scholarly journals CONCERNING SUMMABLE SZLENK INDEX

2018 ◽  
Vol 62 (1) ◽  
pp. 59-73
Author(s):  
RYAN MICHAEL CAUSEY
Keyword(s):  
Banach Space ◽  
Short Proof ◽  
Compact Set ◽  
Direct Sums ◽  
Ideal Norm ◽  
Injective Tensor Product ◽  
Finite Dimensional ◽  
Szlenk Index ◽  
Funct Anal ◽  
Final Consequence

AbstractWe generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak*-compact set. We prove that a weak*-compact set has summable Szlenk index if and only if its weak*-closed, absolutely convex hull does. As a consequence, we offer a new, short proof of a result from Draga and Kochanek [J. Funct. Anal. 271 (2016), 642–671] regarding the behavior of summability of the Szlenk index under c0 direct sums. We also use this result to prove that the injective tensor product of two Banach spaces has summable Szlenk index if both spaces do, which answers a question from Draga and Kochanek [Proc. Amer. Math. Soc. 145 (2017), 1685–1698]. As a final consequence of this result, we prove that a separable Banach space has summable Szlenk index if and only if it embeds into a Banach space with an asymptotic c0 finite dimensional decomposition, which generalizes a result from Odell et al. [Q. J. Math. 59, (2008), 85–122]. We also introduce an ideal norm $\mathfrak{s}$ on the class $\mathfrak{S}$ of operators with summable Szlenk index and prove that $(\mathfrak{S}, \mathfrak{s})$ is a Banach ideal. For 1 ⩽ p ⩽ ∞, we prove precise results regarding the summability of the Szlenk index of an ℓp direct sum of a collection of operators.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

scholarly journals TENSOR EXTENSION PROPERTIES OF C(K)-REPRESENTATIONS AND APPLICATIONS TO UNCONDITIONALITY

2010 ◽  
Vol 88 (2) ◽  
pp. 205-230 ◽  
Author(s):  
CHRISTOPH KRIEGLER ◽  
CHRISTIAN LE MERDY
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unconditional Basis ◽  
Compact Set ◽  
Bounded Operators ◽  
Space Setting ◽  
Uniformly Bounded

AbstractLet K be any compact set. The C*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept ofR-boundedness. Then we apply these results to operators with a uniformly bounded H∞-calculus, as well as to unconditionality on Lp. We show that any unconditional basis on Lp ‘is’ an unconditional basis on L2 after an appropriate change of density.

CLASSES OF SEQUENTIALLY LIMITED OPERATORS

2015 ◽  
Vol 58 (3) ◽  
pp. 573-586
Author(s):  
JAN H. FOURIE ◽  
ELROY D. ZEEKOEI
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Normed Space ◽  
Operator Ideal ◽  
Operator Ideals ◽  
Ideal Norm ◽  
Relevant Operator

AbstractThe purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals.

scholarly journals On the convergence of greedy algorithms for initial segments of the Haar basis

2010 ◽  
Vol 148 (3) ◽  
pp. 519-529 ◽  
Author(s):  
S. J. DILWORTH ◽  
E. ODELL ◽  
TH. SCHLUMPRECHT ◽  
ANDRÁS ZSÁK
Keyword(s):  
Banach Space ◽  
Greedy Algorithm ◽  
General Result ◽  
Initial Segment ◽  
Greedy Algorithms ◽  
Dimensional Banach Space ◽  
Haar Basis ◽  
Finite Dimensional ◽  
Finite Dimensional Banach Space ◽  
Number Of Iterations

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

scholarly journals On a problem of Barnes and Duncan

1991 ◽  
Vol 34 (2) ◽  
pp. 321-323
Author(s):  
R. G. McLean
Keyword(s):  
Banach Algebra ◽  
Free Monoid ◽  
Finite Dimensional ◽  
Image Position ◽  
Funct Anal

Consider the free monoid on a non-empty set P, and let R be the quotient monoid determined by the relations:Let R have its natural involution * in which each element of P is Hermitian. We show that the Banach *-algebra ℓ1(R) has a separating family of finite dimensional *-representations and consequently is *-semisimple. This generalizes a result of B. A. Barnes and J. Duncan (J. Funct. Anal.18 (1975), 96–113.) dealing with the case where P has two elements.

scholarly journals Inductive limit of direct sums of simple TAI algebras

2019 ◽  
pp. 1-26
Author(s):  
Bo Cui ◽  
Chunlan Jiang ◽  
Liangqing Li
Keyword(s):  
Inductive Limit ◽  
Direct Sums ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Rank One ◽  
Tracial Rank ◽  
Ideal Property ◽  
Funct Anal

An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], respectively) is an inductive limit [Formula: see text], where each [Formula: see text] is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404], the second author classified all ATAI algebras by an invariant consisting orderd total [Formula: see text]-theory and tracial state spaces of cut down algebras under an extra restriction that all element in [Formula: see text] are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic [Formula: see text]-group as an addition to the invariant used in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404]. The theorem is proved by reducing the class to the classification theorem of [Formula: see text] algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of inductive limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (see Corollary 4.3).

scholarly journals A categorical reconstruction of crystals and quantum groups at q = 0

2019 ◽  
Vol 70 (3) ◽  
pp. 895-925
Author(s):  
Craig Smith
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Analogous Result ◽  
Crystal Bases ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Monoidal Structure

Abstract The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

scholarly journals Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2066
Author(s):  
Messaoud Bounkhel ◽  
Mostafa Bachar
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Space Setting ◽  
Nonlinear Anal ◽  
Finite Dimensional ◽  
In Trans ◽  
First Time ◽  
The Relationship ◽  
Primal Lower Nice Functions

In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces in [Trans. Am. Math. Soc. 1995, 347, 1269–1294]. Our principal target is to extend some existing characterisations of this class to our Banach space setting and to study the relationship between this concept and the generalised V-prox-regularity of the epigraphs in the sense proposed recently by the authors in [J. Math. Anal. Appl. 2019, 475, 699–29].

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