Unbounded Fredholm Modules and Spectral Flow

Carey, Alan ; Phillips, John

Canadian journal of mathematics, Tome 50 (1998) no. 4, pp. 673-718

Résumé

An odd unbounded (respectively, $p$ -summable) Fredholm module for a unital Banach $*$ -algebra, $A$ , is a pair $(H,D)$ where $A$ is represented on the Hilbert space, $H$ , and $D$ is an unbounded self-adjoint operator on $H$ satisfying:(1) ${{(1+{{D}^{2}})}^{-1}}$ is compact (respectively, Trace $\left( {{\left( 1+{{D}^{2}} \right)}^{-(p/2)}} \right)<\infty $ , and(2) $\{a\in A|\,\,[D,a]\,\,\text{is}\,\,\text{bounded}\}\text{ }$ is a dense $*$ - subalgebra of $A$ .If $u$ is a unitary in the dense $*$ -subalgebra mentioned in (2) then $$uD{{u}^{*}}=D+u[D,{{u}^{*}}]=D+B$$ where $B$ is a bounded self-adjoint operator. The path $$D_{t}^{u}:=(1-t)D+tuD{{u}^{*}}=D+tB$$ is a “continuous” path of unbounded self-adjoint “Fredholm” operators. More precisely, we show that $$F_{t}^{u}:=D_{t}^{u}{{\left( 1+{{\left( D_{t}^{u} \right)}^{2}} \right)}^{-\frac{1}{2}}}$$ is a norm-continuous path of (bounded) self-adjoint Fredholm operators. The spectral flow of this path $\left\{ F_{t}^{u} \right\}$ (or $\{D_{t}^{u}\}$ ) is roughly speaking the net number of eigenvalues that pass through 0 in the positive direction as $t$ runs from 0 to 1. This integer, $$\text{sf}\left( \left\{ D_{t}^{u} \right\} \right):=\text{sf}\left( \left\{ F_{t}^{u} \right\} \right),$$ recovers the pairing of the $K$ -homology class $[D]$ with the $K$ -theory class $[u]$ .We use I.M. Singer's idea (as did E. Getzler in the $\theta $ -summable case) to consider the operator $B$ as a parameter in the Banach manifold, ${{B}_{\text{sa}}}\left( H \right)$ , so that spectral flow can be exhibited as the integral of a closed 1-form on this manifold. Now, for $B$ in our manifold, any $X\,\in \,{{T}_{B}}({{B}_{\text{sa}}}(H))$ is given by an $X$ in ${{B}_{\text{sa}}}(H)$ as the derivative at $B$ along the curve $t\,\mapsto \,B\,+\,tX$ in the manifold. Then we show that for $m$ a sufficiently large half-integer: $$\alpha (X)=\frac{1}{{{{\tilde{C}}}_{m}}}\text{Tr}\left( X{{\left( 1+{{\left( D+B \right)}^{2}} \right)}^{-m}} \right)$$ is a closed 1-form. For any piecewise smooth path $\left\{ {{D}_{t}}=D+{{B}_{t}} \right\}$ with ${{D}_{0}}$ and ${{D}_{1}}$ unitarily equivalent we show that $$\text{sf}\left( \left\{ {{D}_{t}} \right\} \right)=\frac{1}{{{{\tilde{C}}}_{m}}}\int_{\text{0}}^{\text{1}}{\text{T}}\text{r}\left( \frac{d}{dt}\left( {{D}_{t}} \right){{\left( 1+D_{1}^{2} \right)}^{-m}} \right)dt$$ the integral of the 1-form $\alpha $ . If ${{D}_{0}}$ and ${{D}_{1}}$ are not unitarily equivalent, wemust add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form: $$\text{sf}\left( \left\{ {{F}_{t}} \right\} \right)=\frac{1}{{{C}_{n}}}\int_{\text{0}}^{\text{1}}{\text{T}}\text{r}\left( \frac{d}{dt}\left( {{F}_{t}} \right){{\left( 1-{{F}_{t}}^{2} \right)}^{n}} \right)dt$$ for $n\ge \frac{p-1}{2}$ an integer. The unbounded case is proved by reducing to the bounded case via the map $D\mapsto F=D{{(1+{{D}^{2}})}^{-\frac{1}{2}}}$ We prove simultaneously a type II version of our results.

Détail

DOI : 10.4153/CJM-1998-038-x

Mots-clés : 46L80, 19K33, 47A30, 47A55

Bibliographie

[APS] Atiyah, M. F., Patodi, V. K. and Singer, I. M., Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambridge Philos. Soc. 79(1976), 71–99. Google Scholar

[ASS] Avron, J., Seiler, R. and Simon, B., The index of a pair of projections. J. Funct. Anal. 120(1994), 220–237. Google Scholar

[BF] Booß-Bavnbek, B. and Furutani, K., The Maslov Index: A Functional Analytic Definition and the Spectral Flow Formula. (1995), preprint. Google Scholar

[BW] B., Booß-Bavnbek and Wojciechowski, K. P., Elliptic Boundary Problems for Dirac Operators. Birkhäuser, Boston, Basel, Berlin, 1993. Google Scholar

[B1] Breuer, M., Fredholm theories in von Neumann Algebras, I. Math. Ann. 178(1968), 243–254. Google Scholar

[B2] Breuer, M., Fredholm theories in von Neumann Algebras, II. Math. Ann. 180(1969), 313–325. Google Scholar

[CP] Carey, A. L. and Phillips, J., Algebras almost commuting with Clifford algebras in a II∞ factor. K-Theory 4(1991), 445–478. Google Scholar

[C1] Connes, A., Noncommutative differential geometry. Publ. Inst. Hautes Études Sci. Publ. Math. 62(1985), 41–144. Google Scholar

[C2] Connes, A., Noncommutative Geometry. Academic Press, San Diego, 1994. Google Scholar

[CMX] Curto, R., Muhly, P. S. and Xia, J., Toeplitz operators on flows. J. Funct. Anal. 93(1990), 391–450 Google Scholar

[D] Dixmier, J., Formes Linéaires sur un Anneau d’Opérateurs. Bull. Soc. Math. France 81(1953), 9–39. Google Scholar

[DHK] Douglas, R. G., S.Hurder and Kaminker, J., Cyclic cocycles, renormalization and eta-invariants. Invent. Math. 103(1991), 101–179. Google Scholar

[DS] Dunford, N. and Schwartz, J. T., Linear Operators, Part II. Wiley, New York, London, 1963. Google Scholar

[FK] Fack, T. and Kosaki, H., Generalized s-numbers of τ-measurable operators. Pacific J. Math. 123(1986), 269–300. Google Scholar

[G] Getzler, E., The odd Chern character in cyclic homology and spectral flow. Topology 32(1993), 489–507. Google Scholar

[H] Hurder, S., Eta invariants and the odd index theorem for coverings. Contemporary Math. (2) 105(1990), 47–82. Google Scholar

[K] Kato, T., Perturbation Theory for Linear Operators. Springer-Verlag, New York, 1966. Google Scholar

[Kam] Kaminker, J., Operator algebraic invariants for elliptic operators. Proc. Symp. PureMath. (I) 51(1990), 307–314. Google Scholar

[L] Lesch, M., On the index of the infinitesimal generator of a flow. J. Operator Theory 26(1991), 73–92. Google Scholar

[M] Mathai, V., Spectral flow, eta invariants and von Neumann algebras. J. Funct.Anal. 109(1992), 442–456. Google Scholar

[Ped] Pedersen, G. K., C*-Algebras and Their Automorphism Groups. Academic Press, London, New York, San Francisco, 1979. Google Scholar

[Per] Perera, V. S., Real valued spectral flow. Contemporary Math. 185(1993), 307–318. Google Scholar

[P1] Phillips, J., Self-adjoint Fredholm operators and spectral flow. Canad.Math. Bull 39(1996), 460–467. Google Scholar

[P2] Phillips, J., Spectral flow in type I and II factors—A new approach. Fields Inst. Comm., Cyclic Cohomology &amp; Noncommutative Geometry, 17(1997), 137–153. Google Scholar

[PR] Phillips, J. and Raeburn, I. F., An index theorem for Toeplitz operators with noncommutative symbol space. J. Funct. Anal. 120(1994), 239–263. Google Scholar

[RS] Reed, M. and Simon, B., Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, New York, San Francisco, London, 1978. Google Scholar

[S] Spivak, M., A Comprehensive Introduction to Differential Geometry. vol. I, 2nd ed., Publish or Perish Inc., Berkeley, 1979. Google Scholar

[Si] Singer, I. M., Eigenvalues of the Laplacian and Invariants of Manifolds. Proc. International Congress I, Vancouver, 1974. 187–200. Google Scholar

[Su] Sukochev, F. A., Perturbation Estimates for a Certain Operator-Valued Function. Flinders University, 1998. preprint. Google Scholar