Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Marius Junge and Quanhua Xu
by Dale Alspach 17 May '05

17 May '05

This is an announcement for the paper "On the best constants in some non-commutative martingale inequalities" by Marius Junge and Quanhua Xu. Abstract: We determine the optimal orders for the best constants in the non-commutative Burkholder-Gundy, Doob and Stein inequalities obtained recently in the non-commutative martingale theory. Archive classification: Operator Algebras, Functional Analysis; Probability Mathematics Subject Classification: 46L53, 46L51 Citation: Bull. London Math. Soc. 37:243--253, 2005 The source file(s), constant.revised.tex: 33956 bytes, is(are) stored in gzipped form as 0505309.gz with size 11kb. The corresponding postcript file has gzipped size 48kb. Submitted from: qx(a)math.univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.OA/0505309 or http://arXiv.org/abs/math.OA/0505309 or by email in unzipped form by transmitting an empty message with subject line uget 0505309 or in gzipped form by using subject line get 0505309 to: math(a)arXiv.org.

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Abstract of a paper by Quanhua Xu
by Dale Alspach 17 May '05

17 May '05

This is an announcement for the paper "Embedding of $C_q$ and $R_q$ into noncommutative $L_p$-spaces, $1\le p<q\le 2$" by Quanhua Xu. Abstract: We prove that a quotient of subspace of $C_p\oplus_pR_p$ ($1\le p<2$) embeds completely isomorphically into a noncommutative $L_p$-space, where $C_p$ and $R_p$ are respectively the $p$-column and $p$-row Hilbertian operator spaces. We also represent $C_q$ and $R_q$ ($p<q\le2$) as quotients of subspaces of $C_p\oplus_pR_p$. Consequently, $C_q$ and $R_q$ embed completely isomorphically into a noncommutative $L_p(M)$. We further show that the underlying von Neumann algebra $M$ cannot be semifinite. Archive classification: Functional Analysis; Operator Algebras Mathematics Subject Classification: Primary 46L07; Secondary 47L25 The source file(s), embed.tex: 63829 bytes, is(are) stored in gzipped form as 0505307.gz with size 19kb. The corresponding postcript file has gzipped size 96kb. Submitted from: qx(a)math.univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505307 or http://arXiv.org/abs/math.FA/0505307 or by email in unzipped form by transmitting an empty message with subject line uget 0505307 or in gzipped form by using subject line get 0505307 to: math(a)arXiv.org.

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Abstract of a paper by Quanhua Xu
by Dale Alspach 17 May '05

17 May '05

This is an announcement for the paper "Operator space Grothendieck inequalities for noncommutative $L_p$-spaces" by Quanhua Xu. Abstract: We prove the operator space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$-spaces with $2<p<\infty$. One of our results states that given a map $u: E\to F^*$, where $E, F\subset L_p(M)$ ($2<p<\infty$, $M$ being a von Neumann algebra), $u$ is completely bounded iff $u$ factors through a direct sum of a $p$-column space and a $p$-row space. We also obtain several operator space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative $L_p$-space ($2<p<\infty$) with values in a $q$-column space for every $q\in [p', p]$ ($p'$ being the index conjugate to $p$). These results are the $L_p$-space analogues of the recent works on the operator space Grothendieck theorems by Pisier and Shlyakhtenko. The key ingredient of our arguments is some Khintchine type inequalities for Shlyakhtenko's generalized circular systems. One of our main tools is a Haagerup type tensor norm, which turns out particularly fruitful when applied to subspaces of noncommutative $L_p$-spaces ($2<p<\infty$). In particular, we show that the norm dual to this tensor norm, when restricted to subspaces of noncommutative $L_p$-spaces, is equal to the factorization norm through a $p$-row space. Archive classification: Functional Analysis; Operator Algebras Mathematics Subject Classification: Primary 46L07; Secondary 46L50 Remarks: To appear in Duke Math. J The source file(s), gro.tex: 127607 bytes, is(are) stored in gzipped form as 0505306.gz with size 38kb. The corresponding postcript file has gzipped size 172kb. Submitted from: qx(a)math.univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505306 or http://arXiv.org/abs/math.FA/0505306 or by email in unzipped form by transmitting an empty message with subject line uget 0505306 or in gzipped form by using subject line get 0505306 to: math(a)arXiv.org.

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Abstract of a paper by Quanhua Xu
by Dale Alspach 17 May '05

17 May '05

This is an announcement for the paper "A description of $\big(C_p[L_p(M)],\; R_p[L_p(M)]\big)_\theta$" by Quanhua Xu. Abstract: We give a simple explicit description of the norm in the complex interpolation space $(C_p[L_p(M)],\; R_p[L_p(M)])_\theta$ for any von Neumann algebra $M$ and any $1\le p\le\infty$. Archive classification: Functional Analysis; Operator Algebras Mathematics Subject Classification: Primary 46M35 and 46L51; Secondary 46L07 Remarks: To appear in Proc. Edinburgh Math. Soc The source file(s), interpCR.tex: 33942 bytes, is(are) stored in gzipped form as 0505305.gz with size 11kb. The corresponding postcript file has gzipped size 61kb. Submitted from: qx(a)math.univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505305 or http://arXiv.org/abs/math.FA/0505305 or by email in unzipped form by transmitting an empty message with subject line uget 0505305 or in gzipped form by using subject line get 0505305 to: math(a)arXiv.org.

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Abstract of a paper by Teresa Martinez, Jose L. Torrea and Quanhua Xu
by Dale Alspach 17 May '05

17 May '05

This is an announcement for the paper "Vector-valued Littlewood-Paley-Stein theory for semigroups" by Teresa Martinez, Jose L. Torrea and Quanhua Xu. Abstract: We develop a generalized Littlewood-Paley theory for semigroups acting on $L^p$-spaces of functions with values in uniformly convex or smooth Banach spaces. We characterize, in the vector-valued setting, the validity of the one-sided inequalities concerning the generalized Littlewood-Paley-Stein $g$-function associated with a subordinated Poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying Banach space. We show that in the case of the usual Poisson semigroup and the Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on ${\mathbb R}^n$, this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calder\'on-Zygmund singular integral operators. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20; 42B25, 42A61 Remarks: To appear in Adv. Math The source file(s), lpsII.tex: 111765 bytes, is(are) stored in gzipped form as 0505303.gz with size 31kb. The corresponding postcript file has gzipped size 144kb. Submitted from: qx(a)math.univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505303 or http://arXiv.org/abs/math.FA/0505303 or by email in unzipped form by transmitting an empty message with subject line uget 0505303 or in gzipped form by using subject line get 0505303 to: math(a)arXiv.org.

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Abstract of a paper by Eric Ricard and Quanhua Xu
by Dale Alspach 17 May '05

17 May '05

This is an announcement for the paper "Khintchine type inequalities for reduced free products and applications" by Eric Ricard and Quanhua Xu. Abstract: We prove Khintchine type inequalities for words of a fixed length in a reduced free product of $C^*$-algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length $d$ is completely bounded with norm depending linearly on $d$. We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykema's theorem on the stability of exactness under the reduced free product for $C^*$-algebras. We next study the stability of the completely contractive approximation property (CCAP) under reduced free product. Our first result in this direction is that a reduced free product of finite dimensional $C^*$-algebras has the CCAP. The second one asserts that a von Neumann reduced free product of injective von Neumann algebras has the weak-$*$ CCAP. In the case of group $C^*$-algebras, we show that a free product of weakly amenable groups with constant 1 is weakly amenable. Archive classification: Operator Algebras; Functional Analysis Mathematics Subject Classification: Primary 46L09, 46L54; Secondary 47L07, 47L25 The source file(s), kpl.tex: 94450 bytes, is(are) stored in gzipped form as 0505302.gz with size 30kb. The corresponding postcript file has gzipped size 136kb. Submitted from: qx(a)math.univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.OA/0505302 or http://arXiv.org/abs/math.OA/0505302 or by email in unzipped form by transmitting an empty message with subject line uget 0505302 or in gzipped form by using subject line get 0505302 to: math(a)arXiv.org.

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Abstract of a paper by Felix Cabello Sanchez, Jesus M. F. Castillo and Pier Luigi Papini
by Dale Alspach 17 May '05

17 May '05

This is an announcement for the paper "Seven views on approximate convexity and the geometry of K-spaces" by Felix Cabello Sanchez, Jesus M. F. Castillo and Pier Luigi Papini. Abstract: As in Hokusai's series of paintings "Thirty six views of mount Fuji" in which mount Fuji's is sometimes scarcely visible, the central topic of this paper is the geometry of $K$-spaces although in some of the seven views presented $K$-spaces are not easily visible. We study the interplay between the behaviour of approximately convex (and approximately affine) functions on the unit ball of a Banach space and the geometry of Banach K-spaces. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20; 52A05; 42A65; 26B25 Remarks: 2 figures The source file(s), ccp.tex: 61322 bytes, cubo.eps: 19389 bytes, kominek.eps: 82795 bytes, is(are) stored in gzipped form as 0505291.tar.gz with size 38kb. The corresponding postcript file has gzipped size 119kb. Submitted from: castillo(a)unex.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505291 or http://arXiv.org/abs/math.FA/0505291 or by email in unzipped form by transmitting an empty message with subject line uget 0505291 or in gzipped form by using subject line get 0505291 to: math(a)arXiv.org.

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Abstract of a paper by M.Yaskina
by Dale Alspach 17 May '05

17 May '05

This is an announcement for the paper "Non-intersection bodies all of whose central sections are intersection bodies" by M.Yaskina. Abstract: We construct symmetric convex bodies that are not intersection bodies, but all of their central hyperplane sections are intersection bodies. This result extends the studies by Weil in the case of zonoids and by Neyman in the case of subspaces of $L_p$. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 52A20, 52A21, 46B20 Remarks: 10 pages The source file(s), inters8.tex: 33376 bytes, is(are) stored in gzipped form as 0505277.gz with size 10kb. The corresponding postcript file has gzipped size 54kb. Submitted from: yaskinv(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505277 or http://arXiv.org/abs/math.FA/0505277 or by email in unzipped form by transmitting an empty message with subject line uget 0505277 or in gzipped form by using subject line get 0505277 to: math(a)arXiv.org.

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Abstract of a paper by Denny H. Leung and Wee-Kee Tang
by Dale Alspach 11 May '05

11 May '05

This is an announcement for the paper "Extension of functions with small oscillation" by Denny H. Leung and Wee-Kee Tang. Abstract: A classical theorem of Kuratowski says that every Baire one function on a G_\delta subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index \beta(f). We prove a refinement of Kuratowski's theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that \beta_{Y}(f)<\omega^{\alpha}, \alpha < \omega_1, then f has an extension F onto X so that \beta_X(F)is not more than \omega^{\alpha}. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that \beta_{X}(F)is not more than 3. An example is constructed to show that this result is optimal. Archive classification: Classical Analysis and ODEs; Functional Analysis Mathematics Subject Classification: 26A21; 03E15, 54C30 The source file(s), DLeungWTangBaire1Ext.tex: 47118 bytes, is(are) stored in gzipped form as 0505168.gz with size 13kb. The corresponding postcript file has gzipped size 71kb. Submitted from: wktang(a)nie.edu.sg The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.CA/0505168 or http://arXiv.org/abs/math.CA/0505168 or by email in unzipped form by transmitting an empty message with subject line uget 0505168 or in gzipped form by using subject line get 0505168 to: math(a)arXiv.org.

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Abstract of a paper by Marianne Morillon
by Dale Alspach 11 May '05

11 May '05

This is an announcement for the paper "A new proof of James' sup theorem" by Marianne Morillon. Abstract: We provide a new proof of James' sup theorem for (non necessarily separable) Banach spaces. One of the ingredients is the following generalization of a theorem of Hagler and Johnson (1977) : "If a normed space $E$ does not contain any asymptotically isometric copy of $\ell^1(\IN)$, then every bounded sequence of $E'$ has a normalized block sequence pointwise converging to $0$". Archive classification: Functional Analysis Mathematics Subject Classification: 46B ; 03E25 Report Number: ERMIT-MM-07jan2005 The source file(s), envoi.bbl: 2807 bytes, envoi.tex: 35905 bytes, icone-ermit.eps: 24310 bytes, is(are) stored in gzipped form as 0505176.tar.gz with size 19kb. The corresponding postcript file has gzipped size 69kb. Submitted from: Marianne.Morillon(a)univ-reunion.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505176 or http://arXiv.org/abs/math.FA/0505176 or by email in unzipped form by transmitting an empty message with subject line uget 0505176 or in gzipped form by using subject line get 0505176 to: math(a)arXiv.org.

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