Operator algebras and approximate diagonals

Vestnik MGSU 9/2013
  • Myasnikov Aleksey Georgievich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoye shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 16-22

The author argues that two approaches dominate the study of amenable algebras, groups, modules, etc. They are the homological approach and the approach based on the invariance in respect of a particular group of transformations. In the latter case, an invariant mean serves as a convenient instrument. In particular, a mean is determined as a positive finitely additive measure which is identified using the algebra of all subsets of the group in question.In the first part of the article, the author introduces definitions of an inversely amenable module and an inversely amenable C* algebra. The criteria for the inverse amenability for C* algebras is formulated using virtual diagonals constructed with the help of means, which are invariant in respect of components of amenability in a certain space of limited functions. In the further part of the article, the author presents necessary and sufficient conditions of inverse amenability based on the existence of approximate diagonals. Unlike the standard approach applied to describe amenable Banach algebras, the above approach offers a set of invariant means that are more easily perceived by intuition.

DOI: 10.22227/1997-0935.2013.9.16-22

  1. Paterson A.L.T. Amenability. Providence, RI, AMS, 1988, 452 p.
  2. Myasnikov A.G. Amenable Banach L1(G)-modules, Invariant Means and Regularity in the Sense of Arens. Izvestiya vuzov. Matematika [News of Institutions of Higher Education. Mathematics] 1993, no. 37, pp. 69—77.
  3. Myasnikov A.G. Weak Amenability Components of L1(G)-modules, Amenable Groups and Ergodic Theorem. Mathematical Notes. 1999, no. 66, pp. 726—732.
  4. Myasnikov A.G. Amenable L1(G)-modules and amenable S*-algebras. Voprosy matematiki, mekhaniki sploshnykh sred i primeneniya matematicheskikh metodov v stroitelstve [Issues of Mathematics, Mechanics of Continuous Media and Application of Mathematical Methods in Civil Engineering]. Sb. nauchn. tr. [Collection of Research Works]. 2008, no. 11, pp. 101—119.
  5. Paterson A.L.T. Invariant Mean Characterizations of Amenable C*-algebras. Houston J. Math. 1991, vol.17, no. 4, pp. 551—565.
  6. Kaijser S., Sinclair A.M. Projective Tensor Products of C*-algebras. Math. Scand. 1984, no. 55, pp. 161—187.
  7. Greenleaf F.P. Invariant Means on Topological Groups and Their Applications. 1969, New York University, 113 p.
  8. Bodaghi A. Module Amenability of Banach Algebras. Lambert Academic Publishing, 2012, 168 p.
  9. Johnson B.E. Cohomology in Banach Algebras. Mem. Amer. Math. Soc. Providence, 1972, no. 127.
  10. Rosenberg J. Amenability of Crossed Products of C*-algebras. Commun. Math. Phys. 1977, no. 57, pp. 187—191.


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