We introduce the concept of a smooth point of order $n$ of the closed unit ball of a Banach space $E$ and characterize such points for $E = c_0$, $L_p(\mu)$ ($1\leq p \le\infty$), and $C(K)$. We show that every locally uniformly rotund multilinear form and homogeneous polynomial on a Banach space $E$ is generated by locally uniformly rotund linear functionals on $E$. We also classify such points for $E = c_0$, $L_p(\mu)(1\leq p \le\infty)$, and $C(K)$.

Let $f_0, f_1, \dots, f_k$ be $n$--variable polynomials over a finite prime field $\fp$. A proof of the ideal membership $f_0 \in \langle f_1, \dots, f_k \rangle$ in {\em polynomial calculus} is a sequence of polynomials $h_1, \dots, h_t$ such that $h_t = f_0$, and such that every $h_i$ is either an $f_j$, $j \geq 1$, or obtained from $h_1, \dots, h_{i-1}$ by one of the two inference rules: $g_1$ and $g_2$ entail any $\fp$--linear combination of $g_1$, $g_2$, and $g$ entails $g \cdot g'$, for any polynomial $g'$. The degree of the proof is the maximum degree of the $h_i$'s. We give a condition on families $\{ f_{N,0}, \dots, f_{N,k_N}\}_{N<\omega}$ of $n_N$--variable polynomials of bounded degree implying that the minimum degree of polynomial calculus proofs of $f_{N,0}$ from $f_{N,1}, \dots, f_{N,k_N}$ cannot be bounded by an independent constant and, in fact, is $\Omega(\log(\log(N)))$. In particular, we obtain an $\Omega(\log(\log(N)))$ lower bound for the degrees of proofs of $1$ (so called {\em refutations}) of the $(N,m)$--system (defined in \cite{BIKPP}) formalizing a modular counting principle (where $m$ is fixed and not divisible by $p$, and the parameter $N$ is not divisible by $m$), and a similar lower bound for refutations of systems encoding that $N$ is composite (whenever $N$ is a prime). No bounds were previously known for these systems. The same method yields $\Omega(\log(N))$ lower bounds for the degree of coefficient polynomials in Nullstellensatz proofs. The method is based on a new result about a uniform way of generating all submoduli of tabloid moduli.

We first show that for an infinite dimensional Banach space $X$, the unitary spectrum of any superstable operator is countable. In connection with descriptive set theory, we show that if $X$ is separable, then the set of stable operators and the set of power bounded operators are Borel subsets of $L(X)$ (equipped with the strong operator topology), while the set $\mathcal{S}'(X)$ of superstable operators is coanalytic. However, $\mathcal{S}'(X)$ is a Borel set if $X$ is a superreflexive and hereditarily indecomposable space. On the other hand, if $X$ is superreflexive and $X$ has a complemented subspace with unconditional basis or, more generally, if $X$ has a polynomially bounded and not superstable operator, then the set $\mathcal{S}'(X)$ is non Borel.

We consider the problem of determining when the product of two Bergman space Toeplitz operators is a Toeplitz operator. In particular, in the case of the zero product $T_fT_g=0$, we give some conditions that guarantee only the trivial solution.

A Banach space $X$ is said to have the {\it separable lifting property} if for every subspace $Y$ of $X^{**}$ containing $X$ and such that $Y/X$ is separable there exists a bounded linear lifting from $Y/X$ to $Y$. We show that if a sequence of Banach spaces $E_1, E_2, \ldots$ has the joint uniform approximation property and $E_n$ is $c$-complemented in $E_n^{**}$ for every $n$ (with $c$ fixed), then $\eco$ has the separable lifting property. In particular, if $E_n$ is a ${\mathcal{L}}_{p_n, \lambda}$-space for every $n$ ($1 < p_n < \infty$, $\lambda$ independent of $n$), an $L_\infty$ or an $L_1$ space, then $\eco$ has the separable lifting property. We also show that there exists a Banach space $X$ which is not extendably locally reflexive; moreover, for every $n$ there exists an $n$-dimensional subspace $E \hra X^{**}$ such that if $u : X^{**} \raw X^{**}$ is an operator ($=$ bounded linear operator) such that $u (E) \subset X$, then $||(u|_E)^{-1}|| \cdot ||u|| \geq c \sqrt{n}$, where $c$ is a numerical constant.

Let $R$ be a Noetherian integral domain, let $V=\Spec(R)$, and let $I, J$ be nonzero ideals of $R$. Clearly, if $J$ is either a divisor of $I$ or a power of $I$ there is a map $Bl_I(V)\to Bl_J(V)$ of schemes over $V.$ The purpose of this note is to prove, conversely, that if such a map exists, then $J$ must be a fractional ideal divisor of some power of $I$.

We extend two theorems of Krein concerning entire functions of Cartwright class, and give applications for the Bernstein weighted approximation problem.

For a finitely generated module $M$, over a universally catenary local ring, whose symmetric algebra is equidimensional, the ideals generated by the rows of a minimal presentation matrix are shown to have height at most $\mu(M) - \rank M$. Moreover, in the extremal case, they are Cohen-Macaulay ideals if the symmetric algebra is Cohen-Macaulay. Some applications are given to residual intersections of ideals.

Given a locally compact group $G$ let $\mathcal{J}_a(G)$ denote the set of all closed left ideals $J$ in $L^1(G)$ which have the form $J=[L^1(G)*(\delta_e -\mu)]\overline{\vphantom{t}\ }$ where $\mu$ is an absolutely continuous probability measure on $G$. We explore the order structure of $\mathcal{J}_a(G)$ when $\mathcal{J}_a(G)$ is ordered by inclusion. When $G$ is connected and amenable we prove that every nonempty family $\mathcal{F}\subseteq \mathcal{J}_a(G)$ admits both a minimal and a maximal element; in particular, every ideal in $\mathcal{J}_a(G)$ contains an ideal that is minimal in $\mathcal{J}_a(G)$. Furthermore, we obtain that every chain in $\mathcal{J}_a(G)$ is necessarily finite. A natural generalization of these results to almost connected amenable groups is discussed. Our proofs use results from the theory of boundaries of random walks.

In this paper we obtain a tangency principle for hypersurfaces, with not necessarily constant $r$-mean curvature function $H_r $, of an arbitrary Riemannian manifold. That is, we obtain sufficient geometric conditions for two submanifolds of a Riemannian manifold to coincide, as a set, in a neighborhood of a tangency point. As applications of our tangency principle, we obtain, under certain conditions on the function $H_r$, sharp estimates on the size of the greatest ball that fits inside a connected compact hypersurface embedded in a space form of constant sectional curvature $c\leq 0$ and on the size of the smallest ball that encloses the image of an immersion of a compact Riemannian manifold into a Riemannian manifold with sectional curvatures limited from above. This generalizes results of Koutroufiotis, Coghlan-Itokawa, Pui-Fai Leung, Vlachos and Markvorsen. We also generalize a result of Serrin. Our techniques permit us to extend results of Hounie-Leite.

It was shown in \cite{NB} that if $T$ is a contraction in a Hilbert space with finite defect (i.e., $\|T\|\le 1$ and $\operatorname{rank} (I- T^*T) <\infty$), and if the spectrum $\sigma(T)$ does not coincide with the closed unit disk $\overline{\mathbb{D}}$, then the Linear Resolvent Growth condition $$ \|(\la I - T)^{-1} \|\le\frac{C}{\operatorname{dist}(\la,\si(T))},\ \la\in\bc\backslash \si(T) $$ implies that $T$ is similar to a normal operator. The condition $\operatorname{rank}(I - T^*T)<\infty$ measures how close $T$ is to a unitary operator. A natural question is whether this condition can be relaxed. For example, it was conjectured in \cite{NB} that this condition can be replaced by the condition $I - T^*T\in \fS_1$, where $\fS_1$ denotes the trace class. In this note we show that this conjecture is not true, and that, in fact, one cannot replace the condition $\operatorname{rank}(I - T^*T)<\infty$ by any reasonable condition of closeness to a unitary operator.

Let $N \subseteq M$ be von Neumann algebras and let $E:M\to N$ be a faithful normal conditional expectation. In this work it is shown that the similarity orbit ${\cal S}(E)$ of $E$ by the natural action of the invertible group of $G_M$ of $M$ has a natural complex analytic structure and that the map $G_M\to {\cal S}(E)$ given by this action is a smooth principal bundle. It is also shown that if $N$ is finite then ${\cal S}(E)$ admits a Reductive Structure. These results were previously known under the additional assumptions that the index is finite and $N'\cap M \subseteq N$. Conversely, if the orbit ${\cal S}(E)$ has a Homogeneous Reductive Structure for every expectation defined on $M$, then $M$ is finite. For every algebra $M$ and every expectation $E$, a covering space of the unitary orbit ${\cal U}(E)$ is constructed in terms of the connected component of $1$ in the normalizer of $E$. Moreover, this covering space is the universal covering in each of the following cases: (1) $\m$ is a finite factor and $\ind < \infty $; (2) $M$ is properly infinite and $E$ is any expectation; (3) $E$ is the conditional expectation onto the centralizer of a state. Therefore, in these cases, the fundamental group of ${\cal U}(E)$ can be characterized as the Weyl group of $E$.

It is known that a tube over a Kähler submanifold in a complex space form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed $C^{2n-1}$ regular Hopf hypersurface in the complex projective space is a tube over an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed $C^{2n-1}$ regular Hopf hypersurface is a geodesic hypersphere.

We introduce two classes of $m$-homogeneous polynomials defined on Banach spaces, which extend the classes of Dunford-Pettis and Dieudonné linear operators. These extensions allow us to prove that several characterization theorems related to the Dunford-Pettis, Schur, and reciprocal Dunford-Pettis properties, are also valid in the more general case of homogeneous polynomials of any degree $m\in\N$.

Let $\lambda_1,\dots,\lambda_4$ be non-zero with $\lambda_1/\lambda_2$ irrational and negative, and let $\mathcal S$ be the set of values attained by the form $ \lambda_1x_1^3 + \dots + \lambda_4x^3_4 $ when $x_1$ has at most 3 prime divisors and the remaining variables are prime. We prove that most real numbers are close to an element of $\mathcal S$.