Zero product preserving maps on Banach algebras of Lipschitz functions

Let (K,d) be a non-empty, compact metric space and α∈]0,1[. Let A be either lipα(K) or Lipα(K) and let B be a commutative unital Banach algebra. We show that every continuous linear map T:A→B with the property that T(f)T(g)=0 whenever f,g∈A are such that fg=0 is of the form T=wΦ for some invertible element w in B and some continuous epimorphism Φ:A→B.     

A generalization of peripherally-multiplicative surjections between standard operator algebras

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σ_π (T) of T is defined by σ_π (T) = z ∈ σ(T): |z| = max_w∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σ_π (φ(S)ϕ(T)) = σ_π (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A ₁ TA ₂ ⁻¹ and ϕ(T) = A ₂ TA ₁ ⁻¹ for some bijective bounded linear operators A ₁; A ₂ of X onto Y, or of the form φ(T) = B ₁ T*B ₂ ⁻¹ and ϕ(T) = B ₂ T*B ⁻¹ for some bijective bounded linear operators B ₁;B ₂ of X* onto Y.      

Multiplicatively spectrum-preserving maps of function algebras

Let be a compact Hausdorff space and a function algebra. Assume that is the maximal ideal space of . Denoting by the spectrum of an , which in this case coincides with the range of , a result of Molnár is generalized by our Main Theorem: If is a surjective map with the property for every pair of functions , then there exists a homeomorphism such that

for every and every with .

     

Nonlinear conditions for weighted composition operators between Lipschitz algebras

Let T:Lip₀(X)→Lip₀(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:

     

On fully operator Lipschitz functions

Let be the disc algebra of all continuous complex-valued functions on the unit disc holomorphic in its interior. Functions from act on the set of all contraction operators (A1) on Hilbert spaces. It is proved that the following classes of functions from coincide: (1) the class of operator Lipschitz functions on the unit circle ; (2) the class of operator Lipschitz functions on ; and (3) the class of operator Lipschitz functions on all contraction operators. A similar result is obtained for the class of operator C₂-Lipschitz functions from .     

Commuting traces and commutativity preserving maps on triangular algebras

Let A be a triangular algebra. The problem of describing the form of a bilinear map satisfying B(x,x)x=xB(x,x) for all xA is considered. As an application, commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.     

Disjointness preserving operators between little Lipschitz algebras

Given a real number α∈(0,1) and a metric space (X,d), let Lipα(X) be the algebra of all scalar-valued bounded functions f on X such that

     

Analytic and harmonic maps into a topological space

The relationship between the harmonicity and analyticity of a continuous map from the open unit disc to the underlying space of a real algebra is investigated.     

Noncompactness and noncompleteness in isometries of Lipschitz spaces

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip(X,E) and Lip(Y,F), for strictly convex normed spaces E and F and metric spaces X and Y:

(i)

Characterize those base spaces X and Y for which all isometries are weighted composition maps.

(ii)

Give a condition independent of base spaces under which all isometries are weighted composition maps.

(iii)

Provide the general form of an isometry, both when it is a weighted composition map and when it is not.

In particular, we prove that requirements of completeness on X and Y are not necessary when E and F are not complete, which is in sharp contrast with results known in the scalar context.     

Multiplicatively range-preserving maps between Banach function algebras

Let A and B be two Banach function algebras on locally compact Hausdorff spaces X and Y, respectively. Let T be a multiplicatively range-preserving map from A onto B in the sense that (TfTg)(Y)=(fg)(X) for all f,g∈A. We define equivalence relations on appropriate subsets and of X and Y, respectively, and show that T induces a homeomorphism between the quotient spaces of and by these equivalence relations. In particular, if all points in the Choquet boundaries of A and B are strong boundary points, then and are equal to the Choquet boundaries of A and B, respectively, and moreover, there exist a continuous function h on the Choquet boundary of B taking its values in {−1,1} and a homeomorphism φ from the Choquet boundary of B onto the Choquet boundary of A such that Tf(y)=h(y)f(φ(y)) for all f∈A and y in the Choquet boundary of B. For certain Banach function algebras A and B on compact Hausdorff spaces X and Y, respectively, we can weaken the surjectivity assumption and give a representation for maps belonging 2-locally to the family of all multiplicatively range-preserving maps from A onto B.     

Generalizations of weakly peripherally multiplicative maps between uniform algebras

Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y, respectively. For f∈A, the peripheral spectrum of f, denoted by σπ(f)={λ∈σ(f):|λ|=‖f‖}, is the set of spectral values of maximum modulus. A map T:A→B is weakly peripherally multiplicative if σπ(T(f)T(g))∩σπ(fg)≠∅ for all f,g∈A. We show that if T is a surjective, weakly peripherally multiplicative map, then T is a weighted composition operator, extending earlier results. Furthermore, if T₁,T₂:A→B are surjective mappings that satisfy σπ(T₁(f)T₂(g))∩σπ(fg)≠∅ for all f,g∈A, then T₁(f)T₂(1)=T₁(1)T₂(f) for all f∈A, and the map f?T₁(f)T₂(1) is an isometric algebra isomorphism.     

On the Lipschitz perturbation of monotonic functions

Summary A real valued function <InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"16"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"17"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"18"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"19"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"20"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"21"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"22"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"23"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"24"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"25"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"26"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"27"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"28"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"29"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"30"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"31"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"32"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"33"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"34"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>f$ defined on a real interval $I$ is called \emph{$d$-Lipschitz} if it satisfies $|\ell(x)- \ell(y)| \le d(x,y)$ for $x,y\in I$. In this paper, we investigate when a function $p\: I \to \bR$ can be decomposed in the form $p=q+ \ell$, where $q$ is increasing and $\ell$ is $d$-Lipschitz. In the general case when $d\: I^{2} \to \bR$ is an arbitrary semimetric, a function $p\: I \to \bR$ can be written in the form $p=q+ \ell$ if and only if \vspace{-4pt} <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sum_{i=1}^{n}{\big(p(s_{i})-p(t_{i})-d(t_{i},s_{i}) \big)^{+}} \le \sum_{j=1}^{m}{\big(p(v_{j})-p(u_{j})+d(u_{j},v_{j}) \big)} \vspace{-4pt} $$ is fulfilled for all real numbers $t_{1}<s_{1}, \dots, t_{n}<s_{n}$ and $u_{1}<v_{1}, \dots, u_{m}<v_{m}$ in $I$ satisfying the condition \vspace{-4pt} $$ \sum_{i=1}^{n} 1_{\left]t_i,s_i\right]}= \sum_{j=1}^{m} 1_{\left]u_j,v_j\right]}, \vspace{-4pt} $$ where $1_{\left]a,b\right]}$ denotes the characteristic function of the interval $\left]a,b\right]$. In the particular case when $d\: I^{2} \to R$ is a so-called concave semimetric, a function $p\: I \to \bR$ is of the form $p=q+ \ell$ if and only if \vspace{-4pt} $$ 0 \le \sum_{k=1}^{n}{d(x_{2k-1},x_{2k})} + d(x_0,x_{2n+1}) + \sum_{k=0}^{n}{\big(p(x_{2k+1})-p(x_{2k})\big)} \vspace{-4pt} $$ holds for all $x_0\le x_1\ki \cdots\ki x_{2n}\le x_{2n+1}$ in $I$.     

Jordan higher derivable maps on triangular algebras by commutative zero products

In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained.     

解析函数与Lipschitz条件

研究了解析函数与Lipschitz条件,得到了如下两个结果:(i)设D是一平面区域,f(z)在D中解析,00,对任意z∈D有|f′(z)|≤md(z,D)k-1,则f∈Lipk(D)且‖f‖k≤cmk,其中c=c(D)是仅与D有关的常数.     

Isotropic Hypersurfaces and Minimal Extensions of Lipschitz Functions

The existence and uniqueness theorem for isotropic hypersurfaces with prescribed boundary in Lorentzian warped products is proved.The proof is based on minimal Lipschitz extensions of functions. __________ Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 39, No. 3, pp. 28–36, 2005 Original Russian Text Copyright #x00A9; by A. A. Klyachin and V. M. Miklyukov