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$B^p_r(F_n)$ HAS NO NONTRIVIAL IDEMPOTENTS
Part of:
Abstract harmonic analysis
Topological algebras, normed rings and algebras, Banach algebras
Published online by Cambridge University Press: 06 October 2022
Abstract
We show that there is no nontrivial idempotent in the reduced group 
$\ell ^p$-operator algebra 
$B^p_r(F_n)$ of the free group 
$F_n$ on n generators for each positive integer n.
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 108 , Issue 1 , August 2023 , pp. 142 - 149
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
The second author is supported by NSFC (No. 12171156).
References
Baum, P., Connes, A. and Higson, N., ‘Classifying space for proper actions and
$K$
-theory of group
${C}^{\ast }$
-algebras’, in:
${C}^{\ast }$
-Algebras: 1943–1993, Contemporary Mathematics, 167 (ed. Doran, R. S.) (American Mathematical Society, Providence, RI, 1994), 240–291.Google Scholar
$K$
-theory of group
${C}^{\ast }$
-algebras’, in:
${C}^{\ast }$
-Algebras: 1943–1993, Contemporary Mathematics, 167 (ed. Doran, R. S.) (American Mathematical Society, Providence, RI, 1994), 240–291.Google ScholarCohen, J. M. and Figà-Talamanca, A., ‘Idempotents in the reduced
${C}^{\ast }$
-algebra of a free group’, Proc. Amer. Math. Soc. 103(3) (1988), 779–782.Google Scholar
${C}^{\ast }$
-algebra of a free group’, Proc. Amer. Math. Soc. 103(3) (1988), 779–782.Google ScholarConnes, A., ‘Noncommutative differential geometry’, Publ. Math. Inst. Hautes Études Sci. 62 (1985), 41–144.CrossRefGoogle Scholar
Connes, A. and Moscovici, H., ‘Cyclic cohomology, the Novikov conjecture and hyperbolic groups’, Topology 29(3) (1990), 345–388.CrossRefGoogle Scholar
de la Harpe, P., ‘Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint’, C. R. Math. Acad. Sci. Paris 307(14) (1988), 771–774.Google Scholar
Gardella, E. and Thiel, H., ‘Representations of
$p$
-convolution algebras on
${L}_q$
-spaces’, Trans. Amer. Math. Soc. 371(3) (2019), 2207–2236.CrossRefGoogle Scholar
$p$
-convolution algebras on
${L}_q$
-spaces’, Trans. Amer. Math. Soc. 371(3) (2019), 2207–2236.CrossRefGoogle ScholarGardella, E. and Thiel, H., ‘Isomorphisms of algebras of convolution operators’, Ann. Sci. Éc. Norm. Supér. (4), to appear.Google Scholar
Haagerup, U., ‘An example of a nonnuclear
${C}^{\ast }$
-algebra, which has the metric approximation property’, Invent. Math. 50(3) (1978), 279–293.CrossRefGoogle Scholar
${C}^{\ast }$
-algebra, which has the metric approximation property’, Invent. Math. 50(3) (1978), 279–293.CrossRefGoogle ScholarHejazian, S. and Pooya, S., ‘Simple reduced
${L}_p$
-operator crossed products with unique trace’, J. Operator Theory 74(1) (2015), 133–147.CrossRefGoogle Scholar
${L}_p$
-operator crossed products with unique trace’, J. Operator Theory 74(1) (2015), 133–147.CrossRefGoogle ScholarHigson, N. and Kasparov, G.. ‘
$E$
-theory and
${KK}$
-theory for groups which act properly and isometrically on Hilbert space’, Invent. Math. 144(1) (2001), 23–74.CrossRefGoogle Scholar
$E$
-theory and
${KK}$
-theory for groups which act properly and isometrically on Hilbert space’, Invent. Math. 144(1) (2001), 23–74.CrossRefGoogle ScholarJolissaint, P., ‘Rapidly decreasing functions in reduced
${C}^{\ast }$
-algebras of groups’, Trans. Amer. Math. Soc. 317(1) (1990), 167–196.Google Scholar
${C}^{\ast }$
-algebras of groups’, Trans. Amer. Math. Soc. 317(1) (1990), 167–196.Google ScholarLafforgue, V., ‘A proof of property (RD) for cocompact lattices of
${\mathrm{SL}}(3,\mathbb{R})$
and
${\mathrm{SL}}(3,\mathbb{C})$
’, J. Lie Theory 10(2) (2000), 255–267.Google Scholar
${\mathrm{SL}}(3,\mathbb{R})$
and
${\mathrm{SL}}(3,\mathbb{C})$
’, J. Lie Theory 10(2) (2000), 255–267.Google ScholarLafforgue, V., ‘
$K$
-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes’, Invent. Math. 149(1) (2002), 1–95.CrossRefGoogle Scholar
$K$
-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes’, Invent. Math. 149(1) (2002), 1–95.CrossRefGoogle ScholarLiao, B. and Yu, G.,
$`K$
-theory of group Banach algebras and Banach property RD’, Preprint, 2017, arXiv:1708.01982.Google Scholar
$`K$
-theory of group Banach algebras and Banach property RD’, Preprint, 2017, arXiv:1708.01982.Google ScholarMineyev, I. and Yu, G., ‘The Baum–Connes conjecture for hyperbolic groups’, Invent. Math. 149(1) (2002), 97–122.CrossRefGoogle Scholar
Phillips, N. C., ‘Crossed products of
${L}_p$
operator algebras and the
$K$
-theory of Cuntz algebras on
${L}_p$
spaces’, Preprint, 2013, arXiv:1309.6406.Google Scholar
${L}_p$
operator algebras and the
$K$
-theory of Cuntz algebras on
${L}_p$
spaces’, Preprint, 2013, arXiv:1309.6406.Google ScholarPhillips, N. C., ‘Simplicity of reduced group Banach algebras’, Preprint, 2019, arXiv:1909.11278.Google Scholar
Phillips, N. C., ‘Open problems related to operator algebras on
${L}_p$
spaces’. https://tinyurl.com/phillips-pdf.Google Scholar
${L}_p$
spaces’. https://tinyurl.com/phillips-pdf.Google ScholarPimsner, M. and Voiculescu, D., ‘
$K$
-groups of reduced crossed products by free groups’, J. Operator Theory 8(1) (1982), 131–156.Google Scholar
$K$
-groups of reduced crossed products by free groups’, J. Operator Theory 8(1) (1982), 131–156.Google ScholarPuschnigg, M., ‘The Kadison–Kaplansky conjecture for word-hyperbolic groups’, Invent. Math. 149(1) (2002), 153–194.CrossRefGoogle Scholar
Schweitzer, L. B., ‘A short proof that
${M}_n(A)$
is local if
$A$
is local and Fréchet’, Internat. J. Math. 3(4) (1992), 581–589.CrossRefGoogle Scholar
${M}_n(A)$
is local if
$A$
is local and Fréchet’, Internat. J. Math. 3(4) (1992), 581–589.CrossRefGoogle Scholar