Abstract
We introduce determinant andL 2-analytic functions forn-tuples of commuting elements in a semifinite von Neumann algebra. Some fundamental properties of these functions are investigated.
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References
[BoS] J. Bolte and F. Steiner, Determinants of Laplace-like operators on Riemann surfaces, Commun. Math. Phys. 130 (1990), 581–597.
[Bro] L. Brown, Lidskii's theorem in the type II case, Geometric Method in Operator Algebras, Ed. H. Arak, E. Effors, Proc. of the US-Japan Seminar, Kyoto (1981), 1–34, Longman Sci. Tech. NY. 1986.
[BrK] L. Brown and H. Kosaki, Jensen's inequality in semifinite von Neumann algebras, J. Operator Theory, 23 (1990), 3–19.
[CaM] A. Carey and V. Mathai,L 2-analytic torsion invariants, J. Funct. Anal. 107 (1992), 369–386.
[CoM] A. Connes and H. Moscovici, TheL 2-index theorem for homogeneous spaces of Lie groups, Ann. Math. 115 (1982), 291–330.
[Dix] J. Dixmier, von Neumann Algebras, North-Holland Publ. Company, NY. 1981.
[Fac] T. Fack, Sur la notion de valuur caracteristique, J. Operator Theory, 7 (1982), 307–333.
[FaK] T. Fack and H. Kosaki, Generalizeds-numbers of τ-measurable operators, Pacif. J. Math. 123 (1986), 269–300.
[For] R. Forman, Functional determinants and geometry, Invent. Math. 87 (1987), 447–493.
[FuK] B. Fuglede and R. Kadison, Determinant theory in finite factors, Ann. Math. 55 (1952), 520–530.
[GK] I.C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monographs, 8, AMS, Providence, RI, 1969.
[Gon 1] D. Gong,L 2-analytic torisons, equivariant cyclic cohomology and the Novikov conjecture, thesis, SUNY at Stony Brook, 1992.
[GoP] D. Gong and J. Pincus, Torsion invariants for finite von Neumann algebras, to appear.
[Hej] D. A. Hejhal, The Selberg Trace Formula forPSL(2,ℝ), Vol. 2, Lecture Notes in Math., No. 1001, Springer, NY. 1983.
[HoP] E. D'Hoker and D. H. Phong, On determinants of Laplacians on Riemann surfaces, Commun. Math. Phys. 104 1986, 537–545.
[KKW] S. Klimek, W. Kondracki and M. Wisniowski, The ζ-determinant and functional analysis, Bull. Sci. Math. 113 (1989), 175–193.
[Lot] J. Lott, Heat kernel on covering spaces and topological invariants, J. Diff. Geom. 35 (1992), 471–510.
[LüR] W. Lück and M. Rothenberg, Reidemeister torsion and theK-theory of von Neumann algebras,K-Theory, 5 (1991), 213–264.
[MoS] H. Moscovici and R. Stanton, R-torsion and zeta functions for locally symmetric manifolds, Invent. Math. 105 (1991), 185–216.
[NoS] S. P. Novikov and M. Shubin, Morse inequalities and von Neumann invariants of non simplyconnected manifolds, Usp. Mat. Nauk. 41 (1986), 223.
[RaS] D. Ray and I. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 74 (1971), 145–210.
[SaR] P. Sarnak, Determinants of Laplacians, Commun. Math. Phys. 110 (1987), 113–120.
[Shu] M. Shubin, Pseudodifferential Operators and Spectral Theory, Springer, NY. 1987.
[Sim] B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Notes, No. 35, Cambridge Univ. Press, 1979.
[Vor] A. Voros, Spectral functions, special functions and the Selberg Zeta function, Commun. Math. Phys. 110 (1987), 439–465.