## Banach Journal of Mathematical Analysis

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

Marius Măntoiu

#### 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

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

#### 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

#### 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.