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Search: MSC category 19K56 ( Index theory [See also 58J20, 58J22] )

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1. CJM Online first

Phillips, John; Raeburn, Iain
Centre-valued Index for Toeplitz Operators with Noncommuting Symbols
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 $Z\subseteq Z(A)$ of a unital $C^*$-algebra, $A$. We assume a faithful $Z$-valued trace $\tau$ on $A$ left invariant under an action $\alpha:{\mathbf R}\to Aut(A)$ leaving $Z$ pointwise fixed.If $\delta$ is the infinitesimal generator of $\alpha$ and $u$ is invertible in $\operatorname{dom}(\delta)$ then the ``winding operator'' of $u$ is $\frac{1}{2\pi i}\tau(\delta(u)u^{-1})\in Z_{sa}.$ By a careful choice of representations we extend $(A,Z,\tau,\alpha)$ to a von Neumann setting $(\mathfrak{A},\mathfrak{Z},\bar\tau,\bar\alpha)$ where $\mathfrak{A}=A^{\prime\prime}$ and $\mathfrak{Z}=Z^{\prime\prime}.$ Then $A\subset\mathfrak{A}\subset \mathfrak{A}\rtimes{\bf R}$, the von Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$-trace on $\mathfrak{A}\rtimes{\bf R}$. If $P$ is the projection in $\mathfrak{A}\rtimes{\bf R}$ 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 for $u\in A^{-1}$, $T_u=P\tilde\pi(u) P$ 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 the proof 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 $\mathfrak{A}\rtimes{\mathbf R}$ required 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.

Keywords:index ,Toeplitz operator
Categories:46L55, 19K56, 46L80

2. CJM 2005 (vol 57 pp. 225)

Booss-Bavnbek, Bernhelm; Lesch, Matthias; Phillips, John
Unbounded Fredholm Operators and Spectral Flow
We study the gap (= ``projection norm'' = ``graph distance'') topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

Categories:58J30, 47A53, 19K56, 58J32

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