In this paper we investigate the relationship between the properties of a compact set $σ\subseteq \mathbb{C}$ and the structure of the space $BV(σ)$ of functions of bounded variation (in the sense of Ashton and Doust) defined on $σ$. For the subalgebras of absolutely continuous functions on $σ$, it is known that for certain classes of compact sets one obtains a Gelfand--Kolmogorov type result: the function spaces $AC(σ_1)$ and $AC(σ_2)$ are isomorphic if and only if the domain sets $σ_1$ and $σ_2$ are homeomorphic. Our main theorem is that in this case the isomorphism must extend to an isomorphism of the $BV(σ)$ spaces. An application is given to the spectral theory of $AC(σ)$ operators.
Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.
In the paper, it is given isomorphic classification of $F$-spaces of $log$-integrable measurable functions constructed using different measure spaces. At the same time, it is proved that such spaces are non-isometric.
Classical Segal-Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on $\mathbb{R}^n$ with respect to a Gaussian measure, the space of monogenic square-integrable functions on $\mathbb{R}^{n+1}$ with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor…
We show that among compact subsets of the plane which are drawings of linear graphs, two sets $σ$ and $τ$ are homeomorphic if and only if the corresponding spaces of absolutely continuous functions (in the sense of Ashton and Doust) are isomorphic as Banach algebras. This gives an analogue for this class of sets to the well-known result of Gelfand and Kolmogorov for $C(Ω)$ spaces.
Let $K_1$, $K_2$ be compact Hausdorff spaces and $E_1, E_2$ be Banach spaces not containing a copy of $c_0$. We establish lower estimates of the Banach-Mazur distance between the spaces of continuous functions $\mathcal{C}(K_1, E_1)$ and $\mathcal{C}(K_2, E_2)$ based on the ordinals $ht(K_1)$, $ht(K_2)$, which are new even for the case of spaces of real valued functions on ordinal intervals. As a corollary we deduce that $\mathcal{C}(K_1, E_1)$ and $\mathcal{C}(K_2, E_2)$ are not isomorphic if $ht(K_1)$ is substantially different from $ht(K_2)$.
We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.
Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that an affine function with the point of continuity property on X satisfies the minimum principle. As a corollary we obtain a generalization of a theorem by H.B. Cohen and C.H. Chu by proving the following result. Let X,Y be compact convex sets such that every extreme point of X and Y is a weak peak point and let the Banach-Mazur distance between spaces of affine continuous functions on X and Y is smaller then 2. Then ext X is homeomorphic to ext Y.
It is known that the classical Banach--Stone theorem does not extend to the class of $AC(σ)$ spaces of absolutely continuous functions defined on compact subsets of the complex plane. On the other hand, if $σ$ is restricted to the set of compact polygons, then all the corresponding $AC(σ)$ spaces are isomorphic. In this paper we examine the case where $σ$ is the spectrum of a compact operator, and show that in this case one can obtain an infinite family of homeomorphic sets for which the corresponding function spaces are not isomorphic.
Let $G$ be a locally compact group and $ω$ be a continuous weight on $G$. In this paper, we first characterize bicontinuous biseparating algebra isomorphisms between weighted $L^p$-algebras. As a result we extend previous results of Edwards, Strichartz and Parrot on algebra isomorphisms between $L^p$-algebras to the setting of weighted $L^p$-algebras. We then study the automorphisms of certain weighted $L^p$-algebras on integers, applying known results on composition operators on classical function spaces.
This article presents an isomorphism between two operator algebras $L_1$ and $L_2$ where $L_1$ is the set of operators on a space of Hilbert-Schmidt operators and $L_2$ is the set of operators on a tensor product space. We next compare our isomorphism to a well-known result called Choi's isomorphism theorem. The advantage of Choi's isomorphism is that it takes completely positive maps to positive operators. One advantage of our isomorphism is that it applies to infinite dimensional Hilbert spaces, while Choi's isomorphism only holds for finite dimensions. Also, our isomorphism preserves operat…
Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces
我们将有关可分巴拿赫空间之间同构关系复杂性的已知结果改进为Borel可约性，并且我们使用经典空间$c_0$、$\ell_p$和$L_p$、$1 \leq p <2$实现了这一点。更准确地说，我们表明，关系$E_{K_σ}$是Borel可约同构和补充双嵌入之间的$c_0$或$\ell_p, 1 \leq p <2$的子空间。我们证明了关系$E_{K_σ} \otimes =^+$是Borel可约的同构，补双嵌入，和Lipschitz等价的子空间之间的$L_p, 1 \leq p <2$。
We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that $\operatorname{Lip}_0(\mathbb{Z}^d)\simeq\operatorname{Lip}_0(\mathbb{R}^d)$, for all $d\in\mathbb{N}$. More generally, we e.g. show that $\operatorname{Lip}_0(Γ)\simeq \operatorname{Lip}_0(G)$, where $Γ$ is from a large class of finitely generated nilpotent groups and $G$ is its Mal'cev closure; or that $\operatorname{Lip}_0(\ell_p)\simeq\operatorname{Lip}_0(L_p)$, for all $1\leq p<\infty$. We leave a large area for further possib…
We study pairs of Dirichlet forms related by an intertwining order isomorphisms between the associated $L^2$-spaces. We consider the measurable, the topological and the geometric setting respectively. In the measurable setting, we deal with arbitrary (irreducible) Dirichlet forms and show that any intertwining order isomorphism is necessarily unitary (up to a constant). In the topological setting we deal with quasi-regular forms and show that any intertwining order isomorphism induces a quasi-homeomorphism between the underlying spaces. In the geometric setting we deal with both regular Dirich…
The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $Δ$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_0$ which is isomorphic to a closed subspace of a space with a basis and $C(I, X)$ is linearly homeomorphic to $C(Δ, X)$, then $X$ is locally convex, i.e., a Banach space. It is shown that Kalton result is sharp by exhibiting non locally convex quasi Banach spaces X with a basis for which $C(I, X)$ and $C(Δ, X)$ are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to $C(Δ, X)$ …
We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.
