1. CJM 2016 (vol 68 pp. 1023)
 Phillips, John; Raeburn, Iain

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

2. CJM 2005 (vol 57 pp. 225)
 BoossBavnbek, 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) selfadjoint 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 
