On boundaries of Levi-flat hypersurfaces in Cn

Let S be a smooth 2-codimensional real compact submanifold of ${\mathbb{C}}^n$, $n>2$. We address the problem of finding a compact hypersurface M, with boundary S, such that $M?S$ is Levi-flat. We prove the following theorem. Assume that (i) S is nonminimal at every CR point, (ii) every complex point of S is flat and elliptic and there exists at least one such point, (iii) S does not contain complex submanifolds of dimension $n?2$. Then there exists a Levi-flat $(2n?1)$-subvariety $\stackrel{?}{M}?\mathbb{C}\times {\mathbb{C}}^n$ with negligible singularities and boundary $\stackrel{?}{S}$ (in the sense of currents) such that the natural projection $\pi :\mathbb{C}\times {\mathbb{C}}^n\to {\mathbb{C}}^n$ restricts to a CR diffeomorphism between S and $\stackrel{?}{S}$. To cite this article: P. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

The group of invariants of an inner function with finite spectrum

This paper determines the group of continuous invariants corresponding to an inner function Θ with finitely many singularities on the unit circle $\mathbb{T}$; that is, the continuous mappings $g:\mathbb{T}\to \mathbb{T}$ such that $\Theta °g=\Theta$ on $\mathbb{T}$. These mappings form a group under composition.     

The Ramsey property for Banach spaces and Choquet simplices,and applications

We show that the class of finite-dimensional Banach spaces and the class of finite-dimensional Choquet simplices have the Ramsey property. As an application, we show that the group $\mathrm{Aut}(\mathbb{G})$ of surjective linear isometries of the Gurarij space $\mathbb{G}$ is extremely amenable, and that the canonical action $\mathrm{Aut}(\mathbb{P})?\mathbb{P}$ is the universal minimal flow of the group $\mathrm{Aut}(\mathbb{P})$ of affine homeomorphisms of the Poulsen simplex $\mathbb{P}$. This answers questions of Melleray–Tsankov and Conley–Törnquist.     

Subproduct systems over N×N

We develop the theory of subproduct systems over the monoid $\mathbb{N}\times \mathbb{N}$, and the non-self-adjoint operator algebras associated with them. These are double sequences of Hilbert spaces ${\{X(m,n)\}}_{m,n=0}^{\infty }$ equipped with a multiplication given by coisometries from $X(i,j)?X(k,l)$ to $X(i+k,j+l)$. We find that the character space of the norm-closed algebra generated by left multiplication operators (the tensor algebra) is homeomorphic to a complex homogeneous affine algebraic variety intersected with a unit ball. Certain conditions are isolated under which subproduct systems whose tensor algebras are isomorphic must be isomorphic themselves. In the absence of these conditions, we show that two numerical invariants must agree on such subproduct systems. Additionally, we classify the subproduct systems over $\mathbb{N}\times \mathbb{N}$ by means of ideals in algebras of non-commutative polynomials.