Banach August 2005

banach@mathdept.okstate.edu

2 participants
8 discussions

Abstract of a paper by Mukul S. Patel
by Dale Alspach 30 Aug '05

30 Aug '05

This is an announcement for the paper "Noncommmutative Gelfand duality for not necessarily unital $C^*$-algebras, Jordan canonical form, and the existence of invariant subspaces" by Mukul S. Patel. Abstract: Gelfand-Naimark duality (Commutative $C^*$-algebras $\equiv$ Locally compact Hausdorff spaces) is extended to \begin{center}$C^*$-algebras $\equiv$ Quotient maps on locally compact Hausdorff spaces.\end{center} Using this duality, we give for an \emph{arbitrary} bounded operator on a complex Hilbert space of several dimensions, a functional calculus and the existence theorem for nontrivial invariant subspace. Archive classification: Functional Analysis; Operator Algebras Mathematics Subject Classification: 46L05; 47A13; 47A13; 43A40; 22B05 Remarks: Under consideration for publication by Electronic Reasearch The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0508545 or http://arXiv.org/abs/math.FA/0508545 or by email in unzipped form by transmitting an empty message with subject line uget 0508545 or in gzipped form by using subject line get 0508545 to: math(a)arXiv.org.

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Abstract of a paper by A K Mirmostafaee
by Dale Alspach 26 Aug '05

26 Aug '05

This is an announcement for the paper "A note on convex renorming and fragmentability" by A K Mirmostafaee. Abstract: Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a)~If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b)~If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c)~If the Banach space is weakly locally uniformly rotund, then its weak topology is fragmented by a metric which is stronger than the norm topology. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20, 54E99, 54H05 Citation: Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 2, May 2005, The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0508311 or http://arXiv.org/abs/math.FA/0508311 or by email in unzipped form by transmitting an empty message with subject line uget 0508311 or in gzipped form by using subject line get 0508311 to: math(a)arXiv.org.

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Abstract of a paper by Gilles Pisier
by Dale Alspach 26 Aug '05

26 Aug '05

This is an announcement for the paper "Simultaneous similarity, bounded generation and amenability" by Gilles Pisier. Abstract: We prove that a discrete group $G$ is amenable iff it is strongly unitarizable in the following sense: every unitarizable representation $\pi$ on $G$ can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of $\pi$. Analogously a $C^*$-algebra $A$ is nuclear iff any bounded homomorphism $u:\ A\to B(H)$ is strongly similar to a $*$-homomorphism in the sense that there is an invertible operator $\xi$ in the von Neumann algebra generated by the range of $u$ such that $a\to \xi u(a) \xi^{-1}$ is a $*$-homomorphism. An analogous characterization holds in terms of derivations. We apply this to answer several questions left open in our previous work concerning the length $\ell(A,B)$ of the maximal tensor product $A\otimes_{\max} B$ of two unital $C^*$-algebras, when we consider its generation by the subalgebras $A\otimes 1$ and $1\otimes B$. We show that if $\ell(A,B)<\infty$ either for $B=B(\ell_2)$ or when $B$ is the $C^*$-algebra (either full or reduced) of a non Abelian free group, then $A$ must be nuclear. We also show that $\ell(A,B)\le d$ iff the canonical quotient map from the unital free product $A\ast B$ onto $A\otimes_{\max} B$ remains a complete quotient map when restricted to the closed span of the words of length $\le d$. Archive classification: Operator Algebras; Functional Analysis Mathematics Subject Classification: Primary 46 L06. Secondary: 46L07, 46L57 Primary 46 L06. Secondary: The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.OA/0508223 or http://arXiv.org/abs/math.OA/0508223 or by email in unzipped form by transmitting an empty message with subject line uget 0508223 or in gzipped form by using subject line get 0508223 to: math(a)arXiv.org.

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Abstract of a paper by Anna Maria Pelczar
by Dale Alspach 26 Aug '05

26 Aug '05

This is an announcement for the paper "Stabilization of Tsirelson-type norms on $\ell_p$ spaces" by Anna Maria Pelczar. Abstract: We consider classical Tsirelson-type norms of $T[A_r,\theta]$ and their modified versions on $\ell_p$ spaces. We show that for any $1<p<\infty$ there is a constant $\lambda_p$ such that considered Tsirelson-type norms do not $\lambda_p$-distort any of subspaces of $\ell_p$. Archive classification: Functional Analysis Mathematics Subject Classification: 46B03 Remarks: 10 pages The source file(s), stab-tsir.tex: 27412 bytes, is(are) stored in gzipped form as 0508352.gz with size 9kb. The corresponding postcript file has gzipped size 57kb. Submitted from: anna.pelczar(a)im.uj.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0508352 or http://arXiv.org/abs/math.FA/0508352 or by email in unzipped form by transmitting an empty message with subject line uget 0508352 or in gzipped form by using subject line get 0508352 to: math(a)arXiv.org.

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Abstract of a paper by Javier Parcet
by Dale Alspach 26 Aug '05

26 Aug '05

This is an announcement for the paper "Weak type estimates associated to Burkholder's martingale inequality" by Javier Parcet. Abstract: Given a probability space $(\Omega, \mathsf{A}, \mu)$, let $\mathsf{A}_1, \mathsf{A}_2, \ldots$ be a filtration of $\sigma$-subalgebras of $\mathsf{A}$ and let $\mathsf{E}_1, \mathsf{E}_2, \ldots$ denote the corresponding family of conditional expectations. Given a martingale $f = (f_1, f_2, \ldots)$ adapted to this filtration and bounded in $L_p(\Omega)$ for some $2 \le p < \infty$, Burkholder's inequality claims that $$\|f\|_{L_p(\Omega)} \sim_{\mathrm{c}_p} \Big\| \Big( \sum_{k=1}^\infty \mathsf{E}_{k-1}(|df_k|^2) \Big)^{1/2} \Big\|_{L_{p}(\Omega)} + \Big( \sum_{k=1}^\infty \|df_k\|_p^p \Big)^{1/p}.$$ Motivated by quantum probability, Junge and Xu recently extended this result to the range $1 < p < 2$. In this paper we study Burkholder's inequality for $p=1$, for which the techniques (as we shall explain) must be different. Quite surprisingly, we obtain two non-equivalent estimates which play the role of the weak type $(1,1)$ analog of Burkholder's inequality. As application, we obtain new properties of Davis decomposition for martingales. Archive classification: Probability; Functional Analysis Mathematics Subject Classification: 42B25; 60G46; 60G50 Remarks: 19 pages The source file(s), WeakBurk.tex: 66319 bytes, is(are) stored in gzipped form as 0508447.gz with size 18kb. The corresponding postcript file has gzipped size 88kb. Submitted from: javier.parcet(a)uam.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.PR/0508447 or http://arXiv.org/abs/math.PR/0508447 or by email in unzipped form by transmitting an empty message with subject line uget 0508447 or in gzipped form by using subject line get 0508447 to: math(a)arXiv.org.

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Abstract of a paper by Marc A. Rieffel
by Dale Alspach 08 Aug '05

08 Aug '05

This is an announcement for the paper "Lipschitz extension constants equal projection constants" by Marc A. Rieffel. Abstract: For a finite-dimensional Banach space $V$ we define its Lipschitz extension constant, $\cL\cE(V)$, to be the smallest constant $c$ such that for every compact metric space $(Z,\rho)$, every $X \subset Z$, and every $f: X \to V$, there is an extension, $g$, of $f$ to $Z$ such that $L(g) \le cL(f)$, where $L$ denotes the Lipschitz constant. Our main theorem is that $\cL\cE(V) = \cP\cC(V)$ where $\cP\cC(V)$ is the well-known projection constant of $V$. We obtain some consequences, especially when $V = M_n(\bC)$. We also discuss what happens if we also require that $\|g\|_{\infty} = \|f\|_{\infty}$. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 46B20; 26A16 Remarks: 12 pages. Intended for the proceedings of GPOTS05 The source file(s), liparc.tex: 35141 bytes, is(are) stored in gzipped form as 0508097.gz with size 12kb. The corresponding postcript file has gzipped size 61kb. Submitted from: rieffel(a)math.berkeley.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0508097 or http://arXiv.org/abs/math.FA/0508097 or by email in unzipped form by transmitting an empty message with subject line uget 0508097 or in gzipped form by using subject line get 0508097 to: math(a)arXiv.org.

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Abstract of a paper by M. Junge, D. Kutzarova and E. Odell
by Dale Alspach 02 Aug '05

02 Aug '05

This is an announcement for the paper "On asymptotically symmetric Banach spaces" by M. Junge, D. Kutzarova and E. Odell. Abstract: We define and study asymptotically symmetric Banach spaces (a.s.) and its variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse fails. We also show that w.a.s. and w.n.a.s. are not equivalent properties and that Schlumprecht's space S fails to be w.n.a.s. We show that if X is separable and has the property that every normalized weakly null sequence in X has a subsequence equivalent to the unit vector basis of c_0 then X is w.a.s.. We obtain an analogous result if c_0 is replaced by ell_1 and also show it is false if c_0 is replaced by ell_p, 1 < p < infinity. We prove that if 1 less than or equal p < infinity and the norm of the sum of (x_i)_1^n is of the order n^{1/p} for all (x_i)_1^n in the n^{th} asymptotic structure of $X$, then X contains an asymptotic ell_p, hence w.a.s. subspace. Archive classification: Functional Analysis Mathematics Subject Classification: 46B03; 46B20 Remarks: 22 pages, AMSLaTeX The source file(s), jko31.tex: 68725 bytes, is(are) stored in gzipped form as 0508035.gz with size 21kb. The corresponding postcript file has gzipped size 107kb. Submitted from: combs(a)mail.ma.utexas.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0508035 or http://arXiv.org/abs/math.FA/0508035 or by email in unzipped form by transmitting an empty message with subject line uget 0508035 or in gzipped form by using subject line get 0508035 to: math(a)arXiv.org.

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conference at U of Missouri.
by Dale Alspach 01 Aug '05

01 Aug '05

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SECOND ANNOUNCEMENT Seminar on Analysis December 9-10, 2005 in honor of Elias Saab's 60th birthday. The following have accepted to give talks in this meeting: Joe Diestel (Kent State) Gilles Godefroy (University of Paris VI) Steve Hofmann (MU) Alex Koldobsky (MU) Marius Mitrea (MU) Narcisse Randrianantoanina (Miami University, OH) Mark Rudelson (MU) Jerry Uhl (Urbana-Champaign) For more up to date information on this conference please visit http://www.math.missouri.edu/~organizers/ Partial support is available for graduate students and recent Ph.D. recipients. Women and minorities are particularly encouraged to apply. For instructions on how to apply go to http://www.math.missouri.edu/~organizers/financial.htm If you have any questions please contact the organizers by e-mail at organizers(a)math.missouri.edu The Seminar on Analysis is sponsored by the University of Missouri-Columbia College of Arts and Science and the Department of Mathematics. The organizers, Fritz Gesztesy and Dorina Mitrea %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -- %------------------------------------------- % Dorina Mitrea | % University of Missouri-Columbia | % Department of Mathematics | % 202 Math Sciences Bldg | % Columbia, MO 65211 | % | % Phone: (573)-882-4233 | % Fax: (573)-882-1869 | % e-mail: dorina(a)math.missouri.edu | % http://www.math.missouri.edu/ | %--------------------------------------------

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Banach August 2005