zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The uniform separation property and Banach-Stone theorems for lattice-valued Lipschitz functions. (English) Zbl 1202.46043
This paper contributes to a very active area of `local’ Banach-Stone theorems for spaces of vector-valued functions. For an isomorphism $T$ of such spaces, `local conditions’ are assumed on $T$ to ensure that the underlying spaces are isomorphic and $T$ is described in terms of these objects in a canonical way (composition operator). Let $(X,d)$ be a metric space and let $E$ be a nonzero Banach lattice. Let Lip$(X,E)$ denote the Banach space of bounded $E$-valued Lipschitz functions with pointwise order and with respect to the norm $\max\{\text{Lip}(f),\|f\|_{\infty}\}$. For $X,Y$ and $E,F$ in this category, let $A(X,E)$ and $A(Y,F)$ be {\it closed} sublattices (this correction gets noted in “Correction to `The uniform separation property and Banach-Stone theorems for lattice-valued Lipschitz functions’”, ibid. 138, 1535) such that they separate and join points uniformly. Let $ T: A(X,E) \rightarrow A(Y,F) $ be a vector lattice isomorphism that preserves the nowhere vanishing functions in both the directions. Then there exists a bi-Lipschitz map $\phi: Y \rightarrow X$ and a Lipschtz map $T^{\wedge}: Y \rightarrow L(E,F)$ such that $T^{\wedge}$ takes values as lattice isomorphisms and $T(f)(y)=T^{\wedge}(y)(f(\phi(y))$ for $y \in Y$ and $f \in A(X,E)$.

46E40Spaces of vector- and operator-valued functions
46E05Lattices of continuous, differentiable or analytic functions
Full Text: DOI