-12100.
[5] Blackadar, B., Operator algebras. Theory of C ∗-algebras and von Neumann algebras. Operator Algebras and Non-commutative Geometry. III. Encyclopaedia of Mathematical Sciences, 122, Springer-Verlag, Berlin, 2006. https://doi.org/10.1007/3-540-28517-2.
[6] Brown, N. P. and Ozawa, N., C ∗-algebras and finite-dimensional approximations. Graduate Studies in Mathematics, 88, American Mathematical Society, Providence, RI, 2008. https://doi.org/10.1090/gsm/088.
[7] Choi, M. D., The full C ∗-algebra of the free group on two generators . Pacific J. Math. 87(1980), no. 1, 41–48. https://doi.org/10.2140/pjm.1980.87.41.
[8] Davidson, K. R., C ∗-algebras by example. Fields Institute Monograph, 6, American Mathematical Society, 1996. https://doi.org/10.1090/fim/006.
[9] Dixmier, J., C ∗-algebras. North-Holland Mathematical Library, 15, North Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
[10] Eilers, S. and Exel, R., Finite-dimensional representations of the soft torus . Proc. Amer. Math. Soc. 130(2002), no. 3, 727–731. https://doi.org/10.1090/S0002-9939-01-06150-0.
[11] Exel, R. and Loring, T., Finite-dimensional representations of free product C ∗-algebras . Internat. J. Math. 3(1992), no. 4, 469–476. https://doi.org/10.1142/S0129167X92000217.
[12] Fritz, T., Netzer, T., and Thom, A., Can you compute the operator norm? Proc. Amer. Math. Soc. 142(2014), 4265–4276. https://doi.org/10.1090/S0002-9939-2014-12170-8.
[13] Glimm, J., Type I C ∗-algebras . Ann. of Math. 73(1961), 572–612. https://doi.org/10.2307/1970319.
[14] Goodearl, K. R. and Menal, P., Free and residually finite-dimensional C ∗-algebras . J. Funct. Anal. 90(1990), 391–410. https://doi.org/10.1016/0022-1236(90)90089-4.
[15] Grigorchuk, R., Musat, M., and Rørdam, M., Just-infinite C ∗-algebras. arxiv:1604.08774.
[16] Hadwin, D., A lifting characterization of RFD C ∗-algebras . Math. Scand. 115(2014), no. 1, 85–95. https://doi.org/10.7146/math.scand.a-18004.
[17] Hadwin, D. and Shulman, T., Stability of group relations under small Hilbert-Schmidt perturbations. arxiv:1706.08405.
[18] Korchagin, A., Amalgamated free products of commutative C ∗-algebras are residually finite-dimensional . J. Operator Theory 71(2014), no. 2, 507–515. https://doi.org/10.7900/jot.2012jul03.1986.
[19] Loring, T. A., Lifting solutions to perturbing problems in C ∗-algebras. Fields Institute Monographs, 8, American Mathematical Society, Providence, RI, 1997.
[20] Loring, T. and Pedersen, G. K., Projectivity, transitivity, and AF-telescopes . Trans. Amer. Math. Soc. 350(1998), 4313–4339. https://doi.org/10.1090/S0002-9947-98-02353-8.
[21] Loring, T. and Shulman, T., Lifting algebraic contractions in C ∗-algebras . Oper. Theory Adv. Appl. 233(2014), 85–92.
[22] Lubotzky, A. and Shalom, Y., Finite representations in the unitary dual and Ramanujan groups. Discrete geometric analysis: proceedings of the first JAMS Symposium on Discrete Geometric Analysis (Sendai, Japan, 2002). Contemp. Math., 347, American Mathematical Society, Providence, RI, 2004, pp. 173–189. https://doi.org/10.1090/conm/347/06272.
[23] Moore, C. C., Groups with finite-dimensional irreducible representations . Trans. Amer. Math. Soc. 166(1972), 401–410. https://doi.org/10.2307/1996058.
[24] Sakai, S., C ∗-algebras and W ∗-algebras. Classics in Mathematics, Springer-Verlag, Berlin, 1971. https://doi.org/10.1007/978-3-642-61993-9.
[25] Sakai, S., A characterisation of type I C ∗-algebras . Bull. Amer. Math. Soc. 72(1966), 508–512. https://doi.org/10.1090/S0002-9904-1966-11520-3.
[26] Thom, A., Convergent sequences in discrete groups . Canad. Math. Bull. 56(2013), 424–433. https://doi.org/10.4153/CMB-2011-155-3.
[27] Thoma, E., Über unitäre Darstellungen abzählbarer diskreter Gruppen . Math. Ann. 153(1964), 111–138. https://doi.org/10.1007/BF01361180.
[28] Thoma, E., Ein Charakterisierung diskreter Gruppen vom Typ I . Invent. Math. 6(1968), 190–196. https://doi.org/10.1007/BF01404824.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

CAPTCHA *

Skip to the audio challenge
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed

Cancel
Confirm
×