Abstract view
Centrevalued Index for Toeplitz Operators with Noncommuting Symbols


Published:20160513
Printed: Oct 2016
John Phillips,
Department of Mathematics and Statistics , University of Victoria, Victoria, B.C. V8W 2Y2
Iain Raeburn,
Department of Mathematics and Statistics , University of Otago, PO Box 56 , Dunedin 9054, New Zealand
Abstract
We formulate and prove a ``winding number'' index
theorem for certain ``Toeplitz'' operators in the same spirit
as GohbergKrein, Lesch and others. The ``number'' is replaced
by a selfadjoint 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 nonnegative 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 scalarvalued index and the centrevalued index fibers as
a section of the scalarvalued indices.
MSC Classifications: 
46L55, 19K56, 46L80 show english descriptions
Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20] Index theory [See also 58J20, 58J22] $K$theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
46L55  Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20] 19K56  Index theory [See also 58J20, 58J22] 46L80  $K$theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
