arXiv subjects

Narcisse Randrianantoanina

Publications and source records attributed to Narcisse Randrianantoanina.

27 records · Page 2Linked to original sources

arXiv1997

Factorization of operators on $C^*$-algebras

Let $A$ be a $C^*$-algebra. It is shown that every absolutely summing operator from $A$ into $\ell_2$ factors through a Hilbert space operator that belongs to the 4-Schatten- von Neumann class. We also provide finite dimensinal examples that show that one can not improve the 4-Schatten-von Neumann class to $p$-Schatten von Neumann class for any $p<4$. As application, we prove that there exists a modulus of capacity $ε\to N(ε)$ so that if $A$ is a $C^*$-algebra and $T \in Π_1(A,\ell_2)$ with $π_1(T)\leq 1$, then for every $ε>0$, the $ε$-capacity of the image of the unit ball of $A$ under $T$ does not exceed $N(ε)$. This aswers positively a question raised by Pełczynski.

math.FA↗

arXiv1997

Hilbert transform associated with finite maximal subdiagonal algebras

Let $M$ be a von Neumann algebra with a faithful normal finite trace $t$, and $H^\infty$ be a finite, maximal, subdiagonal of $M$. Fundamental theorems on conjugate functions for weak* Dirichlet algebras are shown to be a bounded linear map from $L^p(M,t)$ into $L^p(M,t)$ for $1<p<\infty$, and to be a continuous linear map from $L^1(M,t)$ into $L^{1,\infty}(M,t)$. We also obtain that if a positive operator $a$ is such that $a\log^{+}a \in L^1(M,t)$, then its conjugate belongs to $L^1(M,t)$.

math.FA↗

arXiv1996

Conjugate operators for finite maximal subdiagonal algebras

Let $\M$ be a von Neumann algebra with a faithful normal trace $\T$, and let $H^\infty$ be a finite, maximal, subdiagonal algebra of $\M$. Fundamental theorems on conjugate functions for weak$^*$\!-Dirichlet algebras are shown to be valid for non-commutative $H^\infty$. In particular the conjugation operator is shown to be a bounded linear map from $L^p(\M, \T)$ into $L^p(\M, \T)$ for $1 < p < \infty$, and to be a continuous map from $L^1(\M,\T)$ into $L^{1, \infty}(\M,\T)$. We also obtain that if an operator $a$ is such that $|a|\log^+|a| \in L^1(\M,\T)$ then its conjugate belongs to $L^1(\M,\T)$. Finally, we present some partial extensions of the classical Szegö's theorem to the non-commutative setting.

math.OA↗

arXiv1995

Absolutely Summing Operators on non commutative $C^*$-algebras and applications

Let $E$ be a Banach space that does not contain any copy of $\ell^1$ and $\A$ be a non commutative $C^*$-algebra. We prove that every absolutely summing operator from $\A$ into $E^*$ is compact, thus answering a question of Pełczynski. As application, we show that if $G$ is a compact metrizable abelian group and $Λ$ is a Riesz subset of its dual then every countably additive $\A^*$-valued measure with bounded variation and whose Fourier transform is supported by $Λ$ has relatively compact range. Extensions of the same result to symmetric spaces of measurable operators are also presented.

math.FA↗

arXiv1995

Complemented copies of $\ell_1$ in spaces of vector valued measures and applications

Let $X$ be a Banach space and $(Ω,Σ)$ be a measure space. We provide a characterization of sequences in the space of $X$-valued countably additive measures on $Ω,Σ)$ of bounded variation that generate complemented copies of $\ell_1$. As application, we prove that if a dual Banach space $E^*$ has Pełczyński's property (V*) then so does the space of $E^*$-valued countably additive measures with bounded variation. Another application, we show that for a Banach space $X$, the space $\ell_\infty(X)$ contains a complemented copy of $\ell_1$ if and only if $X$ contains all $\ell_1^n$ uniformly complemented.

math.FA↗

arXiv1995

Some remarks on the Dunford-Pettis property

Let $A$ be the disk algebra, $Ω$ be a compact Hausdorff space and $μ$ be a finite Borel measure in $Ω$. It is shown that the dual of $C(Ω,A)$ has the Dunford-Pettis Property. This proved in particular that the spaces $C(Ω,A)$ and $L^1(μ,A*)$ have the Dunford-Pettis property.

math.FA↗

arXiv1994

Complemented copies of $\ell^1$ and Pelczynski's property (V*) in Bochner function spaces

Let $X$ be a Banach space and $(f_n)_n$ be a bounded sequence in $L^1(X)$. We prove a complemented version of the celebrated Talagrand's dichotomy i.e we show that if $(e_n)_n$ denotes the unit vector basis of $c_0$, there exists a sequence $g_n \in \text{conv}(f_n,f_{n+1},\dots)$ such that for almost every $ω$, either the sequence $(g_n(ω) \otimes e_n)$ is weakly Cauchy in $X \widehat{\otimes}_πc_0$ or it is equivalent to the unit vector basis of $\ell^1$. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of $\ell^1$ in $L^1(X)$. As an application, we show that for a Banach space $X$, the space $L^1(X)$ has Pełczyński's property $(V^*)$ if and only if $X$ does.

math.FA↗

arXiv1994

Pelczynski's property (V) on spaces of vector valued functions

Let $E$ be a separable Banach space and $Ω$ be a compact Hausdorff space. It is shown that the space $C(Ω,E)$ has property (V) if and only if $E$ does. Similar result is also given for Bochner spaces $L^p(μ,E)$ if $1<p<\infty$ and $μ$ is a finite Borel measure on $Ω$.

math.FA↗

arXiv1994

Some properties of space of compact operators

Let $X$ be a separable Banach space, $Y$ be a Banach space and $Λ$ be a subset of the dual group of a given compact metrizable abelian group. We prove that if $X^*$ and $Y$ have the type I-$Λ$-RNP (resp. type II-$Λ$-RNP) then $K(X,Y)$ has the type I-$Λ$-RNP (resp. type II-$Λ$-RNP) provided $L(X,Y)=K(X,Y)$. Some corollaries are then presented as well as results conserning the separability assumption on $X$. Similar results for the NearRNP and the WeakRNP are also presented.

math.FA↗