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BISHOP’S THEOREM AND DIFFERENTIABILITY OF A SUBSPACE OF Cb(K)
, 2007
"... Let K be a Hausdorff space and Cb(K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Fréchet differentiability of subspaces of Cb(K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of Cb( ..."
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Let K be a Hausdorff space and Cb(K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Fréchet differentiability of subspaces of Cb(K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of Cb(K) is a dense Gδ subset of H, if K is compact. This gives a generalized Bishop’s theorem, which says that the closure of the set of strong peak point for H is the smallest closed norming subset of H. The classical Bishop’s theorem was proved for a separating subalgebra H and a metrizable compact space K. In the case that X is a complex Banach space with the RadonNikod´ym property, we show that the set of all strong peak functions in Ab(BX) = {f ∈ Cb(BX) : fB ◦ is holomorphic} is dense. As an application, we show that the X smallest closed norming subset of Ab(BX) is the closure of the set of all strong peak points for Ab(BX). This implies that the norm of Ab(BX) is Gâteaux differentiable on a dense subset of Ab(BX), even though the norm is nowhere Fréchet differentiable when X is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of numerical Shilov boundary.
THE 2CONCAVIFICATION OF A BANACH LATTICE EQUALS THE DIAGONAL OF THE FREMLIN TENSOR SQUARE
"... Abstract. We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach lattice E with itself and the 2concavification of E. 1. Introduction and ..."
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Abstract. We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach lattice E with itself and the 2concavification of E. 1. Introduction and
Isoperimetry of group actions
 Adv. Math
"... We survey the recent developments concerning fixed point properties for group actions on Banach spaces. In the setting of Hilbert spaces such fixed point properties correspond to Kazhdan’s property (T). Here we focus on the general, nonHilbert case, we discuss the methods, examples and several appl ..."
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We survey the recent developments concerning fixed point properties for group actions on Banach spaces. In the setting of Hilbert spaces such fixed point properties correspond to Kazhdan’s property (T). Here we focus on the general, nonHilbert case, we discuss the methods, examples and several applications.
On weak compactness in L1 spaces
, 2007
"... We will use the concept of strong generating and a simple renorming theorem to give new proofs to slight generalizations of some results of Argyros and Rosenthal on weakly compact sets in L1(µ) spaces for finite measures µ. ..."
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We will use the concept of strong generating and a simple renorming theorem to give new proofs to slight generalizations of some results of Argyros and Rosenthal on weakly compact sets in L1(µ) spaces for finite measures µ.
Multiplications and elementary operators in the Banach space setting
 In: Methods in Banach space theory. (Proc. V Conference on Banach spaces, Caceres, September 1318, 2004; J.F.M. Castillo and W.B. Johnson, eds.) London Mathematical Society Lecture Note Series 337 (Cambridge
, 2006
"... This expository survey is mainly dedicated to structural properties of the elementary operators (1.1) EA,B; S 7→ n∑ ..."
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This expository survey is mainly dedicated to structural properties of the elementary operators (1.1) EA,B; S 7→ n∑
Nemirovski’s Inequalities Revisited
, 2008
"... An important tool for statistical research are moment inequalities for sums of independent random vectors. Nemirovski and coworkers (1983, 2000) derived one particular type of such inequalities: For certain Banach spaces (B, ‖ · ‖) there exists a constant K = K(B, ‖ · ‖) such that for arbitrary ..."
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An important tool for statistical research are moment inequalities for sums of independent random vectors. Nemirovski and coworkers (1983, 2000) derived one particular type of such inequalities: For certain Banach spaces (B, ‖ · ‖) there exists a constant K = K(B, ‖ · ‖) such that for arbitrary independent and centered random vectors X1, X2,..., Xn ∈ B, their sum Sn satisfies the inequality IE ‖Sn ‖ 2 ≤ K ∑ n i=1 IE ‖Xi ‖ 2. We present and compare three different approaches to obtain such inequalities: Nemirovski’s results are based on deterministic inequalities for norms. Another possible vehicle are type and cotype inequalities, a tool from probability theory on Banach spaces. Finally, we use a truncation argument plus Bernstein’s inequality to obtain another version of the moment inequality above. Interestingly, all three approaches have their own merits. 1
The universality of ℓ1 as a dual space
 MATHEMATISCHE ANNALEN
, 2010
"... Let X be a Banach space with a separable dual. We prove that X embeds isomorphically into a L ∞ space Z whose dual is isomorphic to ℓ1. If, moreover, U is a space with separable dual, so that U and X are totally incomparable, then we construct such a Z, so that Z and U are totally incomparable. If ..."
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Let X be a Banach space with a separable dual. We prove that X embeds isomorphically into a L ∞ space Z whose dual is isomorphic to ℓ1. If, moreover, U is a space with separable dual, so that U and X are totally incomparable, then we construct such a Z, so that Z and U are totally incomparable. If X is separable and reflexive, we show that Z can be made to be somewhat reflexive.
Lower Bounds in Communication Complexity: A Survey
"... We survey lower bounds in communication complexity. Our focus is on lower bounds that work by first representing the communication complexity measure in Euclidean space. That is to say, the first step in these lower bound techniques is to find a geometric complexity measure such as rank, or the trac ..."
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We survey lower bounds in communication complexity. Our focus is on lower bounds that work by first representing the communication complexity measure in Euclidean space. That is to say, the first step in these lower bound techniques is to find a geometric complexity measure such as rank, or the trace norm that serves as a lower bound to the underlying communication complexity measure. Lower bounds on this geometric complexity measure