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126. CJM 1999 (vol 51 pp. 309)
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 
127. CJM 1999 (vol 51 pp. 147)
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 
128. CJM 1999 (vol 51 pp. 26)
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 
129. CJM 1998 (vol 50 pp. 1138)
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 
130. CJM 1998 (vol 50 pp. 1236)
The behaviour of Legendre and ultraspherical polynomials in $L_p$spaces We consider the analogue of the $\Lambda(p)$problem for
subsets of the Legendre polynomials or more general ultraspherical
polynomials. We obtain the ``best possible'' result that if $2

131. CJM 1998 (vol 50 pp. 673)
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 
132. CJM 1998 (vol 50 pp. 658)
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 
133. CJM 1998 (vol 50 pp. 323)
Purely infinite, simple $C^\ast$algebras arising from free product constructions Examples of simple, separable, unital, purely infinite
$C^\ast$algebras are constructed, including:
\item{(1)} some that are not approximately divisible;
\item{(2)} those that arise as crossed products of any of a certain class of
$C^\ast$algebras by any of a certain class of nonunital endomorphisms;
\item{(3)} those that arise as reduced free products of pairs of
$C^\ast$algebras with respect to any from a certain class of states.
Categories:46L05, 46L45 
134. CJM 1997 (vol 49 pp. 1188)
Factorization in the invertible group of a $C^*$algebra In this paper we consider the following problem:
Given a unital \cs\ $A$ and a collection of elements $S$ in the
identity component of the invertible group of $A$, denoted \ino,
characterize the group of finite products of elements of $S$. The
particular $C^*$algebras studied in this paper are either
unital purely infinite simple or of the form \tenp, where $A$ is
any \cs\ and $K$ is the compact operators on an infinite dimensional
separable Hilbert space. The types of elements used in the
factorizations are unipotents ($1+$ nilpotent), positive invertibles
and symmetries ($s^2=1$). First we determine the groups of finite
products for each collection of elements in \tenp. Then we give
upper bounds on the number of factors needed in these cases. The main
result, which uses results for \tenp, is that for $A$ unital purely
infinite and simple, \ino\ is generated by each of these collections
of elements.
Category:46L05 
135. CJM 1997 (vol 49 pp. 1242)
$1$complemented subspaces of spaces with $1$unconditional bases We prove that if $X$ is a complex strictly monotone sequence
space with $1$un\con\di\tion\al basis, $Y \subseteq X$ has no bands
isometric to $\ell_2^2$ and $Y$ is the range of normone projection from
$X$, then $Y$ is a closed linear span a family of mutually
disjoint vectors in $X$.
We completely characterize $1$complemented subspaces and normone
projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q <
\infty$.
Finally we give a full description of the subspaces that are spanned
by a family of disjointly supported vectors and which are
$1$complemented in (real or complex) Orlicz or Lorentz sequence
spaces. In particular if an Orlicz or
Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p <
\infty$ then the only subspaces
of $X$ which are $1$complemented and disjointly supported are the
closed linear spans of block bases with constant
coefficients.
Categories:46B20, 46B45, 41A65 
136. CJM 1997 (vol 49 pp. 963)
Homomorphisms from $C(X)$ into $C^*$algebras Let $A$ be a simple $C^*$algebra
with real rank zero, stable rank one and weakly
unperforated $K_0(A)$ of countable rank. We show that
a monomorphism $\phi\colon C(S^2) \to A$ can be approximated
pointwise by homomorphisms from $C(S^2)$ into $A$ with
finite dimensional range if and only if certain index
vanishes. In particular, we show that every homomorphism
$\phi$ from $C(S^2)$ into a UHFalgebra can be approximated
pointwise by homomorphisms from $C(S^2)$ into the UHFalgebra
with finite dimensional range. As an application, we show
that if $A$ is a simple $C^*$algebra of real rank zero
and is an inductive limit of matrices over $C(S^2)$ then
$A$ is an AFalgebra. Similar results for tori are also
obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$
for lower dimensional spaces is also studied.
Keywords:Homomorphism of $C(S^2)$, approximation, real, rank zero, classification Categories:46L05, 46L80, 46L35 
137. CJM 1997 (vol 49 pp. 160)
The Classical Limit of Dynamics for Spaces Quantized by an Action of ${\Bbb R}^{\lowercase{d}}$ We have previously shown how to construct a deformation quantization
of any locally compact space on which a vector group acts. Within this
framework we show here that, for a natural class of Hamiltonians, the
quantum evolutions will have the classical evolution as their
classical limit.
Categories:46L60, 46l55, 81S30 
138. CJM 1997 (vol 49 pp. 100)
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 