Banach December 2006

banach@mathdept.okstate.edu

1 participants
4 discussions

Abstract of a paper by Thomas Jech
by Dale Alspach 23 Dec '06

23 Dec '06

This is an announcement for the paper "Algebraic characterizations of measure algebras" by Thomas Jech. Abstract: We present necessary and sufficient conditions for the existence of a countably additive measure on a complete Boolean algebra. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 28 The source file(s), Measure.tex: 31579 bytes, is(are) stored in gzipped form as 0612598.gz with size 9kb. The corresponding postcript file has gzipped size 89kb. Submitted from: jech(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0612598 or http://arXiv.org/abs/math.FA/0612598 or by email in unzipped form by transmitting an empty message with subject line uget 0612598 or in gzipped form by using subject line get 0612598 to: math(a)arXiv.org.

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Abstract of a paper by R Haydon, A Molto and J Orihuela
by Dale Alspach 13 Dec '06

13 Dec '06

This is an announcement for the paper "Spaces of functions with countably many discontinuities" by R Haydon, A Molto and J Orihuela. Abstract: Let $\Gamma$ be a Polish space and let $K$ be a separable and poointwise compact set of real-valued functions on $\Gamma$. It is shown that if each function in $K$ has only countably many discontinuities then $C(K)$ may be equipped with a $T_p$-lower semicontinuous and locally uniformly convex norm, equivalent to the supremum norm. Archive classification: Functional Analysis; General Topology Mathematics Subject Classification: 46B03; 54H05 The source file(s), fewdiscfinal.tex: 56379 bytes, is(are) stored in gzipped form as 0612307.gz with size 18kb. The corresponding postcript file has gzipped size 144kb. Submitted from: richard.haydon(a)bnc.ox.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0612307 or http://arXiv.org/abs/math.FA/0612307 or by email in unzipped form by transmitting an empty message with subject line uget 0612307 or in gzipped form by using subject line get 0612307 to: math(a)arXiv.org.

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Abstract of a paper by Joel A. Tropp
by Dale Alspach 06 Dec '06

06 Dec '06

This is an announcement for the paper "A new proof of the paving property for uniformly bounded matrices" by Joel A. Tropp. Abstract: This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison--Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systematic use of symmetrization and Khintchine inequalities to estimate the norm of some random matrices. The key new ideas are due to Rudelson. Archive classification: Metric Geometry; Functional Analysis; Probability Mathematics Subject Classification: 46B07; 47A11; 15A52 Remarks: 12 pages The source file(s), bdd-ks-v1.bbl: 2693 bytes, bdd-ks-v1.tex: 41646 bytes, macro-file.tex: 8551 bytes, is(are) stored in gzipped form as 0612070.tar.gz with size 15kb. The corresponding postcript file has gzipped size 99kb. Submitted from: jtropp(a)umich.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0612070 or http://arXiv.org/abs/math.MG/0612070 or by email in unzipped form by transmitting an empty message with subject line uget 0612070 or in gzipped form by using subject line get 0612070 to: math(a)arXiv.org.

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Abstract of a paper by Jeff Cheeger and Bruce Kleiner
by Dale Alspach 02 Dec '06

02 Dec '06

This is an announcement for the paper "Differentiating maps into L^1 and the geometry of BV functions" by Jeff Cheeger and Bruce Kleiner. Abstract: This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X--->V, and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V=L^1 where differentiability fails. We establish another kind of differentiability for certain X, including R^n and H, the Heisenberg group with its Carnot-Cartheodory metric. It follows that H does not bi-Lipschitz embed into L^1, as conjectured by J. Lee and A. Naor. When combined with their work, this provides a natural counter example to the Goemans-Linial conjecture in theoretical computer science; the first such counterexample was found by Khot-Vishnoi. A key ingredient in the proof of our main theorem is a new connection between Lipschitz maps to L^1 and functions of bounded variation, which permits us to exploit recent work on the structure of BV functions on the Heisenberg group. Archive classification: Metric Geometry; Differential Geometry; Functional Analysis; Group The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0611954 or http://arXiv.org/abs/math.MG/0611954 or by email in unzipped form by transmitting an empty message with subject line uget 0611954 or in gzipped form by using subject line get 0611954 to: math(a)arXiv.org.

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Banach December 2006