126. CJM 2001 (vol 53 pp. 1031)
 Sampson, G.; Szeptycki, P.

The Complete $(L^p,L^p)$ Mapping Properties of Some Oscillatory Integrals in Several Dimensions
We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot
y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself
for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l}
{a_l(1\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=xy^{r}$,
$0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$
are indicated.
Categories:42B20, 46B70, 47G10 

127. CJM 2001 (vol 53 pp. 809)
 Robertson, Guyan; Steger, Tim

Asymptotic $K$Theory for Groups Acting on $\tA_2$ Buildings
Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where
$\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely
on the affine BruhatTits building $\mathcal{B}$ of $G$ and there is an
induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed
product $C^*$algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$
depends only on $\Gamma$ and is classified by its $K$theory. This article
shows how to compute the $K$theory of $\mathcal{A}(\Gamma)$ and of the
larger class of rank two CuntzKrieger algebras.
Keywords:$K$theory, $C^*$algebra, affine building Categories:46L80, 51E24 

128. CJM 2001 (vol 53 pp. 565)
 Hare, Kathryn E.; Sato, Enji

Spaces of Lorentz Multipliers
We study when the spaces of Lorentz multipliers from $L^{p,t}
\rightarrow L^{p,s}$ are distinct. Our main interest is the case when
$s
Keywords:multipliers, convolution operators, Lorentz spaces, Lorentzimproving multipliers Categories:43A22, 42A45, 46E30 

129. CJM 2001 (vol 53 pp. 546)
 Erlijman, Juliana

MultiSided Braid Type Subfactors
We generalise the twosided construction of examples of pairs of
subfactors of the hyperfinite II$_1$ factor $R$ in [E1]which arise
by considering unitary braid representations with certain
propertiesto multisided pairs. We show that the index for the
multisided pair can be expressed as a power of that for the
twosided pair. This construction can be applied to the natural
exampleswhere the braid representations are obtained in connection
with the representation theory of Lie algebras of types $A$, $B$, $C$,
$D$. We also compute the (first) relative commutants.
Category:46L37 

130. CJM 2001 (vol 53 pp. 592)
 Perera, Francesc

Ideal Structure of Multiplier Algebras of Simple $C^*$algebras With Real Rank Zero
We give a description of the monoid of Murrayvon Neumann equivalence
classes of projections for multiplier algebras of a wide class of
$\sigma$unital simple $C^\ast$algebras $A$ with real rank zero and stable
rank one. The lattice of ideals of this monoid, which is known to be
crucial for understanding the ideal structure of the multiplier
algebra $\mul$, is therefore analyzed. In important cases it is shown
that, if $A$ has finite scale then the quotient of $\mul$ modulo any
closed ideal $I$ that properly contains $A$ has stable rank one. The
intricacy of the ideal structure of $\mul$ is reflected in the fact
that $\mul$ can have uncountably many different quotients, each one
having uncountably many closed ideals forming a chain with respect to
inclusion.
Keywords:$C^\ast$algebra, multiplier algebra, real rank zero, stable rank, refinement monoid Categories:46L05, 46L80, 06F05 

131. CJM 2001 (vol 53 pp. 631)
 Walters, Samuel G.

KTheory of NonCommutative Spheres Arising from the Fourier Automorphism
For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$
(containing the rationals) it is shown that the group $K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where
$A_\theta$ is the rotation C*algebra generated by unitaries $U$, $V$
satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier
automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) =
U^{1}$. More precisely, an explicit basis for $K_0$ consisting of
nine canonical modules is given. (A slight generalization of this
result is also obtained for certain separable continuous fields of
unital C*algebras over $[0,1]$.) The Connes Chern character $\ch
\colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta
\rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense
$G_\delta$ set of parameters $\theta$. The main computational tool in
this paper is a group homomorphism $\vtr \colon K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$
obtained from the Connes Chern character by restricting the
functionals in its codomain to a certain ninedimensional subspace of
$H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$
is fully determined for each $\theta$. (We conjecture that this
subspace is all of $H^{\ev}$.)
Keywords:C*algebras, Ktheory, automorphisms, rotation algebras, unbounded traces, Chern characters Categories:46L80, 46L40, 19K14 

132. CJM 2001 (vol 53 pp. 325)
 Matui, Hiroki

Ext and OrderExt Classes of Certain Automorphisms of $C^*$Algebras Arising from Cantor Minimal Systems
Giordano, Putnam and Skau showed that the transformation group
$C^*$algebra arising from a Cantor minimal system is an $AT$algebra,
and classified it by its $K$theory. For approximately inner
automorphisms that preserve $C(X)$, we will determine their classes in
the Ext and OrderExt groups, and introduce a new invariant for the
closure of the topological full group. We will also prove that every
automorphism in the kernel of the homomorphism into the Ext group is
homotopic to an inner automorphism, which extends Kishimoto's result.
Categories:46L40, 46L80, 54H20 

133. CJM 2001 (vol 53 pp. 355)
 Nica, Alexandru; Shlyakhtenko, Dimitri; Speicher, Roland

$R$Diagonal Elements and Freeness With Amalgamation
The concept of $R$diagonal element was introduced in \cite{NS2},
and was subsequently found to have applications to several problems
in free probability. In this paper we describe a new approach to
$R$diagonality, which relies on freeness with amalgamation.
The class of $R$diagonal elements is enlarged to contain examples
living in nontracial $*$probability spaces, such as the
generalized circular elements of \cite{Sh1}.
Category:46L54 

134. CJM 2001 (vol 53 pp. 51)
 Dean, Andrew

A Continuous Field of Projectionless $C^*$Algebras
We use some results about stable relations to show that some of the
simple, stable, projectionless crossed products of $O_2$ by $\bR$
considered by Kishimoto and Kumjian are inductive limits of type I
$C^*$algebras. The type I $C^*$algebras that arise are pullbacks
of finite direct sums of matrix algebras over the continuous
functions on the unit interval by finite dimensional $C^*$algebras.
Categories:46L35, 46L57 

135. CJM 2001 (vol 53 pp. 161)
 Lin, Huaxin

Classification of Simple Tracially AF $C^*$Algebras
We prove that preclassifiable (see 3.1) simple nuclear tracially AF
\CA s (TAF) are classified by their $K$theory. As a consequence all
simple, locally AH and TAF \CA s are in fact AH algebras (it is known
that there are locally AH algebras that are not AH). We also prove
the following Rationalization Theorem. Let $A$ and $B$ be two unital
separable nuclear simple TAF \CA s with unique normalized traces
satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the
same (ordered and scaled) $K$theory and $K_0 (A)_+$ is locally
finitely generated, then $A \otimes Q \cong B \otimes Q$, where $Q$ is
the UHFalgebra with the rational $K_0$. Classification results (with
restriction on $K_0$theory) for the above \CA s are also obtained.
For example, we show that, if $A$ and $B$ are unital nuclear separable
simple TAF \CA s with the unique normalized trace satisfying the UCT
and with $K_1(A) = K_1(B)$, and $A$ and $B$ have the same rational
(scaled ordered) $K_0$, then $A \cong B$. Similar results are also
obtained for some cases in which $K_0$ is nondivisible such as
$K_0(A) = \mathbf{Z} [1/2]$.
Categories:46L05, 46L35 

136. CJM 2000 (vol 52 pp. 1164)
 Elliott, George A.; Villadsen, Jesper

Perforated Ordered $\K_0$Groups
A simple $\C^*$algebra is constructed for which the Murrayvon
Neumann equivalence classes of projections, with the usual
additioninduced by addition of orthogonal projectionsform the
additive semigroup
$$
\{0,2,3,\dots\}.
$$
(This is a particularly simple instance of the phenomenon of
perforation of the ordered $\K_0$group, which has long been known in
the commutative casefor instance, in the case of the
foursphereand was recently observed by the second author in the
case of a simple $\C^*$algebra.)
Categories:46L35, 46L80 

137. CJM 2000 (vol 52 pp. 920)
 Evans, W. D.; Opic, B.

Real Interpolation with Logarithmic Functors and Reiteration
We present ``reiteration theorems'' with limiting values
$\theta=0$ and $\theta = 1$ for a real interpolation method
involving brokenlogarithmic functors. The resulting spaces
lie outside of the original scale of spaces and to describe them
new interpolation functors are introduced. For an ordered couple
of (quasi) Banach spaces similar results were presented without
proofs by Doktorskii in [D].
Keywords:real interpolation, brokenlogarithmic functors, reiteration, weighted inequalities Categories:46B70, 26D10, 46E30 

138. CJM 2000 (vol 52 pp. 999)
 Mankiewicz, Piotr

Compact Groups of Operators on Subproportional Quotients of $l^m_1$
It is proved that a ``typical'' $n$dimensional quotient $X_n$ of
$l^m_1$ with $n = m^{\sigma}$, $0 < \sigma < 1$, has the property
$$
\Average \int_G \Tx\_{X_n} \,dh_G(T) \geq
\frac{c}{\sqrt{n\log^3 n}} \biggl( n  \int_G \tr T \,dh_G (T)
\biggr),
$$
for every compact group $G$ of operators acting on $X_n$, where
$d_G(T)$ stands for the normalized Haar measure on $G$ and the average
is taken over all extreme points of the unit ball of $X_n$. Several
consequences of this estimate are presented.
Categories:46B20, 46B09 

139. CJM 2000 (vol 52 pp. 789)
 Kamińska, Anna; Mastyło, Mieczysław

The DunfordPettis Property for Symmetric Spaces
A complete description of symmetric spaces on a separable measure
space with the DunfordPettis property is given. It is shown that
$\ell^1$, $c_0$ and $\ell^{\infty}$ are the only symmetric sequence
spaces with the DunfordPettis property, and that in the class of
symmetric spaces on $(0, \alpha)$, $0 < \alpha \leq \infty$, the only
spaces with the DunfordPettis property are $L^1$, $L^{\infty}$, $L^1
\cap L^{\infty}$, $L^1 + L^{\infty}$, $(L^{\infty})^\circ$ and $(L^1 +
L^{\infty})^\circ$, where $X^\circ$ denotes the norm closure of $L^1
\cap L^{\infty}$ in $X$. It is also proved that all Banach dual
spaces of $L^1 \cap L^{\infty}$ and $L^1 + L^{\infty}$ have the
DunfordPettis property. New examples of Banach spaces showing that
the DunfordPettis property is not a threespace property are also
presented. As applications we obtain that the spaces $(L^1 +
L^{\infty})^\circ$ and $(L^{\infty})^\circ$ have a unique symmetric
structure, and we get a characterization of the DunfordPettis
property of some K\"otheBochner spaces.
Categories:46E30, 46B42 

140. CJM 2000 (vol 52 pp. 849)
 Sukochev, F. A.

Operator Estimates for Fredholm Modules
We study estimates of the type
$$
\Vert \phi(D)  \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D  D_0
\Vert^{\alpha}, \quad \alpha = \frac12, 1
$$
where $\phi(t) = t(1 + t^2)^{1/2}$, $D_0 = D_0^*$ is an unbounded
linear operator affiliated with a semifinite von Neumann algebra
$\calM$, $D  D_0$ is a bounded selfadjoint linear operator from
$\calM$ and $(1 + D_0^2)^{1/2} \in \emt$, where $\emt$ is a symmetric
operator space associated with $\calM$. In particular, we prove that
$\phi(D)  \phi(D_0)$ belongs to the noncommutative $L_p$space for
some $p \in (1,\infty)$, provided $(1 + D_0^2)^{1/2}$ belongs to the
noncommutative weak $L_r$space for some $r \in [1,p)$. In the case
$\calM = \calB (\calH)$ and $1 \leq p \leq 2$, we show that this
result continues to hold under the weaker assumption $(1 +
D_0^2)^{1/2} \in \calC_p$. This may be regarded as an odd
counterpart of A.~Connes' result for the case of even Fredholm
modules.
Categories:46L50, 46E30, 46L87, 47A55, 58B15 

141. CJM 2000 (vol 52 pp. 633)
 Walters, Samuel G.

Chern Characters of Fourier Modules
Let $A_\theta$ denote the rotation algebrathe universal $C^\ast$algebra
generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where
$\theta$ is a fixed real number. Let $\sigma$ denote the Fourier
automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{1}$,
and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote
the associated $C^\ast$crossed product. It is shown that there is a
canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each
$\theta$ given by nine canonical modules. The unbounded trace functionals
of $B_\theta$ (yielding the Chern characters here) are calculated to obtain
the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when
$\theta$ is irrational. The Chern characters of the nine modulesand more
importantly, the Fourier moduleare computed and shown to involve techniques
from the theory of Jacobi's theta functions. Also derived are explicit
equations connecting unbounded traces across strong Morita equivalence, which
turn out to be noncommutative extensions of certain theta function equations.
These results provide the basis for showing that for a dense $G_\delta$ set
of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is
generated by the nine classes constructed here.
Keywords:$C^\ast$algebras, unbounded traces, Chern characters, irrational rotation algebras, $K$groups Categories:46L80, 46L40 

142. CJM 1999 (vol 51 pp. 850)
 Muhly, Paul S.; Solel, Baruch

Tensor Algebras, Induced Representations, and the Wold Decomposition
Our objective in this sequel to \cite{MSp96a} is to develop extensions,
to representations of tensor algebras over $C^{*}$correspondences, of
two fundamental facts about isometries on Hilbert space: The Wold
decomposition theorem and Beurling's theorem, and to apply these to
the analysis of the invariant subspace structure of certain subalgebras
of CuntzKrieger algebras.
Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theorem Categories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35 

143. CJM 1999 (vol 51 pp. 745)
 Echterhoff, Siegfried; Quigg, John

Induced Coactions of Discrete Groups on $C^*$Algebras
Using the close relationship between coactions of discrete groups and
Fell bundles, we introduce a procedure for inducing a $C^*$coaction
$\delta\colon D\to D\otimes C^*(G/N)$ of a quotient group $G/N$ of a
discrete group $G$ to a $C^*$coaction $\Ind\delta\colon\Ind D\to \Ind
D\otimes C^*(G)$ of $G$. We show that induced coactions behave in many
respects similarly to induced actions. In particular, as an analogue of
the well known imprimitivity theorem for induced actions we prove that
the crossed products $\Ind D\times_{\Ind\delta}G$ and $D\times_{\delta}G/N$
are always Morita equivalent. We also obtain nonabelian analogues of a
theorem of Olesen and Pedersen which show that there is a duality between
induced coactions and twisted actions in the sense of Green. We further
investigate amenability of Fell bundles corresponding to induced coactions.
Category:46L55 

144. CJM 1999 (vol 51 pp. 566)
 Ferenczi, V.

Quotient Hereditarily Indecomposable Banach Spaces
A Banach space $X$ is said to be {\it quotient hereditarily
indecomposable\/} if no infinite dimensional quotient of a subspace
of $X$ is decomposable. We provide an example of a quotient
hereditarily indecomposable space, namely the space $X_{\GM}$
constructed by W.~T.~Gowers and B.~Maurey in \cite{GM}. Then we
provide an example of a reflexive hereditarily indecomposable space
$\hat{X}$ whose dual is not hereditarily indecomposable; so
$\hat{X}$ is not quotient hereditarily indecomposable. We also
show that every operator on $\hat{X}^*$ is a strictly singular
perturbation of an homothetic map.
Categories:46B20, 47B99 

145. CJM 1999 (vol 51 pp. 309)
 Leung, Denny H.; Tang, WeeKee

Symmetric sequence subspaces of $C(\alpha)$, II
If $\alpha$ is an ordinal, then the space of all ordinals less than or
equal to $\alpha$ is a compact Hausdorff space when endowed with the
order topology. Let $C(\alpha)$ be the space of all continuous
realvalued functions defined on the ordinal interval $[0,
\alpha]$. We characterize the symmetric sequence spaces which embed
into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy
$(E_\alpha)$ of symmetric sequence spaces is constructed so that, for
each countable ordinal $\alpha$, $E_\alpha$ embeds into
$C(\omega^{\omega^\alpha})$, but does not embed into
$C(\omega^{\omega^\beta})$ for any $\beta < \alpha$.
Categories:03E13, 03E15, 46B03, 46B45, 46E15, 54G12 

146. CJM 1999 (vol 51 pp. 147)
 Suárez, Daniel

Homeomorphic Analytic Maps into the Maximal Ideal Space of $H^\infty$
Let $m$ be a point of the maximal ideal space of $\papa$ with
nontrivial Gleason part $P(m)$. If $L_m \colon \disc \rr P(m)$ is the
Hoffman map, we show that $\papa \circ L_m$ is a closed subalgebra
of $\papa$. We characterize the points $m$ for which $L_m$ is a
homeomorphism in terms of interpolating sequences, and we show that in
this case $\papa \circ L_m$ coincides with $\papa$. Also, if
$I_m$ is the ideal of functions in $\papa$ that identically vanish
on $P(m)$, we estimate the distance of any $f\in \papa$ to $I_m$.
Categories:30H05, 46J20 

147. CJM 1999 (vol 51 pp. 26)
 Fabian, Marián; Mordukhovich, Boris S.

Separable Reduction and Supporting Properties of FrÃ©chetLike Normals in Banach Spaces
We develop a method of separable reduction for Fr\'{e}chetlike
normals and $\epsilon$normals to arbitrary sets in general Banach
spaces. This method allows us to reduce certain problems involving
such normals in nonseparable spaces to the separable case. It is
particularly helpful in Asplund spaces where every separable subspace
admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable
reduction method in Asplund spaces, we provide a new direct proof of a
nonconvex extension of the celebrated BishopPhelps density theorem.
Moreover, in this way we establish new characterizations of Asplund
spaces in terms of $\epsilon$normals.
Keywords:nonsmooth analysis, Banach spaces, separable reduction, FrÃ©chetlike normals and subdifferentials, supporting properties, Asplund spaces Categories:49J52, 58C20, 46B20 

148. CJM 1998 (vol 50 pp. 1236)
149. CJM 1998 (vol 50 pp. 1138)
 Chalov, P. A.; Terzioğlu, T.; Zahariuta, V. P.

Compound invariants and mixed $F$, $\DF$power spaces
The problems on isomorphic classification and quasiequivalence of bases
are studied for the class of mixed $F$, $\DF$power series spaces,
{\it i.e.} the spaces of the following kind
$$
G(\la,a)=\lim_{p \to \infty} \proj \biggl(\lim_{q \to \infty}\ind
\Bigl(\ell_1\bigl(a_i (p,q)\bigr)\Bigr)\biggr),
$$
where $a_i (p,q)=\exp\bigl((p\la_i q)a_i\bigr)$, $p,q \in \N$, and
$\la =( \la_i)_{i \in \N}$, $a=(a_i)_{i \in \N}$ are
some sequences of positive numbers. These spaces, up to isomorphisms,
are basis subspaces of tensor products of power series spaces of $F$ and
$\DF$types, respectively. The $m$rectangle characteristic
$\mu_m^{\lambda,a}(\delta,\varepsilon; \tau,t)$, $m \in \N$ of the
space $G(\la,a)$ is defined as the number of members of the sequence
$(\la_i, a_i)_{i \in \N}$ which are contained in the union of $m$
rectangles $P_k = (\delta_k, \varepsilon_k] \times (\tau_k, t_k]$,
$k = 1,2, \ldots, m$. It is shown that each $m$rectangle characteristic
is an invariant on the considered class under some proper definition of an
equivalency relation. The main tool are new compound invariants, which
combine some version of the classical approximative dimensions (Kolmogorov,
Pe{\l}czynski) with appropriate geometrical and interpolational operations
under neighborhoods of the origin (taken from a given basis).
Categories:46A04, 46A45, 46M05 

150. CJM 1998 (vol 50 pp. 673)
 Carey, Alan; Phillips, John

Fredholm modules and spectral flow
An {\it odd unbounded\/} (respectively, $p${\it summable})
{\it Fredholm module\/} for a unital Banach $\ast$algebra, $A$, is a pair $(H,D)$
where $A$ is represented on the Hilbert space, $H$, and $D$ is an unbounded
selfadjoint operator on $H$ satisfying:
\item{(1)} $(1+D^2)^{1}$ is compact (respectively, $\Trace\bigl((1+D^2)^{(p/2)}\bigr)
<\infty$), and
\item{(2)} $\{a\in A\mid [D,a]$ is bounded$\}$ is a dense
$\ast$subalgebra of $A$.
If $u$ is a unitary in the dense $\ast$subalgebra mentioned in (2) then
$$
uDu^\ast=D+u[D,u^{\ast}]=D+B
$$
where $B$ is a bounded selfadjoint operator. The path
$$
D_t^u:=(1t) D+tuDu^\ast=D+tB
$$
is a ``continuous'' path of unbounded selfadjoint ``Fredholm'' operators.
More precisely, we show that
$$
F_t^u:=D_t^u \bigl(1+(D_t^u)^2\bigr)^{{1\over 2}}
$$
is a normcontinuous path of (bounded) selfadjoint Fredholm
operators. The {\it spectral flow\/} of this path $\{F_t^u\}$ (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,
$$
\sf(\{D_t^u\}):=\sf(\{F_t^u\}),
$$
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,
$B_{\sa}(H)$, 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 $X\in T_B(B_{\sa}(H))$ is given by an $X$ in $B_{\sa}(H)$ as the
derivative at $B$ along the curve $t\mapsto B+tX$ in the manifold.
Then we show that for $m$ a sufficiently large halfinteger:
$$
\alpha (X)={1\over {\tilde {C}_m}}\Tr \Bigl(X\bigl(1+(D+B)^2\bigr)^{m}\Bigr)
$$
is a closed $1$form. For any piecewise smooth path $\{D_t=D+B_t\}$ with
$D_0$ and $D_1$ unitarily equivalent we show that
$$
\sf(\{D_t\})={1\over {\tilde {C}_m}} \int_0^1\Tr \Bigl({d\over {dt}}
(D_t)(1+D_t^2)^{m}\Bigr)\,dt
$$
the integral of the $1$form $\alpha$. If $D_0$ and $D_1$ are not unitarily
equivalent, we must add a pair of correction terms to the righthand
side. We also prove a bounded finitely summable version of the form:
$$
\sf(\{F_t\})={1\over C_n}\int_0^1\Tr\Bigl({d\over dt}(F_t)(1F_t^2)^n\Bigr)\,dt
$$
for $n\geq{{p1}\over 2}$ an integer. The unbounded case is proved by
reducing to the bounded case via the map $D\mapsto F=D(1+D^2
)^{{1\over 2}}$. We prove simultaneously a type II version of our
results.
Categories:46L80, 19K33, 47A30, 47A55 
