Banach Journal of Mathematical Analysis

Symmetry and inverse closedness for Banach $^*$-algebras associated to discrete groups

Marius Măntoiu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A discrete group $\mathrm{G}$ is called rigidly symmetric if for every $C^*$-algebra $\mathcal{A}$ the projective tensor product $\ell^1(\mathrm{G})\widehat\otimes A$ is a symmetric Banach $^*$-algebra. For such a group we show that the twisted crossed product $\ell^1_{\alpha,\omega}(\mathrm{G};\mathcal{A})$ is also a symmetric Banach $^*$-algebra, for every twisted action $(\alpha,\omega)$ of $\mathrm{G}$ in a $C^*$-algebra $\mathcal{A}$. We extend this property to other types of decay, replacing the $\ell^1$-condition. We also make the connection with certain classes of twisted kernels, used in a theory of integral operators involving group $2$-cocycles. The algebra of these kernels is studied, both in intrinsic and in represented version.

Article information

Source
Banach J. Math. Anal., Volume 9, Number 2 (2015), 289-310.

Dates
First available in Project Euclid: 19 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1419001118

Digital Object Identifier
doi:10.15352/bjma/09-2-19

Mathematical Reviews number (MathSciNet)
MR3296119

Zentralblatt MATH identifier
1328.47079

Subjects
Primary: 47L65: Crossed product algebras (analytic crossed products)
Secondary: 22D15: Group algebras of locally compact groups 47D34 43A20: $L^1$-algebras on groups, semigroups, etc.

Keywords
Discrete group crossed product kernel symmetric Banach algebra weight

Citation

Măntoiu, Marius. Symmetry and inverse closedness for Banach $^*$-algebras associated to discrete groups. Banach J. Math. Anal. 9 (2015), no. 2, 289--310. doi:10.15352/bjma/09-2-19. https://projecteuclid.org/euclid.bjma/1419001118


Export citation

References

  • R. Balan, The noncommutative Wiener lemma, linear independence and spectral properties of the algebra of time-frequency shift operators, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3921–3941.
  • A.G. Baskakov, Wiener's theorem and asymptotic estimates for elements of inverse matrices, Funktsional. Anal. i Prilozhen., 2 (1990), no. 4, 222–224.
  • I. Beltiţă and D. Beltiţă, Inverse-closed algebras of integral operators on locally compact groups, Ann. H. Poincaré, (to appear), arXiv:1303.5346 [math.FA].
  • C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Inc. 1988.
  • F.P. Boca, Rotation $C^*$-algebras and almost Mathieu operators, Theta Series in Advanced Mathematics, 2001.
  • R.C. Busby and H.A. Smith, Representations of twisted group algebras, Trans. Amer. Math. Soc. 149 (1970), 503–537.
  • B. Farrell and T. Strohmer, Inverse-closedness of a Banach algebra of integral operators on the Heisenberg group, J. Operator Theory 64 (2010), no. 1, 189–205.
  • H.G. Feichtinger, Banach convolution algebras of Wiener type, in: Functions, series, operators, Proc. int. conf., Budapest 1980, Vol. I, Colloq. Math. Soc. Janos Bolyai 35 (1983), 509-524.
  • G. Fendler, K. Gröchenig and M. Leinert, Symmetry of weighted $L^1$-algebras and the GRS-condition, Bull. London Math. Soc. 38 (2006), no. 4, 625–635.
  • G. Fendler, K. Gröchenig and M. Leinert, Convolution-dominated operators on discrete groups, Integral equations Operator Theory 61 (2008), no. 4, 493–509.
  • I. Gohberg, M.A. Kaashoek and H.J. Woerdeman, The band method for positive and strictly contractive extension problems: an alternative version and new applications, Integral equations Operator Theory 12 (1989), no. 3, 343–382.
  • K. Gröchenig, Wiener's lemma: theme and variations. An introduction to spectral invariance, In B. Forster and P. Massopust, editors, Four short courses on harmonic analysis, Appl. Num. Harm. Anal. Birkhäuser, Boston, 2010.
  • K. Gröchenig and M. Leinert, Wiener's lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), no. 1, 1–18.
  • K. Gröchenig and M. Leinert, Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2695–2711.
  • K. Gröchenig and M. Leinert, Inverse-closed Banach subalgebras of higher-dimensional noncommutative tori, preprint arXiv.
  • M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), no. 1, 53–78.
  • A.K. Holzherr, Discrete groups whose multiplier representations are type I, J. Austral. Math. Soc. (Series A) 31 (1981), no. 4, 486–495.
  • A. Kleppner, Multiplier representations of discrete groups. Proc. Amer. Math. Soc. 88 (1983), no. 2, 371–375.
  • I. Krishtal, Wiener's lemma: Pictures at an exhibition, Rev. Un. Mat. Argentina 52 (2011), no. 2, 61–79.
  • W. Kugler, On the symmetry of generalized $L^1$-algebras, Math. Z. 168 (1979), no. 3, 241–262.
  • V.G. Kurbatov, Algebras of difference and integral operators, Funktsional. Anal. i Prilozhen. 24 (1990), 87–88.
  • H. Leptin, Verallgemeinerte $L^1$-Algebren und projektive Darstellungen lokal kompakter Gruppen, I, Invent. Math. 3 (1967), no. 4, 257–281.
  • H. Leptin and D. Poguntke, Symmetry and nonsymmetry for locally compact groups, J. Funct. Anal. 33 (1979), no. 2, 119–134.
  • M. Lindner, Fredholmness and index of operators in the Wiener algebra are independent of the underlying space, Oper. Matrices 2 (2008), no. 2, 297–306.
  • J. Packer and I. Raeburn, Twisted crossed products of $\,C^*$-algebras I, Math. Proc. Cambridge Phyl. Soc. 106 (1989), no. 2, 293–311.
  • J. Packer and I. Raeburn, Twisted crossed products of $\,C^*$-algebras II, Math. Ann. 287 (1990), no 1, 595–612.
  • T.W. Palmer, Banach algebras and the general theory of $^*$-algebras, Vol. I. Algebras and Banach algebras, Encyclopedia of Mathematics and its Applications, 49. Cambridge University Press, Cambridge, 1994.
  • T.W. Palmer, Banach algebras and the general theory of $^*$-algebras, Vol. 2. $*$-algebras, Encyclopedia of Mathematics and its Applications, 79. Cambridge University Press, Cambridge, 2001.
  • D. Poguntke, Rigidly symmetric $L^1$-group algebras, Sem. Sophus Lie 2 (1992), no. 2, 189–197.
  • D. Poguntke, Private communication.
  • J. Stewart and S. Watson, Which amalgams are convolution algebras?, Proc. Amer. Math. Soc. 93 (1985), no. 4, 621–627.
  • D. Williams, Crossed products of $\,C^*$-algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007.