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Centre-valued Index for Toeplitz Operators with Noncommuting Symbols

  • John Phillips (a1) and Iain Raeburn (a2)


We formulate and prove a “winding number” index theorem for certain “Toeplitz” operators in the same spirit as Gohberg–Krein, Lesch and others. The “number” is replaced by a self-adjoint operator in a subalgebra $Z\,\subseteq \,Z\left( A \right)$ of a unital $C*$ -algebra, $A$ . We assume a faithful $Z$ -valued trace $\tau $ on $A$ left invariant under an action $\alpha :\,\text{R}\,\to \,\text{Aut}\left( A \right)$ leaving $Z$ pointwise fixed. If $\delta $ is the infinitesimal generator of $\alpha $ and $u$ is invertible in $\text{dom}\left( \delta \right)$ , then the “winding operator” of $u$ is $\frac{1}{2\pi i}\tau \left( \delta \left( u \right){{u}^{-1}} \right)\,\in \,{{Z}_{sa}}$ . By a careful choice of representations we extend $\left( A,\,Z,\,\tau ,\,\alpha \right)$ to a von Neumann setting $\left( \mathfrak{A},\,\mathfrak{Z},\,\bar{\tau },\,\bar{\alpha } \right)$ where $\mathfrak{A}\,=\,{A}''$ and $\mathfrak{Z}\,=\,{Z}''$ . Then $A\,\subset \,\mathfrak{A}\,\subset \,\mathfrak{A}\,\rtimes \,\mathbf{R}$ , the von Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$ -trace on $\mathfrak{A}\,\rtimes \,\mathbf{R}$ . If $P$ is the projection in $\mathfrak{A}\,\rtimes \,\mathbf{R}$ corresponding to the non-negative spectrum of the generator of $\mathbf{R}$ inside $\mathfrak{A}\,\rtimes \,\mathbf{R}$ and $\tilde{\pi }:\,A\,\to \,\mathfrak{A}\,\rtimes \,\mathbf{R}$ is the embedding, then we define ${{T}_{u}}\,=\,P\tilde{\pi }\left( u \right)P\,\text{for}\,u\,\in \,{{A}^{-1}}$ and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$ -valued index of ${{T}_{u}}$ is the negative of the winding operator. In outline the proof follows that of the scalar case done previously by the authors. The main difficulty is making sense of the constructions with the scalars replaced by $\mathfrak{Z}$ in the von Neumann setting. The construction of the dual $\mathfrak{Z}$ -trace on $\mathfrak{A}\,\rtimes \,\mathbf{R}$ requires the nontrivial development of a $\mathfrak{Z}$ -Hilbert algebra theory. We show that certain of these Fredholm operators fiber as a “section” of Fredholm operators with scalar-valued index and the centre-valued index fibers as a section of the scalar-valued indices.



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[AP] Anderson, J. andPaschke, W., The rotation algebra. Houston J. Math. 15(1989), 126.
[Arv] Arveson, W. , Subalgebras of C*-algebras. Acta Math. 123(1969), 141224. http://dx.doi.Org/10.1007/BF02392388
[Brl] Breuer, M., Fredholm theories in von Neumann algebras, I. Math. Ann. 178(1968), 243254. http://dx.doi.Org/10.1007/BF01350663
[Br2] Breuer, M., Fredholm theories in von Neumann algebras, II. Math. Ann., 180(1969), 313325. http://dx.doi.Org/10.1016/0022-1236(90)90133-6
[Co] Connes, A., Noncommutative differential geometry. Publ. Math. Inst. Hautes Etudes Sci. 62 (1985), 41144.
[CMX] Curto, R., Muhly, P. S., and Xia, J., Toeplitz operators onflows. J. Funct. Anal. 93(1990), 391450.
[Dix] Dixmier, J., Les algèbres d'opérateurs dans l'espace hilbertien (Algèbres de von Neumann), Gauthier-Villars, Paris, 1969.
[DM] H.|Dym and H. P.|McKean, Fourier series and integrals. Academic Press, New York, 1972.
[H] Hôrmander, L., The Weyl calculus of pseudodifferential operators. Comm. Pure Appl. Math. 32(1979), 359443.
[Ji] Ji, R., Toeplitz operators on noncommutative tori and their real-valued index.Proc. Sympos. Pure Math. 51, Amer. Math. Soc. Providence, RI, (1990), pp. 153158.
[K] Kaplansky, I., Modules over operator algebras. Amer. J. Math. 75(1953), 839858.
[L] Lance, E. C., Hilbert C*-modules. London Math. Soc. Lecture Notes Series 210, Cambridge University Press, Cambridge, 1995.
[Le] Lesch, M., On the index of the infinitesimal generator of a flow. J. Operator Theory 26(1991), 7392.
[PR] Packer, J. A. and Raeburn, I., On the structure of twisted group C* -algebras. Trans. Amer. Math. Soc. 334(1992), 685718.
[Pa] Paschke, W., Inner product modules over B* -algebras. Trans. Amer. Math. Soc, 182(1973), 443468.
[Ped] Pedersen, G. K., C*-algebras and their automorphism groups. Academic Press, London, 1979.
[Ph] Phillips, J., Spectral flow in type I and II factors-a new approach. Fields Inst. Commun., 17(1997), 137153.
[PhR] Phillips, J. and Raeburn, I., An index theorem for Toeplitz operators with noncommutative symbol space. J. Funct. Anal. 120(1994), 239263. http://dx.doi.Org/10.1006/jfan.1994.1032
[R] Rieffel, M., Morita equivalence for C* -algebras and W*-algebras. J. Pure Appl. Algebra 5(1974), 5196. http://dx.doi.Org/10.1016/0022-4049(74)90003-6
[T] Tomiyama, J., On the projection of norm one in W*-algebras. Proc. Japan Acad. Ser. A Math.Sci. 33(1957), 608612. http://dx.doi.Org/10.3792/pja/1195524885
[U] Umegaki, H., Conditional expectation in an operator algebra I. Tohôku Math. J. 6(1954), 358362.
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