On the extension of bi-Lipschitz mappings (original) (raw)

C0 and bi-Lipschitz K -equivalence of mappings

Mathematische Zeitschrift, 2011

In this paper we investigate the classification of mappings up to K-equivalence. We give several results of this type. We study semialgebraic deformations up to semialgebraic C 0 K-equivalence and bi-Lipschitz K-equivalence. We give an algebraic criterion for bi-Lipschitz K-triviality in terms of semi-integral closure (Theorem 3.5). We also give a new proof of a result of Nishimura: we show that two germs of smooth mappings f, g : R n → R n , finitely determined with respect to K-equivalence are C 0-K-equivalent if and only if they have the same degree in absolute value.

C 0 and bi-Lipschitz {\mathcal{K}}$$ -equivalence of mappings

Mathematische Zeitschrift, 2011

Riesz and quasi-compact endomorphisms of Lipschitz algebras

Houston Journal of Mathematics, 2010

In this paper, we study Riesz and quasi-compact endomorphisms of Lipschitz algebras. For a unital endomorphism of a little Lipschitz algebra, we establish a lower bound for its essential spectral radius. Also, we show that this lower bound can be attained by imposing some extra assumption.

On open and closed morphisms between semialgebraic sets

ams.org

In this work we study how open and closed semialgebraic maps between two semialgebraic sets extend, via the corresponding spectral maps, to the Zariski and maximal spectra of their respective rings of semialgebraic and bounded semialgebraic functions.

Homomorphisms on algebras of Lipschitz functions

2010

We characterize a class of *-homomorphisms on Lip * (X, B(H)), a noncommutative Banach *-algebra of Lipschitz functions on a compact metric space and with values in B(H). We show that the zero map is the only multiplicative *-preserving linear functional on Lip * (X, B(H)). We also establish the algebraic reflexivity property of a class of *-isomorphisms on Lip * (X, B(H)).

Amenability, locally finite spaces, and bi-lipschitz embeddings

We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, clearly extends amenability to any locally finite space. Therefore, we start making a comparison between this property and other notions of amenability for locally finite metric spaces that have been proposed by Gromov, Lafontaine and Pansu, by Ceccherini-Silberstein, Grigorchuk and de la Harpe and by Block and Weinberger. We discuss possible applications of the property SN in the study of embedding a metric space into another one. In particular, we propose three results: we prove that a certain class of metric graphs that are isometrically embeddable into Hilbert spaces must have the property SN. We also show, by a simple example, that this result is not true replacing property SN with amenability. As a second result, we prove that many spaces with uniform bounded geometry having a bi-lipschitz embedding into Euclidean spaces must have the property SN. Finally, we prove a Bourgain-like theorem for metric trees: a metric tree with uniform bounded geometry and without property SN does not have bi-lipschitz embeddings into finite-dimensional Hilbert spaces.

Surjectivity of linear operators and semialgebraic global diffeomorphisms

2021

We prove that a CinftyC^{\infty}Cinfty semialgebraic local diffeomorphism of mathbbRn\mathbb{R}^nmathbbRn with non-properness set having codimension greater than or equal to 222 is a global diffeomorphism if n−1n-1n−1 suitable linear partial differential operators are surjective. Then we state a new analytic conjecture for a polynomial local diffeomorphism of mathbbRn\mathbb{R}^nmathbbRn. Our conjecture implies a very known conjecture of Z. Jelonek. We further relate the surjectivity of these operators with the fibration concept and state a general global injectivity theorem for semialgebraic mappings which turns out to unify and generalize previous results of the literature.

Compact endomorphisms of certain analytic Lipschitz algebras

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2005

Let X be a compact plane set. A(X) denotes the uniform algebra of all continuous complex-valued functions on X which are analytic on intX. For 0 < α ≤ 1, Lipschitz algebra of order α, Lip(X, α) is the algebra of all complex-valued functions f on X for which p α (f) = sup{ |f (x)−f (y)| |x−y| α : x, y ∈ X, x = y} < ∞. Let Lip A (X, α) = A(X) Lip(X, α), and Lip n (X, α) be the algebra of complex-valued functions on X whose derivatives up to order n are in Lip(X, α). Lip A (X, α) under the norm f = f X +p α (f), and Lip n (X, α) for a certain plane set X under the norm f = n k=0 f (k) X +pα(f (k)) k! are natural Banach function algebras, where f X = sup x∈X |f (x)|. In this note we study endomorphisms of algebras Lip A (X, α) and Lip n (X, α) and investigate necessary and sufficient conditions for which these endomorphisms to be compact. Finally, we determine the spectra of compact endomorphisms of these algebras.