76. 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 

77. 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 

78. CJM 1998 (vol 50 pp. 538)
79. CJM 1998 (vol 50 pp. 658)
 Symesak, Frédéric

Hankel operators on pseudoconvex domains of finite type in ${\Bbb C}^2$
The aim of this paper is to study small Hankel operators $h$ on the
Hardy space or on weighted Bergman spaces, where $\Omega$ is a
finite type domain in ${\Bbbvii C}^2$ or a strictly pseudoconvex
domain in ${\Bbbvii C}^n$. We give a sufficient condition on the
symbol $f$ so that $h$ belongs to the Schatten class ${\cal S}_p$,
$1\le p<+\infty$.
Categories:32A37, 47B35, 47B10, 46E22 

80. CJM 1998 (vol 50 pp. 290)
 Davidson, Kenneth R.; Popescu, Gelu

Noncommutative disc algebras for semigroups
We study noncommutative disc algebras associated to the free
product of discrete subsemigroups of $\bbR^+$. These algebras are
associated to generalized Cuntz algebras, which are shown to be
simple and purely infinite. The nonselfadjoint subalgebras
determine the semigroup up to isomorphism. Moreover, we establish
a dilation theorem for contractive representations of these
semigroups which yields a variant of the von Neumann inequality.
These methods are applied to establish a solution to the truncated
moment problem in this context.
Category:47D25 

81. CJM 1998 (vol 50 pp. 99)
 Izuchi, Keiji; Matsugu, Yasuo

$A_\phi$invariant subspaces on the torus
Generalizing the notion of invariant subspaces on
the 2dimensional torus $T^2$, we study the structure
of $A_\phi$invariant subspaces of $L^2(T^2)$. A
complete description is given of $A_\phi$invariant
subspaces that satisfy conditions similar to those
studied by Mandrekar, Nakazi, and Takahashi.
Categories:32A35, 47A15 

82. CJM 1997 (vol 49 pp. 1117)
 Hu, Zhiguo

The von Neumann algebra $\VN(G)$ of a locally compact group and quotients of its subspaces
Let $\VN(G)$ be the von Neumann algebra of a locally
compact group $G$. We denote by $\mu$ the initial
ordinal with $\abs{\mu}$ equal to the smallest cardinality
of an open basis at the unit of $G$ and $X= \{\alpha;
\alpha < \mu \}$. We show that if $G$ is nondiscrete
then there exist an isometric $*$isomorphism $\kappa$
of $l^{\infty}(X)$ into $\VN(G)$ and a positive linear
mapping $\pi$ of $\VN(G)$ onto $l^{\infty}(X)$ such that
$\pi\circ\kappa = \id_{l^{\infty}(X)}$ and $\kappa$ and
$\pi$ have certain additional properties. Let $\UCB
(\hat{G})$ be the $C^{*}$algebra generated by
operators in $\VN(G)$ with compact support and
$F(\hat{G})$ the space of all $T \in \VN(G)$ such that
all topologically invariant means on $\VN(G)$ attain the
same value at $T$. The construction of the mapping $\pi$
leads to the conclusion that the quotient space $\UCB
(\hat{G})/F(\hat{G})\cap \UCB(\hat{G})$ has
$l^{\infty}(X)$ as a continuous linear image if $G$ is
nondiscrete. When $G$ is further assumed to be
nonmetrizable, it is shown that $\UCB(\hat{G})/F
(\hat{G})\cap \UCB(\hat{G})$ contains a linear
isomorphic copy of $l^{\infty}(X)$. Similar results are
also obtained for other quotient spaces.
Categories:22D25, 43A22, 43A30, 22D15, 43A07, 47D35 

83. CJM 1997 (vol 49 pp. 736)
84. CJM 1997 (vol 49 pp. 100)
 Lance, T. L.; Stessin, M. I.

Multiplication Invariant Subspaces of Hardy Spaces
This paper studies closed subspaces $L$
of the Hardy spaces $H^p$ which are $g$invariant ({\it i.e.},
$g\cdot L \subseteq L)$ where $g$ is inner, $g\neq 1$. If
$p=2$, the Wold decomposition theorem implies that there is
a countable ``$g$basis'' $f_1, f_2,\ldots$ of
$L$ in the sense that $L$ is a direct sum of spaces
$f_j\cdot H^2[g]$ where $H^2[g] = \{f\circ g \mid f\in H^2\}$.
The basis elements $f_j$ satisfy the
additional property that $\int_T f_j^2 g^k=0$,
$k=1,2,\ldots\,.$ We call such functions $g$$2$inner.
It also
follows that any $f\in H^2$ can be factored $f=h_{f,2}\cdot
(F_2\circ g)$ where $h_{f,2}$ is $g$$2$inner and $F$ is
outer, generalizing the classical Riesz factorization.
Using $L^p$ estimates for the canonical decomposition of
$H^2$, we find a factorization $f=h_{f,p} \cdot (F_p \circ
g)$ for $f\in H^p$. If $p\geq 1$ and $g$ is a finite
Blaschke product we obtain, for any $g$invariant
$L\subseteq H^p$, a finite $g$basis of $g$$p$inner
functions.
Categories:30H05, 46E15, 47B38 
