The promise made at the beginning of Chapter 6 that βX is to be used to characterize the maximal ideals in C(X) and in C*(X), will be fulfilled in this chapter. The key to the description of the maximal ideals in C*(X) has already been given: C*(X) is isomorphic with C(βX), and the maximal ideals in the latter ring are in one-one correspondence with the points of βX.

Characterization of Maximal Ideals

Weaving Schauder frames

This paper gives an affirmative answer to Tingley’s problem in the spaces L p ( μ ) where 0 p 1 by showing that every isometry from the unit sphere of L p ( μ ) onto the unit sphere of X which is a subspace of an L p ( ν ) -space can be extended to a linear isometry on the whole space L p ( μ ) . Especially, by employing the technique underlying the proof of the case 0 p 1 , we can prove that every surjective isometry between the unit spheres of L 1 ( μ ) and a Banach space F can also be linearly and isometrically extended to the whole space.

On extension of isometries on the unit spheres of Lp-spaces for 0<p≤1

We survey some recent studies of linear zero product or orthogonality preservers between C∗/W ∗-algebras, their dual or predual spaces, and holomorphic disjointness preservers of C∗-algebras. Such maps are expected to provide algebra or linear Jordan (∗-) homomorphisms between the underlying operator algebras. We also study orthogonality preservers between Hilbert C∗-modules and Fourier algebras. A few open problems are stated.

Linear disjointness preservers of operator algebras and related structures

Disjointness preservers of AW⁎-algebras

We establish spherical variants of the Gleason–Kahane–Zelazko and Kowalski–S lodkowski theorems, and we apply them to prove that every weak-2-local isometry between two uniform algebras is a linear map. Among the consequences, we solve a couple of problems posed by O. Hatori, T. Miura, H. Oka, and H. Takagi in 2007. Another application is given in the setting of weak-2-local isometries between Lipschitz algebras by showing that given two metric spaces E and F such that the set Iso((Lip(E), k·k), (Lip(F), k·k)) is canonical, then every weak-2-local Iso((Lip(E), k · k), (Lip(F), k · k))-map ∆ from Lip(E) to Lip(F) is a linear map, where k · k can indistinctly stand for kfkL := max{L(f), kfk∞} or kfks := L(f) + kfk∞.

/pdf/weak-2-local-isometries-on-uniform-algebras-and-lipschitz-370cs4veio.pdf

Weak-2-local isometries on uniform algebras and Lipschitz algebras

We construct separable universal injective and projective lattices for the class of all separable Banach lattices.

Separable universal Banach lattices

2-Local standard isometries on vector-valued Lipschitz function spaces

In the present paper we study unconditionally $p$-converging operators and Dunford-Pettis property of order $p$. New characterizations of unconditionally $p$-converging operators and Dunford-Pettis property of order $p$ are established. Six quantities are defined to measure how far an operator is from being unconditionally $p$-converging. We prove quantitative versions of relationships of completely continuous operators,unconditionally $p$-converging operators and unconditionally converging operators. We further investigate possible quantifications of the Dunford-Pettis property of order $p$.

Unconditionally $p$-converging operators and Dunford-Pettis Property of order $p$

Under the right conditions on a compact metric space $X$ and on a Banach space $E$, we give a description of the $2$-local (standard) isometries on the Banach space $\hbox{Lip}(X,E)$ of vector-valued Lipschitz functions from $X$ to $E$ in terms of a generalized composition operator, and we study when every $2$-local (standard) isometry on $\hbox{Lip}(X,E)$ is both linear and surjective.

Lei Li

Papers

Weak-2-local isometries on uniform algebras and Lipschitz algebras

Separable universal Banach lattices

2-Local standard isometries on vector-valued Lipschitz function spaces

Unconditionally $p$-converging operators and Dunford-Pettis Property of order $p$

2-local standard isometries on vector-valued Lipschitz function spaces