ALPS: Classes of stable and semipositive matrices

The purpose of this paper is to characterize and interrelate various degrees of stability and semipositivity for real square matrices. The standard conditions for three major classes of matrices are made both stronger and weaker, and the resulting classes are examined. The major classes are diagonally stable, stable, and semipositive matrices, denoted by $\text{A,L,}$ and $\text{S,}$ respectively. Their relationship to the classes of matrices whose principal minors are positive (denoted by $\text{P}$ and non-negative (denoted by $\text{P}$₀) is also presented.     

Extreme points in sets of positive linear maps on B(H)

Three main results are obtained: (1) If $\text{D}$ is an atomic maximal Abelian subalgebra of $\text{B}$($\text{H}$), $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto $\text{D}$ and h is a complex homomorphism on $\text{D}$, then h ° $\text{P}$ is a pure state on $\text{B}$($\text{H}$). (2) If {P_n} is a sequence of mutually orthogonal projections with rank(P_n) = n and ∑ P_n = I, $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto {P_n}″ given by P(T)=∑trace_n(T)P_n and h is a homomorphism on {P_n}″ such that h(P_n) = 0 for all n then h ° $\text{P}$ induces a type II_∞ factor representation of the Calkin algebra. (3) If $\text{M}$ is a nonatomic maximal Abelian subalgebra of $\text{B}$($\text{H}$) then there is an atomic maximal Abelian subalgebra $\text{D}$ of $\text{B}$($\text{H}$) and a large family {Φ_α} of ¹-homomorphisms from $\text{D}$ onto $\text{M}$ such that for each α, Φ_α ° $\text{P}$ is an extreme point in the set of projections from $\text{B}$($\text{H}$) onto $\text{M}$. (Here $\text{P}$ denotes the projection of $\text{B}$($\text{H}$) onto $\text{D}$.)     

Transcendence order over Qp in Cp

Let (K, ∥ · ∥) be a valued transcendence degree 1 extension of $\text{Q}$_p. An element x ∈ K transcendental over $\text{Q}$_p is said to have order ≤a (a > 0) if there exists C_x > 0 such that every polynomial P(X) ∈ $\text{Q}$_p [X] satisfies

$\text{?}\text{log;}\text{(P(x))? ?}\text{log}\text{∥P∥+c}{}_{\mathrm{x}}\text{(}\text{deg}\text{P)}{}^{\mathrm{a}}$

when ∥ · ∥ is the Gauss norm on $\text{Q}$_p[X]. No x ∈ $\text{C}$_p can have order ≤α if α < 1 but we construct some x ∈ $\text{C}$_p with order ≤ 1. Furthermore, we prove order ≤α is stable by algebraic extension.     

The contraction of K to NM

By studying the contraction of K to $\text{N}\text{M}$ inside the Iwasawa decomposition of a semisimple Lie group of real rank 1, we realize the representations of $\text{NM}$ as limits of representations of K, obtaining in particular, limiting formulas for matrix entries.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

Structure of H-classes in the semigroup of nonnegative matrices

This paper investigates the structure of the $\text{H}$-classes in the semigroup N_n of nonnegative matrices. We obtain two sets of equivalent conditions for any two matrices A,B to satisfy $\text{A}\text{H}\text{B in N}{}_{\mathrm{n}}$. We establish a one-to-one and onto correspondence between the $\text{H}$-class $\text{H}$_A and the group W_A0 of the greatest cone independent submatrix A₀ of A. We find W_A0 can be made up from the groups of the connective submatrices of A₀.     

On some properties of convolution operators in K′1 and S′

Let $\text{H}$′ be either the space $\text{K}$′₁ of distributions of exponential growth or the space $\text{S}$′ of tempered distributions, and let $\text{O}$′_C($\text{H}$′:$\text{H}$′) be the space of convolution operators in $\text{H}$′. In each case $\text{H}$′ is the dual of a space $\text{H}$ of C^∞-functions which are in $\text{O}$′_C($\text{H}$′:$\text{H}$′). We establish necessary and sufficient conditions on the Fourier transform $\text{S}\text{?}$ of ? of S ? $\text{O}$′_C($\text{H}$′:$\text{H}$′) in order that every distribution u? $\text{O}$′_C($\text{H}$′:$\text{H}$′) with S1u?$\text{H}$ be in $\text{H}$. If $\text{H}$′ = $\text{K}$′₁, the condition is equivalent to S×H′¹=H′¹.     

A note on generalizations of strict diagonal dominance for real matrices

We investigate classes of real square matrices possessing some weakened from of strict diagonal dominance of a real matrix whose diagonal entries are all positive. The intersection of each one of these classes with the set of all real matrices, with nonpositive off-diagonal elements, coincides with the set of all nonsingular M- matrices.     

An analogue of the thom isomorphism for crossed products of a C1 algebra by an action of R

In this paper we show the existence and uniqueness of a natural isomorphism ø^j_α of K_j(A) with K_j+1(A ?_α$\text{R}$), j ? $\text{Z}$/2 where (A, $\text{R}$, α) is a $\text{C1}$ dynamical $\text{R}$-system, K is the functor of topological K theory and A ?_α$\text{R}$ is the crossed product of A by the action of $\text{R}$. The Pimsner-Voiculescu exact sequence is obtained as a corollary. We show that given an α-invariant trace τ on A, with dual trace $\text{}\text{?}$gt, one has $\text{}\text{?}\text{gtø}{}^1{}_{\mathrm{\alpha }}\text{[u] = (}\text{1}\text{2}\text{iπ) τ(δ(u)u1)}$ for any unitary u in the domain of the derivation δ of A associated to the action α. Finally, we show that the crossed product of C(S³) (continuous functions on the 3 sphere) by a minimal diffeomorphism is a simple $\text{C1}$ algebra with no nontrivial idempotent.     

Rickart C1 algebras,II

The theory of inner-outer factorization in the Hardy spaces H^p in the unit disc $\text{D}$ is well known and has many applications. It does not carry over to the spaces H^p on the polydisc $\text{D}$ⁿ or the ball $\text{B}$ⁿ when n > 1. However, for Lumer's Hardy spaces (LH)^p on any simply connected complex analytic manifold, we introduce the notions of internal and external functions and prove that every f? (LH)^p has a factorization f = I_ε × E_ε, where I_ε is internal and E_ε is external, and E_ε? (LH)^p?ε, for any ε > 0. The factorization is not unique and an example of Rudin shows that the ε is needed, at least when $\text{p =}\text{2}\text{m}$, where m is an integer.     

Sur l'η-expansion infinieOn infinite η-expansion

We give two characterizations of the ordering on Böhm trees induced by the D_∞ model, one of which formalizes a continuity property of infinite η-expansion: $\text{A}\text{?}{}_{\mathrm{\infty }}\text{B}$ if for any finite approximant A of $\text{A}$ there exists a finite approximant B of $\text{B}$ such that A is a sub-tree of B, modulo finitely many η-equalities and finitely many infinite η-expansions of variables. To cite this article: P.-L. Curien, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 77–82     

Equivariant embeddings of principal Z-bundles into complex vector bundles

Let π: E→X be a principal $\text{Z}$_n-bundle and p:V→X an m-dimensional complex vector bundle over, say, a connected CW-complex X. An equivariant embedding of π into p is an embedding h:E → V commuting with projections such that h(e · z)=zh(e) for all eεE and $\text{zε}\text{Z}{}_{\mathrm{n}}\text{?S}{}^1\text{?}\text{Z}$. We compute the primary obstruction $\text{cεH}{}^{\mathrm{2m}}\text{(X;}\text{Z}\text{)}$ to embedding π equivariantly into p. If dim X?2m, then c=0 if and only if π admits an equivariant embedding into p. If dim X>2m and π embeds equivariantly into p, then c=0. Other embedding criteria exist in case p is the trivial m-plane bundle ε^m. We use these criteria for a discussion of the classification of the equivalence classes of principal $\text{Z}$-bundles that admit equivariant embeddings into ε^m. Finally, we offer an example of a principal $\text{Z}$-bundle that admit an ordinary but not an equivariant embedding into ε¹.     

Zéros des fonctions L et formes toriquesZeroes of L-functions and toric forms

An algebraic number field K defines a maximal torus T of the linear group G=GL_n. Let χ be a character of the idele class group of K, satisfying suitable assumptions. The χ-toric form are the functions defined on $\text{G}{}_{\mathrm{<mtext>Q</mtext>}}\text{Z}{}_{\mathrm{<mtext>A</mtext>}}\text{?G}{}_{\mathrm{<mtext>A</mtext>}}$ such that the Fourier coefficient corresponding to χ with respect to the subgroup induced by T is zero. The Riemann hypothesis is equivalent to certain conditions concerning some spaces of toric forms, constructed from Eisenstein series. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of zeroes of L(s,χ) on the critical line. To cite this article: G. Lachaud, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 219–222.     

Codimension one spheres in Rn with double tangent balls

In contrast to the situation in $\text{R}$³, where a 2-sphere with double tangent balls at each point must be tamely embedded in $\text{R}$³, there exist wild (n?1)-spheres in $\text{R}$ⁿ for n>3 with this same geometric property. However, if the sphere Σ is tame moduio a subset X that lies in a polyhedron P that is tame in Σ, the dimension of P is less than n?2, n>4, and Σ has double tangent balls over X, then Σ must be tame in $\text{R}$ⁿ. Also if the tangent balls extend over P and are pairwise congruent, the dimensional restriction on P can be dropped. Examples are given to support the necessity of the hypotheses of the included theorems.     

Properties of Hida processes on R2. 2. Prediction and interpolation problems for processes on R2

The properties of N-Hida processes Part 1 (B. Prum, 1984, J. Multivar. Anal.15, 336–360) are studied when the indices set is $\text{R}$². First, the past of a point (s, t) of $\text{R}$² is extended to $\text{G}$_st = σ{γ_uv, u ≤ s or v ≤ t}. The dimension of the linear space generated by the conditional expectations of an N-Hida process γ_z when z goes over a p × q lattice is bounded by N(p + q ? 1). The same problem is then considered when the expectations are taken conditionally to the field generated by the process outside of a rectangle, and the bound of the dimension of the linear space generated on a lattice is also given. Special attention is devoted to the case when γ_z is a combination of strong martingales.     

Sur la cohomologie des algèbres de type AnOn the cohomology of type An algebras

We are interested here in the Hochschild cohomology of tensor triangular algebras $\text{T}$. We describe in particular a spectral sequence, whose terms are parametrized by the lengths of the trajectories of the quiver associated with $\text{T}$, and which converges to $\text{HH}{}^1\text{(}\text{T}\text{)}$, the Hochschild cohomology of $\text{T}$. Differentials at the first level are sums of cup products. To cite this article: S. Dourlens, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 527–532.     

On the ∂-Neumann problem for generalized functions

Using the new theory of generalized functions developed by one of the authors the $\text{?}$ equation in $\text{C}$ⁿ is studied. In particular it is proven that if G is any generalized function on $\text{C}$ (in the above sense) then there is a generalized function S on $\text{C}$ such that $\text{?S}\text{?}\text{z}\text{?}\text{= G}$. Several other results are proven valid in polydiscs of $\text{C}$ⁿ, for which differential forms whose coefficients are generalized functions are introduced.     

A maximum principle in Rn + 1

In this paper we establish maximum principles of the Cauchy problem for hyperbolic equations in $\text{R}$³ and $\text{R}$^{n + 1}(n ? 2). Our maximum principles generalize the results of Weinberger [5], and Sather [3, 4] for a class of equations such that the coefficients can be allowed to depend upon t, as well, in {x₁, x₂, t}-space and {x₁, x₂,…, x_n, t}-space. Throughout this paper, the influence of the work of Douglis [1] is apparent. See [2].     

Z3-invariants of real and imaginary quadratic fields

Let $\text{Q}$((?m)^{$\text{1}\text{2}$}) and $\text{Q}$((3m)^{$\text{1}\text{2}$}) be a pair of quadratic fields, m > 0, and let λ^?, μ^?; λ⁺, μ⁺ be the respective Iwasawa invariants of the basic $\text{Z}$₃-extensions of these fields. A generalization of a result of Scholz shows that λ^? ≥ λ⁺ and if μ^? = 0, then μ⁺ = 0.