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 $I_m$.

We develop a method of separable reduction for Fr\'{e}chet-like 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 Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals.

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
Categories:42C10, 33C45, 46B07

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).

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.

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 ${\cal S}_p$, $1\le p<+\infty$.

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 non-unital 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.

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.

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 norm-one 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 norm-one 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.

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 UHF-algebra can be approximated pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra 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 AF-algebra. Similar results for tori are also obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$ for lower dimensional spaces is also studied.

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.

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.