Triviality of outer derivations in lp(G) for one class of groups
- Authors: Arutyunov A.A.1, Naianzin A.V.2
-
Affiliations:
- V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences
- Moscow Institute of Physics and Technology (National Research University)
- Issue: Vol 70, No 3 (2024)
- Pages: 356-374
- Section: Articles
- URL: https://journals.rudn.ru/CMFD/article/view/41137
- DOI: https://doi.org/10.22363/2413-3639-2024-70-3-356-374
- EDN: https://elibrary.ru/PFEHJE
Cite item
Full Text
Abstract
In this paper, we study derivations in group rings completed by various types of norms. The main attention is paid to the class of groups in which conjugations act in a controlled manner in some sense. Using the method of identifying derivations and characters on some category, we obtain an alternative way of proving that for this class of groups all derivations are inner.
About the authors
A. A. Arutyunov
V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences
Author for correspondence.
Email: andronick.arutyunov@gmail.com
Moscow, Russia
A. V. Naianzin
Moscow Institute of Physics and Technology (National Research University)
Email: naianzin.av@phystech.edu
Dolgoprudny, Russia
References
- Арутюнов А.А. Алгебра дифференцирований в некоммутативных групповых алгебрах// Тр. МИАН. -2020.-308.- С. 28-41.-doi: 10.4213/tm4048.
- Арутюнов А.А., Мищенко А.С., Штерн А.И. Деривации групповых алгебр// Фундам. и прикл. мат.- 2016.- 21, № 6.-С. 65-78.
- Арутюнов А.А., Мищенко А.С. Гладкая версия проблемы Джонсона о деривациях групповых алгебр// Мат. сб.- 2019.- 210, № 6.- С. 3-29.- doi: 10.4213/sm9119.
- Наймарк М.А. Нормированные кольца.- М.: Физматлит, 2010.
- Alekseev A., Arutyunov A., Silvestrov S. On (σ,τ)-derivations of group algebra as category characters// В сб.: «Non-Commutative and Non-Associative Algebra and Analysis Structures». -Cham: Springer, 2023.- С. 81-99.-doi: 10.1007/978-3-031-32009-5_5.
- Arutyunov A. A combinatorial view on derivations in bimodules// Adv. Syst. Sci. Appl. -2023.-23, № 2. -С. 178-183.-doi: 10.25728/assa.2023.23.2.1408.
- Bader U., Gelander T., Monod N. A fixed point theorem for L1 spaces// Invent. Math. -2010.-189.- С. 143-148.
- Dales H.G. Banach Algebras and Automatic Continuity.- Oxford: Oxford Univ. Press, 2001.
- Deitmar A., Echterhoff S. Principles of Harmonic Analysis.-Cham, etc.: Springer, 2014.
- Hurley P., Hurley T. Codes from zero-divisors and units in group rings// Int. J. Inform. Coding Theory.- 2009.-1, № 1.-С. 57-87.- doi: 10.1504/IJICOT.2009.024047.
- Johnson B.E. Cohomology in Banach Algebras.-Providence: Am. Math. Soc., 1972.
- Johnson B.E. The derivation problem for group algebras of connected locally compact groups// J. London Math. Soc.- 2001.- 63, № 2.-С. 441-452.- doi: 10.1112/S002461070000185X.
- Johnson B.E., Ringrose J.R. Derivations of operator algebras and discrete group algebras// Bull. London Math. Soc.- 1969.- 1, № 1.- С. 70-74.-doi: 10.1112/blms/1.1.70.
- Johnson B.E., Sinclair A.M. Continuity of derivations and a problem of Kaplansky// Am. J. Math.- 1968.-90, № 4.- С. 1067-1073.-doi: 10.2307/2373290.
- Losert V. The derivation problem for group algebras// Ann. Math.- 2008.- 168, № 1.-С. 221-246.- doi: 10.4007/annals.2008.168.221.
- Mishchenko A.S. Derivations of group algebras and Hochschild cohomology// В сб.: «Differential Equations on Manifolds and Mathematical Physics: Dedicated to the Memory of Boris Sternin». - Cham: Birkh¨auser, 2021.-С. 263-272.-doi: 10.1007/978-3-030-37326-9_16.
- Mishchenko A.S. Geometric description of the Hochschild cohomology of group algebras//В сб.: «Topology, Geometry, and Dynamics: V.A. Rokhlin-Memorial».-Providence: Am. Math. Soc., 2021.- С. 267-280.
- Murphy G. Aspects of the theory of derivations// Banach Center Publ. - 1994.- 30, № 1.-С. 267-275.
- Murphy W. The nonvanishing first Hochschild cohomology of twisted finite simple group algebras// ArXiv. -2022.-2207.03698 [math.GR].
- Neumann B.H. Groups covered By permutable subsets// J. London Math. Soc.- 1954.- s1-29, № 2.- С. 236-248.-doi: 10.1112/jlms/s1-29.2.236.
- Palmer Th. W. Banach Algebras and the General Theory of *-Algebras: Volume 1: Algebras and Banach Algebras.- Cambridge: Cambridge University Press, 1994.
- Pierce R.S. Associative Algebras.- New York-Heidelberg-Berlin: Springer, 1982.
- Rickart C.E. General Theory of Banach Algebras.-Princeton: Van Nostrand, 1960.
- Sakai Sh. On a conjecture of Kaplansky// Tohoku Math. J.-1960.-12.- С. 31-33.
- Sakai Sh. Derivations of W*-algebras// Ann. Math.- 1966.- 83, № 2.- С. 273-279.-doi: 10.2307/1970432.
- Todea C.-C. Nontriviality of the first Hochschild cohomology of some block algebras of finite groups// ArXiv. -2022.-2206.09108 [math.KT].
