Lipschitz Algebras Research Articles

Character Amenability of Lipschitz Algebras

AbstractLet be a locally compact metric space and let 𝒜 be any of the Lipschitz algebras , , or . In this paper, we show, as a consequence of rather more general results on Banach algebras, that 𝒜 is C-character amenable if and only if is uniformly discrete.

Certain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces

We study an interesting class of Banach function algebras of in nitely di erentiable functions onperfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, calledLipschitz algebras of in nitely di erentiable functions and denoted by Lip(X;M; ), where X is aperfect, compact plane set, M = fMng1n=0 is a sequence of positive numbers such that M0 = 1 and(m+n)!Mm+n ( m!Mm)( n!Mn) for m; n 2 N [ f0g and 2 (0; 1]. Let d = lim sup( n!Mn)1n and Xd = fz 2 C :dist(z;X)  dg. Let LipP;d(X;M; )[LipR;d(X;M; )] be the subalgebra of all f 2 Lip(X;M; )that can be approximated by the restriction to Xd of polynomials [rational functions with poles o Xd]. We show that the maximal ideal space of LipP;d(X;M; ) is cXd, the polynomially convex hullof Xd, and the maximal ideal space of LipR;d(X;M; ) is Xd, for certain compact plane sets.. Usingsome formulae from combinatorial analysis, we nd the maximal ideal space of certain subalgebrasof Lipschitz algebras of in nitely di erentiable functions.

On closed ideals in the big Lipschitz algebras of analytic functions

In this paper we study the closed ideals in the big Lipschitz algebras of analytic functions on the unit disk. More precisely we give the smallest closed ideal with given hull and inner factor.

Some Dense Linear Subspaces of Extended Little Lipschitz Algebras

Let be a compact metric space. In 1987, Bade, Curtis, and Dales obtained a sufficient condition for density of a subspace of little Lipschitz algebra in this algebra and in particular showed that is dense in , whenever . Let be a compact subset of . We define new classes of Lipchitz algebras for and for , consisting of those continuous complex-valued functions on such that and , respectively. In this paper we obtain a sufficient condition for density of a linear subspace of extended little Lipschitz algebra in this algebra and in particular show that is dense in , whenever .

Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras

Let X and K be compact plane sets with K⊆X. We define A(X, K) = {f ∈ C(X) : f|K ∈ A(K)}, where A(K) = {g ∈ C(X) : g is analytic on int(K)}. For α ∈ (0,1], we define Lip(X, K, α) = {f ∈ C(X) : pα,K(f) = sup{|f(z) − f(w)|/|z − w|α : z, w ∈ K, z ≠ w} < ∞} and LipA(X, K, α) = A(X, K)∩Lip(X, K, α). It is known that LipA(X, K, α) is a natural Banach function algebra on X under the norm | | f | |Lip(X,K,α) = | | f | |X + pα,K(f), where | | f | |X = sup {|f(x)| : x ∈ X}. These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of LipA(X1, K1, α1) to natural Banach function subalgebras of LipA(X2, K2, α2) and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of LipA(X, K, α).

Weakly peripherally multiplicative surjections of pointed Lipschitz algebras

Weakly peripherally multiplicative surjections of pointed Lipschitz algebras

Blaschke products and nonideal ideals in higher order Lipschitz algebras

We investigate certain ideals (associated with Blaschke products) of the analytic Lipschitz algebra $A^\alpha$, with $\alpha>1$, that fail to be ideal spaces. The latter means that the ideals in question are not describable by any size condition on the function's modulus. In the case where $\alpha=n$ is an integer, we study this phenomenon for the algebra $H^\infty_n=\{f:f^{(n)}\in H^\infty\}$ rather than for its more manageable Zygmund-type version. This part is based on a new theorem concerning the canonical factorization in $H^\infty_n$.

Compact and quasicompact homomorphisms between differentiable Lipschitz algebras

In this note we consider homomorphisms between differentiable Lipschitz algebras $Lip^n(X,\alpha)$ ($0<\alpha \leq 1$) and $lip^n(X,\alpha)$ ($0<\alpha <1$), where $X$ is a perfect compact plane set. We give sufficient conditions implying the compactness and power compactness of these homomorphisms. Moreover, we investigate under what conditions a quasicompact homomorphism between these algebras is power compact. We also give a necessary condition for a homomorphism between these algebras to be quasicompact and in certain cases to be power compact. Finally, using these results, by giving an example we show that there exists a quasicompact homomorphism between these algebras which is not power compact.

Nonlinear conditions for weighted composition operators between Lipschitz algebras

Let T : Lip 0 ( X ) → Lip 0 ( Y ) be a surjective map between pointed Lipschitz ∗ -algebras, where X and Y are compact metric spaces. On the one hand, we prove that if T satisfies the non-symmetric norm ∗ -multiplicativity condition: ‖ T ( f ) T ( g ) ¯ − 1 ‖ ∞ = ‖ f g ¯ − 1 ‖ ∞ ( f , g ∈ Lip 0 ( X ) ) , then T is of the form T ( f ) = τ ⋅ ( η ⋅ ( f ○ φ ) + ( 1 − η ) ⋅ ( f ○ φ ) ¯ ) ( f ∈ Lip 0 ( X ) ) , where η and τ are functions on Y such that η ( Y ) ⊆ { 0 , 1 } and τ ( Y ) ⊆ { α ∈ K : | α | = 1 } , and φ : Y → X is a base point preserving Lipschitz homeomorphism. On the other hand, if T satisfies the weakly peripherally ∗ -multiplicativity condition: Ran π ( f g ¯ ) ∩ Ran π ( T ( f ) T ( g ) ¯ ) ≠ ∅ ( f , g ∈ Lip 0 ( X ) ) , where Ran π ( f ) denotes the peripheral range of f , then T can be expressed as T ( f ) = τ ⋅ ( f ○ φ ) ( f ∈ Lip 0 ( X ) ) , with τ and φ as above. As a consequence, we obtain similar descriptions for surjective maps between Lipschitz ∗ -algebras Lip ( X ) .

Compact Endomorphisms of Infinitely Differentiable Lipschitz Algebras

Compact Endomorphisms of Infinitely Differentiable Lipschitz Algebras

Closed ideals in analytic weighted Lipschitz algebras

We obtain a complete description of closed ideals in weighted Lipschitz algebras Λ ω of analytic functions on the unit disk satisfying the following condition | f ( z ) − f ( w ) | ω ( | z − w | ) = o ( 1 ) ( as | z − w | → 0 ) , where ω is a modulus of continuity satisfying some regularity conditions. In particular the closed ideals of the algebras Λ χ α , where χ α ( t ) : = 1 ( | log ( t ) | + 1 ) α , α > 0 , are standard and this answers Shirokov's question [N.A. Shirokov, Closed ideals of algebras of B p q α -type, Izv. Akad. Nauk SSSR Mat. 46 (6) (1982) 1316–1333 (in Russian), p. 587].

Lipschitz algebras and peripherally-multiplicative maps

Let X be a compact metric space and let Lip(X) be the Banach algebra of all scalarvalued Lipschitz functions on X, endowed with a natural norm. For each f ∈ Lip(X), σ π (f) denotes the peripheral spectrum of f. We state that any map Φ from Lip(X) onto Lip(Y) which preserves multiplicatively the peripheral spectrum: $$ \sigma _\pi (\Phi (f)\Phi (g)) = \sigma _\pi (fg),\forall f,g \in Lip(X), $$ is a weighted composition operator of the form Φ(f) = τ · (f ○ φ) for all f ∈ Lip(X), where τ: Y → {−1, 1} is a Lipschitz function and φ: Y → X is a Lipschitz homeomorphism. As a consequence of this result, any multiplicatively spectrum-preserving surjective map between Lip(X)-algebras is of the form above.

Generalized notions of amenability, II

This paper continues the investigation of the first two authors begun in part I. It is shown that approximate amenability and approximate contractibility are the same properties, as are uniform approximate amenability and amenability. Bounded approximate contractibility and bounded approximate amenability are characterized by the existence of suitable operator bounded approximate identities for the diagonal ideal. Results are given on Banach sequence algebras, Lipschitz algebras and Beurling algebras, and on the crucial role of approximate identities. A new proof is given for a result due to N. Grønbæk on characterizing of amenability for Beurling algebras.

On the maximal ideal space of extended analytic lipschitz algebras

Let X be a compact, plane set and let K be a compact subset of X. We introduce new classes of Lipschitz algebras Lip(X, K, α), lip(X, K, α), consisting of those continuous functions f on X such that f|K ∈ Lip(K, α), lip(K, α), and their analytic subalgebras LipA(X, K, α) = Lip(X, K, α) ∩ A(X, K) and lipA(X, K, α) = lip(X, K, α)∩ A(X, K), where 0 < α ≤ 1 and A(X, K) is the algebra of all continuous complex-valued functions on X, which are analytic on the interior of K. We show that the maximal ideal spaces of these extended Lipschitz algebras coincide with X.

Disjointness preserving operators between little Lipschitz algebras

Given a real number α ∈ ( 0 , 1 ) and a metric space ( X , d ) , let Lip α ( X ) be the algebra of all scalar-valued bounded functions f on X such that p α ( f ) = sup { | f ( x ) − f ( y ) | / d ( x , y ) α : x , y ∈ X , x ≠ y } < ∞ , endowed with any one of the norms ‖ f ‖ = max { p α ( f ) , ‖ f ‖ ∞ } or ‖ f ‖ = p α ( f ) + ‖ f ‖ ∞ . The little Lipschitz algebra lip α ( X ) is the closed subalgebra of Lip α ( X ) formed by all those functions f such that | f ( x ) − f ( y ) | / d ( x , y ) α → 0 as d ( x , y ) → 0 . A linear mapping T : lip α ( X ) → lip α ( Y ) is called disjointness preserving if f ⋅ g = 0 in lip α ( X ) implies ( T f ) ⋅ ( T g ) = 0 in lip α ( Y ) . In this paper we study the representation and the automatic continuity of such maps T in the case in which X and Y are compact. We prove that T is essentially a weighted composition operator T f = h ⋅ ( f ○ φ ) for some nonvanishing little Lipschitz function h and some continuous map φ. If, in addition, T is bijective, we deduce that h is a nonvanishing function in lip α ( Y ) and φ is a Lipschitz homeomorphism from Y onto X and, in particular, we obtain that T is automatically continuous and T −1 is disjointness preserving. Moreover we show that there exists always a discontinuous disjointness preserving linear functional on lip α ( X ) , provided X is an infinite compact metric space.

Compact endomorphisms of certain analytic Lipschitz algebras

Compact endomorphisms of certain analytic Lipschitz algebras

Ideals in big Lipschitz algebras of analytic functions

For $0<\gamma\le1$, let ${\mit\Lambda}_{\gamma}^+$ be the big Lipschitz algebra of functions analytic on the open unit disc ${\mathbb D}$ which satisfy a Lipschitz condition of order $\gamma$ on $\overline{\mathbb D}$. For a closed set $E$ on the unit ci

APPROXIMATELY MULTIPLICATIVE FUNCTIONALS ON ALGEBRAS OF SMOOTH FUNCTIONS

Let φ be ab ounded linear functional on A ,w hereA is a commutative Banach algebra, then the bilinear functional ˇ φ is defined as ˇ φ(a, b )= φ(ab) − φ(a)φ(b )f or eacha and b in A .I f thenorm of ˇ φ is small then φ is approximately multiplicative, and it is of interest whether or not � ˇ φ� being small implies that φ is near to a multiplicative functional. If this property holds for a commutative Banach algebra then A is an AMNM algebra (approximately multiplicative functionals are near multiplicative functionals). The main result of the paper shows that C N [0, 1] M (the complex valued functions defined on [0, 1] M with all Nth order partial derivatives continuous) is AMNM. It is also shown that a similar proof can be applied to certain Lipschitz algebras.

BANACH ALGEBRAS WEAK* GENERATED BY THEIR IDEMPOTENTS

AbstractFor a closed set $E$ contained in the closed unit interval, we show that the big Lipschitz algebra $\varLambda_{\gamma}(E)$ $(0\lt\gamma\lt1)$ is sequentially weak$^{\ast}$ generated by its idempotents if and only if it is weak$^{\ast}$ generated by its idempotents if and only if the little Lipschitz algebra $\lambda_{\gamma}(E)$ is generated by its idempotents, and we describe a class of perfect symmetric sets for which this holds. Moreover, we prove that $\varLambda_1(E)$ is sequentially weak$^{\ast}$ generated by its idempotents if and only if $E$ is of measure zero. Finally, we show that the quotient algebras$$ \mathcal{A}_{\beta}/\overline{J_{\beta}(E)}^{\text{weak}^{\ast}} $$of the Beurling algebras need not be weak$^{\ast}$ generated by their idempotents, when $E$ is of measure zero and $\beta\ge\tfrac{1}{2}$.AMS 2000 Mathematics subject classification: Primary 46J10; 46J30; 26A16; 42A16

The category of compact metric spaces and its functional analytic duals

A Lipschitz algebra Lip(X,dx) over a compact metric space (X,dx) consists of all complex valued continuous functions on (X,dx) which are Lipschitz with respect to dx and the standard metric on the complex plane C (absolute value). The norm on Lip(X,dx) is given by ||/|| = sup{|/(x)| : x e X} + sup{|/(z) - f(y)\/dx(x,y) : x,y G X & x + y}. We show that the category CLip in which objects are Lipschitz algebras and morphisms are algebra homomorphisms is dual to the category CMet in which objects are compact metric spaces and morphisms are Lipschitz maps. Let (X, d) be any metric space, and let Y = {(x,y) e X x X : x ^ y}. De Leeuw derivation defined by the metric d is the operator D : Cf,(X) -> Ct,(Y) be defined by (Df)(x,y) = (f(y) - f(x))/d(x,y) for (x,y) G Y. We consider the category CDer in which objects are pairs (C(X), DX), where (X, dx) is a compact metric space and Dx is the corresponding de Leeuw derivation, and morphisms are all homomorphisms v : C(X) -> C(Y) for which / 6 Domain(Dx) implies vf e Domain(Dy). We show that CDer is equivalent to CLip, and that CDer is dual to CMet.