Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConvexity and smoothness of Banach spaces with numerical index one
154
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~1). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $ell_1$.
rate research
Read More
For every $alpha<omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $omega^{alphaomega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $omega^{alphaomega}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.
Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $mathcal{H}_1$ which contains $mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.
The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is presented for functions defined on $mathbb{R}^N$. In this article we present a more general and simplified construction of the Bochner integral on abstract measure spaces. An extension of the construction to functions with values in a locally convex space is also considered.
We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.
We continue our investigation, from cite{dh}, of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a $C^*$-algebra is purely infinite if and only if any of its ultrapower is. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a Cuntz-like Banach $*$-algebra which is purely infinite, but does not have purely infinite ultrapowers. Our proof of being purely infinite is combinatorial, but direct, and so differs from the proof for the Cuntz algebra. We use an indirect method (and not directly computing norm estimates) to show that this algebra does not have purely infinite ultrapowers.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
