Let $L$ denote the sub-Laplacian on the Heisenberg group $\mathbb{H}_n$ and $T_r^\lambda := (1 - r L)^\lambda_+$ the corresponding Bochner-Riesz operator. Let $Q$ denote the homogeneous dimension and $D$ the Euclidean dimension of $\mathbb{H}_n$. We prove convergence a.e. of the Bochner-Riesz means $T_r^\lambda f$ as $r \rightarrow 0$ for $\lambda > 0$ and for all $f \in L^p(\mathbb{H}_n)$, provided that \frac{Q-1}{Q} \Big(\frac{1}{2} - \frac{\lambda}{D-1} \Big) < 1/p \le 1/2. Our proof is based on explicit formulas for the operators $\partial_{\omega^a}$ with $a \in \mathbb{C}$, defined on the dual of $\mathbb{H}_n$ by $\partial_{\omega^a} \widehat{f} := \widehat{\omega^a f}$, which may be of independent interest. Here $\omega$ is given by $\omega(z,u) := |z|^2 - 4iu$ for all $(z,u) \in \mathbb{H}_n$.

2000 Mathematical Subject Classification: 22E30, 43A80.

We investigate in this paper the complementation of copies of $c_0(I)$ in some classes of Banach spaces (in the class of weakly compactly generated (WCG) Banach spaces, in the larger class $\mathcal{V}$ of Banach spaces which are subspaces of some $C(K)$ space with $K$ a Valdivia compact, and in the Banach spaces $C([1, \alpha ])$, where $\alpha$ is an ordinal) and the embedding of $c_0(I)$ in the elements of the class $\mathcal{C}$ of complemented subspaces of $C(K)$ spaces. Two of our results are as follows:

(i) in a Banach space $X \in \mathcal{V}$ every copy of $c_0(I)$ with $\# I < \aleph _{\omega}$ is complemented;

(ii) if $\alpha _0 = \aleph _0$, $\alpha _{n+1} = 2^{\alpha _n}$, $n \geq 0$, and $\alpha = \sup \{\alpha _n : n \geq 0\}$ there exists a WCG Banach space with an uncomplemented copy of $c_0(\alpha )$.

So, under the generalized continuum hypothesis (GCH), $\aleph _{\omega}$ is the greatest cardinal $\tau$ such that every copy of $c_0(I)$ with $\# I < \tau$ is complemented in the class $\mathcal{V}$. If $T : c_0(I) \to C([1,\alpha ])$ is an isomorphism into its image, we prove that:

(i) $c_0(I)$ is complemented, whenever $\| T \| ,\| T^{-1} \| < (3/2)^{\frac 12}$;

(ii) there is a finite partition $\{I_1, \dots , I_k\}$ of $I$ such that each copy $T(c_0(I_k))$ is complemented.

Concerning the class $\mathcal{C}$, we prove that an already known property of $C(K)$ spaces is still true for this class, namely, if $X \in \mathcal{C}$, the following are equivalent:

(i) there is a weakly compact subset $W \subset X$ with ${\rm Dens}(W) = \tau$;

(ii) $c_0(\tau )$ is isomorphically embedded into $X$.

This yields a new characterization of a class of injective Banach spaces.

2000 Mathematical Subject Classification: 46B20, 46B26.

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras.

2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.

Bohr's inequality says that if $f(z) = \sum^{\infty}_{n = 0} a_n z^n$ is a bounded analytic function on the closed unit disc, then $\sum^{\infty}_{n = 0} \lvert a_n\rvert r^n \leq \Vert f\Vert_{\infty}$ for $0 \leq r \leq 1/3$ and that $1/3$ is sharp. In this paper we give an operator-theoretic proof of Bohr's inequality that is based on von Neumann's inequality. Since our proof is operator-theoretic, our methods extend to several complex variables and to non-commutative situations.

We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains, for the non-commutative disc algebra ${\mathcal A}_n$, and for the reduced (respectively full) group C*-algebra of the free group on $n$ generators.

We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non-unital version of von Neumann's inequality.

2000 Mathematical Subject Classification: 47A20, 47A56.

We extend the notion of Ribaucour transformation from classical surface theory to the theory of holonomic submanifolds of pseudo-Riemannian space forms with arbitrary dimension and codimension, that is, submanifolds with flat normal bundle admitting a global system of principal coordinates. Bianchi gave a positive answer to the question of whether among the Ribaucour transforms of a surface with constant mean or Gaussian curvature there exist other surfaces with the same property. Our main achievement is to solve the same problem for the class of holonomic submanifolds with constant sectional curvature.

2000 Mathematical Subject Classification: 53B25, 58J72.

This is the fifth in a series of papers constructing explicit examples of special Lagrangian submanifolds in ${\mathbb C}^m$. A submanifold of ${\mathbb C}^m$ is ruled if it is fibred by a family of real straight lines in ${\mathbb C}^m$. This paper studies ruled special Lagrangian 3-folds in ${\mathbb C}^3$, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko, and Bryant. Special Lagrangian cones in ${\mathbb C}^3$ are automatically ruled, and each ruled special Lagrangian 3-fold is asymptotic to a unique special Lagrangian cone. We study the family of ruled special Lagrangian 3-folds N asymptotic to a fixed special Lagrangian cone N0. We find that this depends on solving a linear equation, so that the family of such N has the structure of a vector space. We also show that the intersection $\Sigma$ of N0 with the unit sphere ${\mathcal S}^5$ in ${\mathbb C}^3$ is a Riemann surface, and construct a ruled special Lagrangian 3-fold N asymptotic to N0 for each holomorphic vector field w on $\Sigma$. As corollaries of this we write down two large families of explicit special Lagrangian 3-folds in ${\mathbb C}^3$ depending on a holomorphic function on $\mathbb C$, which include many new examples of singularities of special Lagrangian 3-folds. We also show that each special Lagrangian T2-cone N0 can be extended to a 2-parameter family of ruled special Lagrangian 3-folds asymptotic to N0, and diffeomorphic to $T^2\times{\mathbb R}$.

2000 Mathematical Subject Classification: 53C38, 53D12.