Lipschitz Algebras Research Articles

Order isomorphisms between real little Lipschitz algebras with Lipschitz involution

Order isomorphisms between real little Lipschitz algebras with Lipschitz involution

Measurability, spectral densities, and hypertraces in noncommutative geometry

We introduce, in the dual Macaev ideal of compact operators of a Hilbert space, the spectral weight \rho(L) of a positive, self-adjoint operator L having discrete spectrum away from zero. We provide criteria for its measurability and unitarity of its Dixmier traces ( \rho(L) is then called spectral density) in terms of the growth of the spectral multiplicities of L or in terms of the asymptotic continuity of the eigenvalue counting function N_L . Existence of meromorphic extensions and residues of the \zeta -function \zeta_L of a spectral density are provided under summability conditions on spectral multiplicities. The hypertrace property of the states \Omega_L(\cdot)=\mathrm{Tr}_\omega(\cdot\rho(L)) on the norm closure of the Lipschitz algebra \mathcal{A}_L follows if the relative multiplicities of L vanish faster than its spectral gaps or if N_L is asymptotically regular.

Exotic cyclic cohomology classes and Lipschitz algebras

We study the noncommutative geometry of algebras of Lipschitz continuous and H\"older continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair non-trivially with higher algebraic $K$-theory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes-Karoubi's relative sequence in $K$-theory.

The Regularity of the Vector-Valued Lipschitz Algebras

The Regularity of the Vector-Valued Lipschitz Algebras

Surjective isometries on the Banach algebra of continuously differentiable maps with values in Lipschitz algebra

Let $${\text {Lip}}(I)$$ be the Banach algebra of all Lipschitz functions on the closed unit interval I with the norm $$\Vert f\Vert _L=\Vert f\Vert _\infty +L(f)$$ for $$f\in {\text {Lip}}(I)$$ , where L(f) is the Lipschitz constant of f. We denote by $$C^{1}(I, {\text {Lip}}(I))$$ the Banach algebra of all continuously differentiable functions F from I to $${\text {Lip}}(I)$$ equipped with the norm $$\Vert F\Vert _{\Sigma }=\sup _{s\in I}\Vert F(s)\Vert _L+\sup _{t\in I}\Vert D(F)(t)\Vert _L$$ for $$F\in C^{1}(I, {\text {Lip}}(I))$$ . In this paper, we prove that if T is a surjective, not necessarily linear, isometry on $$C^{1}(I, {\text {Lip}}(I))$$ , then $$T-T(0)$$ is a weighted composition operator or its complex conjugation. Among other things, any surjective complex linear isometry on $$C^{1}(I, {\text {Lip}}(I))$$ is of the following form: $$c_{1}F(\tau _1(s),\tau _2(x))$$ , where $$c_{1}$$ is a complex number of modulus 1, and $$\tau _1$$ and $$\tau _2$$ are isometries of I onto itself.

On finitely summable Fredholm modules from Smale spaces

We prove that all K K -homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K K -duality between the Ruelle algebras. Finally, we obtain general results on stability under holomorphic functional calculus and construct Lipschitz algebras on étale groupoids.

Disjointness preserving Fredholm operators between little Lipschitz algebras

Disjointness preserving Fredholm operators between little Lipschitz algebras

The character space of vector-valued Lipschitz algebras

In this paper, we determine the character space of the vector-valued Lipschitz algebra Lipd(X, E), where (X, d) is any metric space and E is a certain unital semisimple commutative *-Banach algebra.

Closed ideals in weighted big Lipschitz algebras of analytic functions

Closed ideals in weighted big Lipschitz algebras of analytic functions

On Vector-Valued Banach Function Algebras

We consider vector-valued Banach function algebras on a compact Hausdorff space. Then, we define the subalgebras generated by vector-valued polynomials and rational functions, and determine their maximal ideal spaces and Silov boundaries. We finally make use the results for a certain category of these algebras such as vector-valued Lipschitz algebras, vector-valued Dales–Davie algebras (algebras of vector-valued differentiable functions) and the algebras of vector-valued differentiable Lipschitz functions.

BSE Properties of Some Banach Function Algebras

In this paper, BSE properties of some Banach function algebras are studied. We show that Lipschitz algebras Lipα (X, d) and Dales-Davie algebras D(X, M) are BSE-algebras for certain underlying plane sets X. Moreover, we investigate BSE properties of certain subalgebras of Lipα(X, d)suchas LipA (X, α), Lipn (X, α) and Lip(X, M, α). BSE properties of Bloch type spaces $${{\cal B}_\alpha}$$ and Zygmund type spaces Zα are also investigated in different cases of α ∈ ℝ.

The BSE concepts for vector-valued Lipschitz algebras

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ (K,d) $\end{document}</tex-math></inline-formula> be a compact metric space, <inline-formula><tex-math id="M2">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> be a commutative semisimple Banach algebra and <inline-formula><tex-math id="M3">\begin{document}$ 0&lt;\alpha\leq 1 $\end{document}</tex-math></inline-formula>. The overall purpose of the present paper is to demonstrate that all BSE concepts of <inline-formula><tex-math id="M4">\begin{document}$ {\rm Lip}_\alpha(K,\mathcal A) $\end{document}</tex-math></inline-formula> are inherited from <inline-formula><tex-math id="M5">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> and vice versa. Recently, the authors proved in the case that <inline-formula><tex-math id="M6">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> is unital, <inline-formula><tex-math id="M7">\begin{document}$ {\rm Lip}_\alpha(K,\mathcal A) $\end{document}</tex-math></inline-formula> is a BSE-algebra if and only if <inline-formula><tex-math id="M8">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> is so. In this paper, we generalize this result for an arbitrary commutative semisimple Banach algebra <inline-formula><tex-math id="M9">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula>. Furthermore, we investigate the BSE-norm property for <inline-formula><tex-math id="M10">\begin{document}$ {\rm Lip}_\alpha(K,\mathcal A) $\end{document}</tex-math></inline-formula> and prove that <inline-formula><tex-math id="M11">\begin{document}$ {\rm Lip}_\alpha(K,\mathcal A) $\end{document}</tex-math></inline-formula> belongs to the class of BSE-norm algebras if and only if <inline-formula><tex-math id="M12">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> is owned by this class. Moreover, we prove that for any natural number <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M14">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>, if all continuous bounded functions on <inline-formula><tex-math id="M15">\begin{document}$ \Delta({\rm Lip}_\alpha(K,\mathcal A)) $\end{document}</tex-math></inline-formula> are <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula>-BSE-functions, then <inline-formula><tex-math id="M17">\begin{document}$ K $\end{document}</tex-math></inline-formula> is finite. As a result, we obtain that <inline-formula><tex-math id="M18">\begin{document}$ {\rm Lip}_{\alpha}(K,\mathcal A) $\end{document}</tex-math></inline-formula> is a BSE-algebra of type I if and only if <inline-formula><tex-math id="M19">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> is a BSE-algebra of type I and <inline-formula><tex-math id="M20">\begin{document}$ K $\end{document}</tex-math></inline-formula> is finite. Furthermore, in according to a result of Kaniuth and Ülger, which disapproves the BSE-property for <inline-formula><tex-math id="M21">\begin{document}$ {\rm lip}_{\alpha}K $\end{document}</tex-math></inline-formula>, we show that for any commutative semisimple Banach algebra <inline-formula><tex-math id="M22">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M23">\begin{document}$ {\rm lip}_{\alpha}(K,\mathcal A) $\end{document}</tex-math></inline-formula> fails to be a BSE-algebra, as well. Finally, we concentrate on the classical Lipschitz algebra <inline-formula><tex-math id="M24">\begin{document}$ {\rm Lip}_\alpha X $\end{document}</tex-math></inline-formula>, for an arbitrary metric space (not necessarily compact) <inline-formula><tex-math id="M25">\begin{document}$ (X,d) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M26">\begin{document}$ \alpha&gt;0 $\end{document}</tex-math></inline-formula>, when <inline-formula><tex-math id="M27">\begin{document}$ {\rm Lip}_\alpha X $\end{document}</tex-math></inline-formula> separates the points of <inline-formula><tex-math id="M28">\begin{document}$ X $\end{document}</tex-math></inline-formula>. In particular, we show that <inline-formula><tex-math id="M29">\begin{document}$ {\rm Lip}_\alpha X $\end{document}</tex-math></inline-formula> is a BSE-algebra, as well as a BSE-norm algebra.

A SPHERICAL VERSION OF THE KOWALSKI–SŁODKOWSKI THEOREM AND ITS APPLICATIONS

AbstractLi et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let $\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties: (a) $\Delta $ is 1-homogeneous (that is, $\Delta (\lambda x)=\lambda \Delta (x)$ for all $x \in A$ , $\lambda \in \mathbb C$ ); (b) $\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$ . Then $\Delta $ is linear and there exists $\lambda _{0} \in \mathbb {T}$ such that $\lambda _{0}\Delta $ is multiplicative. In this note we prove that if (a) is relaxed to $\Delta (0)=0$ , then $\Delta $ is complex-linear or conjugate-linear and $\overline {\Delta (\mathbf {1})}\Delta $ is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.

The Maximal Ideal Space of Extended Di erentiable Lipschitz Algebras

The Maximal Ideal Space of Extended Di erentiable Lipschitz Algebras

Additive jointly separating maps and ring homomorphisms

Let X and Y be compact Hausdorff spaces, E and F be real or complex normed spaces and be a subspace of . For a function , let be the cozero set of f. A pair of additive maps is said to be jointly separating if whenever . In this paper, first, we give a partial description of additive jointly separating maps between certain spaces of vector-valued continuous functions (including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions). Then we apply the results to characterize continuous ring homomorphisms between certain Banach algebras of vector-valued continuous functions. In particular, the results provide some generalizations of the recent results on unital homomorphisms between vector-valued Lipschitz algebras, with a different approach.

Weighted Composition Operators Between Extended Lipschitz Algebras on Compact Metric Spaces

‎In this paper, we provide a complete description of weighted composition operators between extended Lipschitz algebras on compact metric spaces. We give necessary and sufficient conditions for the injectivity and the sujectivity of these operators. We also obtain some sufficient conditions and some necessary conditions for a weighted composition operator between these spaces to be compact.

Approximate biflatness and Johnson pseudo-contractibility of some Banach algebras

We study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space $X$, the Lipschitz algebras ${\rm Lip}_{\alpha}(X)$ and ${\rm lip}_{\alpha}(X)$ are approximately biflat if and only if $X$ is finite, provided that $0<\alpha<1$. We give a necessary and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible. We also show that some triangular Banach algebras are not approximately biflat.

Isometric Composition Operators on Lipschitz Spaces

Given pointed metric spaces X and Y, we characterize the basepoint-preserving Lipschitz maps $$\phi $$ from Y to X inducing an isometric composition operator $$C_\phi $$ between the Lipschitz spaces $$\mathrm {Lip}_0(X)$$ and $$\mathrm {Lip}_0(Y)$$, whenever X enjoys the peak property. This gives an answer to a question posed by Weaver in his book [Lipschitz algebras. Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018].

2-Local uniform isometries between complex Lipschitz algebras

Let (X, d) be a metric space and let $$\mathrm{Lip}(X,d) $$ denote the complex algebra of all complex-valued bounded functions f on X for which f is a Lipschitz function on $$\mathrm{(X,d)}$$. In this paper we give a complete description of all 2-local real and complex uniform isometries between $$\mathrm{Lip}(X,d) $$ and $$\mathrm{Lip(Y},\rho \mathrm{)}$$, where (X, d) and $$(Y,\rho )$$ are compact metric spaces. In particular, we show that every 2-local real (complex, respectively) uniform isometry from $$\mathrm{Lip}(X,d) $$ to $$\mathrm{Lip(Y,}\rho \mathrm{)}$$ is a surjective real (complex, respectively) linear uniform isometry.

Derivations and cohomologies of Lipschitz algebras

For a compact metric space (M, d), $${\mathrm {Lip}}M$$ denotes the Banach algebra of all complex-valued Lipschitz functions on (M, d). Motivated by a classical result of de Leeuw, we give a canonical construction of a compact Hausdorff space $${\hat{M}}$$ and a continuous surjection $$\pi :{\hat{M}} \rightarrow M$$ which may viewed as a metric analogue of the unit sphere bundle over a Riemannian manifold. It is shown that, for each $$n \ge 1$$ the continuous Hochschild cohomology $${\mathrm {H}}^{n}({\mathrm {Lip}}M, C({\hat{M}}))$$ has the infinite rank as a $${\mathrm {Lip}}M$$-module, if the metric space (M, d) admits a local geodesic structure, for example, if M is a compact Riemannian manifold or a non-positively curved metric space. Here $$C({\hat{M}})$$ denotes the algebra of all complex-valued continuous functions on $${\hat{M}}$$. On the other hand, if the coefficient $$C({\hat{M}})$$ is replaced with C(M), then it is shown that $${\mathrm {H}}^{1}({\mathrm {Lip}}M,C(M)) = 0$$ for each compact Lipschitz manifold M.