Document Type : Research Paper

Author

Department of mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Abstract

‎This is a survey of a variety of equivariant (co)homology theories for operator algebras‎. ‎We briefly discuss a background on equivariant Hochschild cohomology‎. ‎We discuss a notion of equivariant $L^2$-cohomology and equivariant $L^2$-Betti numbers for subalgebras of a von Neumann algebra‎. ‎For graded $C^*$-algebras (with grading over a group) we elaborate on a notion of graded $L^2$-cohomology and its relation to equivariant $L^2$-cohomology‎.

Keywords

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###### ##### References
[1] M. Atiyah, F. Bott, On the periodicity theorem for complex vector bundles, Acta Math. 112 (1964), 229-247.
[2] M. Atiyah, F. Bott, Periodicity and the index of elliptic operators, The Quarterly J. Math. 19 (1) (1968), 113-140.
[3] S. K. Berberian, The maximal ring of quotients of a finite von Neumann algebra, Rocky Mountain J. Math. 12 (1) (1982), 149-164.
[4] N. Berline, M. Vergne, Fourier transforms of orbits of the coadjoint representation, Representation Theory of Reductive Groups. 40 (1983), 53-67.
[5] A. Borel, Seminar on Transformation Groups, Annals of Mathematics Studies, Princeton University Press, Princeton, 1960.
[6] H. Cartan, Notions de algebre differentielle; application aux groupes de Lie et aux varietes ou opere un groupe de Lie, Colloque de Topologie. 2 (1950), 15-27.
[7] A. Connes, Correspondences, unpublished.
[8] A. Connes, V. Jones, Property T for von Neumann algebras, Bull. London Math. Soc. 17 (1985), 57-62.
[9] A. Connes, D. Shlyakhtenko, L2-homology for von Neumann algebras, Reine Angew. Math. 586 (2005), 125-168.
[10] J. Dixmier, Von Neumann Algebras. North-Holland, Amsterdam, 1981.
[11] R. Exel, Amenability for Fell bundles, Reine Angew. Math. 492 (1997), 41-73.
[12] Y. Felix, S. Halperin, J. C. Thomas, Rational Homotopy Theory, Springer-Verlag, New York, 2001.
[13] G. Hochschild, On the cohomology groups of an associative algebra, Annals of Mathematics. 46 (1) (1945), 58-67.
[14] K. K. Jensen, Foundations of an equivariant cohomology theory of Banach algebras I, Advances in Math. 117 (1996), 52-146.
[15] K. K. Jensen, Foundations of an equivariant cohomology theory of Banach algebras II, Advances in Math. 147 (1999), 173-259.
[16] W. Luck, L2-Invariants: Theory and Applications to Geometry and K-Theory, Springer-Verlag, Berlin, 2002.
[17] F. J. Murray, J. von Neumann, On rings of operators, Ann. of Math. 37 (2) (1936), 116-229.
[18] E. Nelson, Notes on non-commutative integration, Journal of Funct. Anal. 15 (1974), 103-116.
[19] H. Poincare, Analysis situs, Journal de lEcole Polytechnique. 1 (1895), 1-123.
[20] I. Raeburn, On graded C-algebras, Bull. Aust. Math. Soc. 97 (2018), 127-132.
[21] A. Thom, L2-cohomology for von Neumann algebras, Geom. Funct. Anal. 18 (2008), 251-270.
[22] L. W. Tu, What is...equivariant cohomology?, Notices of the AMS, 58 (3) (2011), 423-426.
[23] M. Vergne, Applications of equivariant cohomology, Eur. Math. Soc. I (2007), 635-664.
[24] Y. Utumi, On quotient rings, Osaka Math. J. 8 (1956), 1-18.
[25] E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692.