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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 self-adjoint 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 self-adjoint operator. The path $$ D_t^u:=(1-t) D+tuDu^\ast=D+tB $$ is a ``continuous'' path of unbounded self-adjoint ``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 norm-continuous path of (bounded) self-adjoint 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 half-integer: $$ \alpha (X)={1\over {\tilde {C}_m}}\Tr \Bigl(X\bigl(1+(D+B)^2\bigr)^{-m}\Bigr) $$ 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 right-hand side. We also prove a bounded finitely summable version of the form: $$ \sf(\{F_t\})={1\over C_n}\int_0^1\Tr\Bigl({d\over dt}(F_t)(1-F_t^2)^n\Bigr)\,dt $$ for $n\geq{{p-1}\over 2}$ 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)

Froese, Richard
Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions
The purpose of this note is to provide a simple proof of the sharp polynomial upper bound for the resonance counting function of a Schr\"odinger operator in odd dimensions. At the same time we generalize the result to the class of super-exponentially decreasing potentials.

Categories:47A10, 47A40, 81U05

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 nonself-adjoint 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 2-dimensional 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 non-metrizable, 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)

Fendler, Gero
Dilations of one parameter Semigroups of positive Contractions on $L^{\lowercase {p}}$ spaces
It is proved in this note, that a strongly continuous semigroup of (sub)positive contractions acting on an $L^p$-space, for $1
Categories:47D03, 22D12, 43A22

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
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