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On amenability of group algebras, I

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We study amenability of algebras and modules (based on the notion of almost-invariant finite-dimensional subspace), and apply it to algebras associated with finitely generated groups.

We show that a group G is amenable if and only if its group ring \( \mathbb{K} \) G is amenable for some (and therefore for any) field \( \mathbb{K} \).

Similarly, a G-set X is amenable if and only if its span \( \mathbb{K} \) X is amenable as a \( \mathbb{K} \) G-module for some (and therefore for any) field \( \mathbb{K} \).

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  1. G. Elek, The amenability of affine algebras, Journal of Algebra 264 (2003), 469–478. MR MR1981416 (2004d:16043)

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Erschler, On isoperimetric profiles of finitely generated groups, Geometriae Dedicata 100 (2003), 157–171. MR MR2011120 (2004j:20087)

    Article  MATH  MathSciNet  Google Scholar 

  3. R. I. Grigorchuk and A. Żuk, The lamplighter group as a group generated by a 2-state automaton, and its spectrum, Geometriae Dedicata 87 (2001), 209–244. MR MR1866850 (2002j:60009)

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Gromov, Asymptotic invariants of infinite groups, in Geometric group theory, Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR MR1253544 (95m:20041)

    Google Scholar 

  5. M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I, Mathematical Physics, Analysis and Geometry 2 (1999), 323–415. MR MR1742309 (2001j:37037)

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Grünbaum, Convex Polytopes, second ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003, Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR MR1976856 (2004b:52001)

    Google Scholar 

  7. M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Annals of Mathematics. Second Series 154 (2001), 115–138. MR MR1847590 (2002e:20010)

    MATH  MathSciNet  Google Scholar 

  8. J. von Neumann, Zur allgemeinen theorie des masses, Fundamenta Mathematicae 13 (1929), no. 1, 73–116 and 333, Collected works, vol. I, pages 599–643. MR MR1847590 (2002e:20010)

    MATH  Google Scholar 

  9. A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR MR961261 (90e:43001)

    MATH  Google Scholar 

  10. J. M. Rosenblatt, A generalization of Følner’s condition, Mathematica Scandinavica 33 (1973), 153–170. MR MR0333068 (48 #11393)

    MATH  MathSciNet  Google Scholar 

  11. R. Schneider, On Steiner points of convex bodies, Israel Journal of Mathematics 9 (1971), 241–249. MR MR0278187 (43 #3918)

    Article  MATH  MathSciNet  Google Scholar 

  12. N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR MR1218884 (95f:43008)

    Google Scholar 

  13. A. Vershik, Amenability and approximation of infinite groups, Selecta Mathematica Sovietica 2 (1982), no. 4, 311–330, Selected translations. MR MR721030 (86g:43006)

    MATH  MathSciNet  Google Scholar 

  14. S. Wagon, The Banach-Tarski Paradox, Cambridge University Press, Cambridge, 1993, With a foreword by Jan Mycielski, Corrected reprint of the 1985 original. MR MR1251963 (94g:04005)

    Google Scholar 

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Correspondence to Laurent Bartholdi.

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Bartholdi, L. On amenability of group algebras, I. Isr. J. Math. 168, 153–165 (2008).

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