On James' Quasi-Reflexive Banach Space as a Banach Algebra

1980 ◽  
Vol 32 (5) ◽  
pp. 1080-1101 ◽  
Author(s):  
Alfred D. Andrew ◽  
William L. Green
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Reflexive Banach Space ◽  
Equivalent Norm ◽  
New Techniques ◽  
Codimension One ◽  
Gelfand Transform ◽  
Schauder Bases ◽  
Second Dual ◽  
Arens Multiplication

In [4] and [5], R. C. James introduced a non-reflexive Banach space J which is isometric to its second dual. Developing new techniques in the theory of Schauder bases, James identified J**, showed that the canonical image of J in J** is of codimension one, and proved that J** is isometric to J.In Section 2 of this paper we show that J, equipped with an equivalent norm, is a semi-simple (commutative) Banach algebra under point wise multiplication, and we determine its closed ideals. We use the Arens multiplication and the Gelfand transform to identify J**, which is in fact just the algebra obtained from J by adjoining an identity.

scholarly journals c0 can be renormed to have the fixed point property for affine nonexpansive mappings

Filomat ◽  
2018 ◽  
Vol 32 (16) ◽  
pp. 5645-5663 ◽  
Author(s):  
Veysel Nezir ◽  
Nizami Mustafa
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Reflexive Banach Space ◽  
Nonexpansive Mappings ◽  
Equivalent Norm ◽  
Positive Answer ◽  
Fixed Point Property ◽  
Null Sequence ◽  
Point Property ◽  
Usual Norm

P.K. Lin gave the first example of a non-reflexive Banach space (X,||?||) with the fixed point property (FPP) for nonexpansive mappings and showed this fact for (l1,||?||1) with the equivalent norm ||?|| given by ||x|| = sup k?N 8k/1+8k ?1,n=k |xn|, for all x = (xn)n?N ? l1. We wonder (c0, ||?||1) analogue of P.K. Lin?s work and we give positive answer if functions are affine nonexpansive. In our work, for x = (?k)k ? c0, we define |||x||| := lim p?? sup ?k?N ?k (?1,j=k |?j|p/2j)1/p where ?k ?k 3, k is strictly increasing with ?k > 2, ?k ? N, then we prove that (c0,|||?|||) has the fixed point property for affine |||?|||-nonexpansive self-mappings. Next, we generalize this result and show that if ?(?) is an equivalent norm to the usual norm on c0 such that lim sup n ?(1/n ?n,m=1 xm + x) = lim sup n ?(1/n ?n,m=1 xm) + ?(x) for every weakly null sequence (xn)n and for all x ? c0, then for every ? > 0, c0 with the norm ||?||? = ?(?)+?|||?||| has the FPP for affine ||?||?-nonexpansive self-mappings.

scholarly journals A smooth, non-reflexive second conjugate space

1976 ◽  
Vol 15 (1) ◽  
pp. 129-131 ◽  
Author(s):  
M.A. Smith
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Equivalent Norm ◽  
Conjugate Space

It is shown that a separable, quasi-reflexive Banach space of deficiency one admits an equivalent norm such that its second conjugate space is smooth; this answers a question raised by Ivan Singer [Bull. Austral. Math. Soc. 12 (1975), 407–416.

scholarly journals Arens regularity and retractions

1983 ◽  
Vol 24 (1) ◽  
pp. 17-21
Author(s):  
Nilgün Arikan
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Regularity Theory ◽  
Recent Survey ◽  
Normed Algebra ◽  
Natural Embedding ◽  
Arens Regularity ◽  
Second Dual ◽  
Image Position

In this paper a characterisation of the regularity of a normed algebra A is given in terms of retractions onto A** from A4*. The second dual A** of a normed algebra A possesses two natural Banach algebra multiplications, say ° and *. Each of ° and * extends the original algebra multiplication on A; see (2). An algebra A is called regular if and only if F * G = F ° G for all F, G ∈ A**. See (1). The existing results in the Arens regularity theory can be found in a recent survey (2). Denoting the nth dual of A by An*, and en the natural embedding of An* in its second dual A(n+2)*, we can naturally represent the second dual A** of A as a Banach space retract of A4* in two different ways:Our main results say that A** is in fact a Banach algebra retract of A4* (i.e. the maps involved are homomorphisms) in either of these cases if and only if A is regular.

The numerical range in the second dual of a Banach algebra

1981 ◽  
Vol 89 (2) ◽  
pp. 301-307
Author(s):  
R. R. Smith
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Numerical Range ◽  
Special Property ◽  
Empty Interior ◽  
Banach Theorem ◽  
Second Dual ◽  
Algebra Theory ◽  
Unital Banach Algebra ◽  
Selection Of

An elementary consequence of the Hahn-Banach theorem is that every Banach space Y is ω*-dense in its second dual Y**, so that every element y ∈ Y** is the w*-limit of a net {ya}α ∈ Λ from Y. There is, of course, a great deal of choice in the selection of such a net, and so one may impose extra conditions on the net related to some special property of the limit point, and then ask for existence. The object of this paper is to present such a result in the context of a unital Banach algebra and its second dual , and then to give two applications to Banach algebra theory. The theorem to be proved is this: if the numerical range W(a) of an element in has non-empty interior then a is the ω*-limit of a net {aa}α ∈ Λ from whose numerical ranges are contained in W(a), while if W(a) has empty interior then the numerical ranges W(aα) are contained in a shrinking set of neighbourhoods of W(a).

Convolution and the second dual of a Banach algebra

1969 ◽  
Vol 65 (3) ◽  
pp. 597-599 ◽  
Author(s):  
John S. Pym
Keyword(s):  
Banach Algebra ◽  
Commutative Banach Algebra ◽  
Second Dual ◽  
Arens Multiplication

We shall point out a connexion between convolution (as defined in (5); see also (7)) and the Arens multiplication on the second dual of a Banach algebra (1, 2). This enables us to demonstrate in an easy way the existence of a commutative Banach algebra whose second dual is not commutative. Examples can also be found in (3, 9) and elsewhere.

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

scholarly journals ASYMPTOTIC PROPERTIES OF KOLMOGOROV WIDTHS

2010 ◽  
Vol 82 (1) ◽  
pp. 10-17
Author(s):  
MIKHAIL I. OSTROVSKII
Keyword(s):  
Banach Space ◽  
Asymptotic Behavior ◽  
Banach Spaces ◽  
Asymptotic Properties ◽  
Codimension One ◽  
Kolmogorov Widths

AbstractWe consider two problems concerning Kolmogorov widths of compacts in Banach spaces. The first problem is devoted to relations between the asymptotic behavior of the sequence of n-widths of a compact and of its projections onto a subspace of codimension one. The second problem is devoted to comparison of the sequence of n-widths of a compact in a Banach space 𝒴 and of the sequence of n-widths of the same compact in other Banach spaces containing 𝒴 as a subspace.

Darboux chart on projective limit of weak symplectic Banach manifold

2015 ◽  
Vol 12 (07) ◽  
pp. 1550072 ◽  
Author(s):  
Pradip Mishra
Keyword(s):  
Banach Space ◽  
Symplectic Structure ◽  
Reflexive Banach Space ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Banach Manifold ◽  
Boundedness Condition ◽  
Projective System ◽  
Banach Manifolds

Suppose M be the projective limit of weak symplectic Banach manifolds {(Mi, ϕij)}i, j∈ℕ, where Mi are modeled over reflexive Banach space and σ is compatible with the projective system (defined in the article). We associate to each point x ∈ M, a Fréchet space Hx. We prove that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure.

scholarly journals Uniqueness of the maximal ideal of operators on the ℓp-sum of ℓ∞n (n ∈ ) for 1 < p < ∞

2016 ◽  
Vol 160 (3) ◽  
pp. 413-421 ◽  
Author(s):  
TOMASZ KANIA ◽  
NIELS JAKOB LAUSTSEN
Keyword(s):  
Banach Space ◽  
Recent Result ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Maximal Ideal ◽  
American Mathematical Society ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Unique Maximal Ideal

AbstractA recent result of Leung (Proceedings of the American Mathematical Society, 2015) states that the Banach algebra ℬ(X) of bounded, linear operators on the Banach space X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓ1 contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓp and X = (⊕n∈$\mathbb{N}$ ℓ1n)ℓp whenever p ∈ (1, ∞).

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