Jyh-Shyang Jeang, Ngai-Ching Wong
Suppose $X$ and $Y$ are locally compact Hausdorff spaces, $E$ and $F$ are Banach spaces and $F$ is strictly convex. We show that every linear isometry $T$ from $C_0(X,E)$ {\em into} $C_0(Y,F)$ is essentially a weighted composition operator $Tf(y) = h(y) (f(φ(y)))$. This supplements results of Jerison (when $T$ is onto) and Cambern (when $X,Y$ are compact).
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We prove that if the group of isometries of C(X,E) is algebraically reflexive, then the group of n-isometries is also algebraically reflexive. Here, X is a compact Hausdorff space and E is a Banach space. As a corollary to this, we establish the algebraic reflexivity of the set of generalized bi-circular projections on C(X,E).
Surjective Hausdorff Isometries of Hyperspaces of Bounded Closed Convex Sets
本文证明实 Banach 空间中有界闭凸集超空间的满射 Hausdorff 等距必由底层空间的满射仿射等距诱导，将 Gruber-Lettl 结果从有限维推广到任意实 Banach 空间。
We say that a map $T: S_X\rightarrow S_Y$ between the unit spheres of two real normed-spaces $X$ and $Y$ is a phase-isometry if it satisfies \begin{eqnarray*} \{\|T(x)+T(y)\|, \|T(x)-T(y)\|\}=\{\|x+y\|, \|x-y\|\} \end{eqnarray*} for all $x,y\in S_X$. In the present paper, we show that there is a phase function $\varepsilon:S_X\rightarrow \{-1,1\}$ such that $\varepsilon \cdot T$ is an isometry which can be extended a linear isometry on the whole space $X$ whenever $T$ is surjective, $X=C(K)$ ($K$ is a compact Hausdorff space) and $Y$ is an arbitrary Banach space. Additionally, if $T$ is a phas…
Surjective isometries on function spaces with derivatives
本文研究了带导数的复Banach函数空间上的满射等距问题，通过刻画可能非线性的满射等距映射，统一了特定函数空间间满射等距的已有结果。
Mazur-Ulam Theorem With Gromov-Hausdorff Distance
本文在 Gromov-Hausdorff 距离框架下，研究了 Banach 空间线性等距的充要条件，并给出了有限维情况下的简化证明与等距判据。
In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces $A$ and $B$ of $C_0(X,E)$ and $C_0(Y,F)$ where $X$ and $Y$ are locally compact Hausdorff spaces and $E$ and $F$ are normed spaces, not assumed to be neither strictly convex nor complete. We show that for a class of normed spaces $F$ satisfying a new defined property related to their $T$-sets, such an isometry is a (generalized) weighted composition operator up to a translation. Then we apply the result to study surjective isometries between $A$ and $B$ whenever $A$ and $B$ are equi…
In the recent paper \cite{Hos}, surjective isometries, not necessarily linear, $T: {\rm AC}(X,E) \longrightarrow {\rm AC}(Y,F)$ between vector-valued absolutely continuous functions on compact subsets $X$ and $Y$ of the real line, has been described. The target spaces $E$ and $F$ are strictly convex normed spaces. In this paper, we assume that $X$ and $Y$ are compact Hausdorff spaces and $E$ and $F$ are normed spaces, which are not assumed to be strictly convex. We describe (with a short proof) surjective isometries $T: (A,\|\cdot\|_A) \longrightarrow (B,\|\cdot\|_B)$ between certain normed su…
Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence
本文在Latremoliere的propinquity框架下，从度量角度研究了谱三元组的等距群与双Lipschitz自同构集，证明了其紧致性，并给出了归纳极限下等距群作用在协变Gromov-Hausdorff propinquity意义下收敛的充要条件。
Let $f: S(E) \to S(B)$ be a surjective isometry between the unit spheres of two weakly compact JB$^*$-triples not containing direct summands of rank smaller than or equal to 3. Suppose $E$ has rank greater than or equal to 5. Applying techniques developed in JB$^*$-triple theory, we prove that $f$ admits an extension to a surjective real linear isometry $T: E\to B$. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C$^*$-algebras $A$ and $B$ (and in particular when $A=K(H)$ and $B=K(H')$) extends to a surjective real linear isometry from $A$…
Lattice renormings of $C_0(X)$ spaces
该研究针对局部紧Polish空间上的$C_0(X)$空间，证明满足条件的格等距群可通过等价格重赋范成为其全体格等距群，并将结果推广到任意局部紧Polish群的表示问题。
The 2-locality problem of diameter-preserving maps between C(X)-spaces is addressed in this paper. For any compact Hausdorff space X with at least three points, we give an example of a 2-local diameter-preserving map on C(X) which is not linear. However, we show that for first countable compact Hausdorff spaces X and Y, every 2-local diameter-preserving map from C(X) to C(Y) is linear and surjective up to constants in some sense. This yields the 2-algebraic reflexivity of isometries with respect to the diameter norms on the quotient spaces.
Phase-isometries between the positive cones of the Banach space of continuous real-valued functions
本文研究连续实值函数空间正锥间的满射相位等距，证明其必由底空间同胚诱导，并可扩张为全空间上的满射线性等距。
On phase-isometries between the unit spheres of the Banach space of continuous real-valued functions
本文研究了Banach空间连续实值函数单位球之间的相位等距映射，证明其可表示为加权复合算子的变体。
Let us assume that we are given two metric spaces, where the Hausdorff dimension of the first space is strictly smaller than the one of the second space. Suppose further that the first space has sigma-finite measure with respect to the Hausdorff measure of the corresponding dimension. We show for quite general metric spaces that for any measurable surjection from the first onto the second space, there is a set of measure zero that is mapped to a set of positive measure (both measures are the Hausdorff measures corresponding to the Hausdorff dimension of the first space). We also study more gen…
On extremal nonexpansive mappings
本文研究赋范空间非空有界闭凸子集上非扩张映射的极值性，证明满射等距映射在具有Radon-Nikodym性质的巴拿赫空间及C(K)空间中为极值，并表明Baire范畴意义下典型非扩张映射接近极值。
Group homomorphisms induced by isometries
本文研究局部紧群上殆周期函数空间的线性等距映射诱导的群同态，分别在保持有限维酉表示和保持三角多项式的条件下，给出了等距映射的具体结构刻画。
Characterization of surjective isometries: the real case
该研究刻画了两类空间上的满实线性等距映射形式，确定对应因子的唯一性，还推导了Mityagin问题的交换与非交换等距形式。
Let $M$ be a separable compact Hausdorff space with $\dim M\le 2$ and $θ\colon M\to M$ be a homeomorphism with prime period $p$ ($p\ge 2$). Set $M_θ=\{x\in M| θ(x)=x\}\not=\varnothing$ and $M_0=M\backslash M_θ$. Suppose that $M_0$ is dense in $M$ and $\mathrm H^2(M_0/θ,\Z)\cong 0$, $\mathrm H^2(χ(M_0/θ),\Z)\cong 0$. Let $M'$ be another separable compact Hausdorff space with $\dim M'\le 2$ and $θ'$ be the self--homeomorphism of $M'$ with prime period $p$. Suppose that $M_0'=M'\backslash M_{θ'}'$ is dense in $M'$. Then $C(M)\times_θ\Z_p\cong C(M')\times_{θ'}\Z_p$ iff there is a homeomorphism $F$…
Let $M$ and $N$ be unital Jordan-Banach algebras, and let $M^{-1}$ and $N^{-1}$ denote the sets of invertible elements in $M$ and $N$, respectively. Suppose that $\mathfrak{M}\subseteq M^{-1}$ and $\mathfrak{N}\subseteq N^{-1}$ are clopen subsets of $M^{-1}$ and $N^{-1}$, respectively, which are closed for powers, inverses and products of the form $U_{a} (b)$. In this paper we prove that for each surjective isometry $Δ: \mathfrak{M}\to \mathfrak{N}$ there exists a surjective real-linear isometry $T_0: M\to N$ and an element $u_0$ in the McCrimmon radical of $N$ such that $Δ(a) = T_0(a) +u_0$ f…
Invariant strictly convex renormings
本文研究Banach空间上关于群等距作用不变的严格凸等价范数的存在性，开发构造工具并证明保持性结果，同时肯定回答了关于$c$的全线性等距群不变严格凸重范化问题，并指出群顺从性的作用。
This paper contains the following results: a) Suppose that X is a non-trivial Banach space and L is a non-empty locally compact Hausdorff space without any isolated points. Then each linear operator T: C_{0}(L,X)\to C_{0}(L,X), whose range does not contain C_{00} isomorphically, satisfies the Daugavet equality ||I+T||=1+||T||. b) Let Γbe a non-empty set and X, Y be Banach spaces such that X is reflexive and Y does not contain c_{0} isomorphically. Then any continuous linear operator T: c_{0}(Γ,X)\to Y is weakly compact.
Let G be a compact group acting on a Banach space L by means of linear isometries. The action of G on L induces a natural continuous action on cc(L), the hyperspace of all compact convex subsets of L endowed with the Hausdorff metric topology. The main result of this paper states that the G-space cc(L) is a G-AE. Under some extra assumptions, this result can be extended to CB(L), the hyperspace of all closed and bounded convex subsets of L.
Let $X$ and $Y$ be completely regular spaces and $E$ and $F$ be Hausdorff topological vector spaces. We call a linear map $T$ from a subspace of $C(X,E)$ into $C(Y,F)$ a \emph{Banach-Stone map} if it has the form $Tf(y) = S_{y}(f(h(y))$ for a family of linear operators $S_{y} : E \to F$, $y \in Y$, and a function $h: Y \to X$. In this paper, we consider maps having the property: \cap^{k}_{i=1}Z(f_{i}) \neq\emptyset\iff\cap^{k}_{i=1}Z(Tf_{i}) \neq \emptyset, where $Z(f) = \{f = 0\}$. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces…
Let $X$ be a topological space. A subset of $C(X)$, the space of continuous real-valued functions on $X$, is a partially ordered set in the pointwise order. Suppose that $X$ and $Y$ are topological spaces, and $A(X)$ and $A(Y)$ are subsets of $C(X)$ and $C(Y)$ respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between $A(X)$ and $A(Y)$. Under some general assumptions on $A(X)$ and $A(Y)$, and when $X$ and $Y$ are compact Hausdorff, it is shown that existence of an order isomorphism between $A(X)$ and $A(Y)$ gives rise to an asso…
Suppose that $\{T_{a}:a\in G\}$ is a group of uniformly $L$-Lipschitzian mappings with bounded orbits $\left\{T_{a}x:a\in G\right\}$ acting on a hyperconvex metric space $M$. We show that if $L<\sqrt{2}$, then the set of common fixed points $Fix \, G$ is a nonempty Hölder continuous retract of $M$. As a consequence, it follows that all surjective isometries acting on a bounded hyperconvex space have a common fixed point. A fixed point theorem for $L$-Lipschitzian involutions and some generalizations to the case of $λ$-hyperconvex spaces are also given.
Observations on some classes of operators on $C(K,X)$
本文研究$C(K,X)$上的几类算子，证明算子的弱Dunford-Pettis等性质与其延拓的等价性，以及强有界极限类算子与其表示测度的蕴含关系，并指出散紧空间下该蕴含变为等价。
Suppose that $Q$ is a weak$^{\ast }$ compact convex subset of a dual Banach space with the Radon-Nikodým property. We show that if $(S,Q)$ is a nonexpansive and norm-distal dynamical system, then there is a fixed point of $S$ in $Q$ and the set of fixed points is a nonexpansive retract of $Q.$ As a consequence we obtain a nonlinear extension of the Bader-Gelander-Monod theorem concerning isometries in $L$-embedded Banach spaces. A similar statement is proved for weakly compact convex subsets of a locally convex space, thus giving the nonlinear counterpart of the Ryll-Nardzewski theorem.
Banach algebras associated to twisted étale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces
本文引入与扭曲étale群胚及扭曲逆半群作用相关的Banach代数，提供统一框架，证明分解定理，建立范数估计与层次结构，覆盖非Hausdorff情形，且结果对复C*代数有新贡献。
Let $X$ and $Y$ be Banach spaces such that the ideal of operators which factor through $Y$ has codimension one in the Banach algebra $\mathscr{B}(X)$ of all bounded operators on $X$, and suppose that $Y$ contains a complemented subspace which is isomorphic to $Y\oplus Y$ and that $X$ is isomorphic to $X\oplus Z$ for every complemented subspace $Z$ of $Y$. Then the $K_0$-group of $\mathscr{B}(X)$ is isomorphic to the additive group $\mathbb{Z}$ of integers. A number of Banach spaces which satisfy the above conditions are identified. Notably, it follows that $K_0(\mathscr{B}(C([0,ω_1])))\cong\ma…
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