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 phase-isometry between the unit spheres of $C(K)$ and $C(Ω)$, where $K$ and $Ω$ are compact Hausdorff spaces, we prove that there is a homeomorphism $φ: Ω\rightarrow K$ such that $T(f)\in\{f\circ φ,-f\circ φ\}$ for all $f\in S_{C(K)}$. This also can be seen as a Banach-Stone type representation for phase-isometries in $C(K)$ spaces.
We prove that an onto isometry between unit spheres of finite-dimensional polyhedral Banach spaces extends to a linear isometry of the corresponding spaces.
Let $Γ$ be an infinite set equipped with the discrete topology. We prove that the space $\ell_{\infty}(Γ),$ of all complex-valued bounded functions on $Γ$, satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of $\ell_{\infty}(Γ)$ onto the unit sphere of an arbitrary complex Banach space $X$ admits a unique extension to a surjective real linear isometry from $\ell_{\infty}(Γ)$ to $X$.
Isometries of Unit Spheres of SFS-Spaces and a Geometric Approach to Tingley's Problem
Institute for Advances Study in Mathematics, Harbin Institute of Technologies（哈尔滨工业大学数学前沿研究院）；Suzhou Research Institute of Harbin Institute of Technology（哈尔滨工业大学苏州研究院）；V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院V.I.罗曼诺夫斯基数学研究所）；Karakalpak State University（卡拉卡尔帕克国立大学）
We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, which one-complemented subspaces inherit, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces, including that they are uniformly convex and uniformly smooth, and that they form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known r…
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 $A$…
There are considered isometries on a Hilbert space. By the Wold theorem any isometry can be decomposed into a unitary operator and a unilateral shift. For a pair of isometries, even commuting, a maximal subspace reducing one isometry to a unitary operator might not reduce the other isometry. In the paper are considered pairs of commuting isometries which are completely non doubly commuting. For such pairs there are no nontrivial subspaces reducing both isometries and one of them to a unitary operator. The results describe a unitary part of an isometry in such a pair.
We prove that every surjective isometry from the unit sphere of the space $K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$, onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a surjective real linear isometry from $K(H)$ onto $Y$. This is probably the first example of an infinite dimensional non-commutative C$^*$-algebra containing no unitaries and satisfying the Mazur--Ulam property. We also prove that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy the Mazur--Ulam property.
We prove that every JBW$^*$-triple $M$ with rank one or rank bigger than or equal to three satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$. We also show that the same conclusion holds if $M$ is not a JBW$^*$-triple factor, or more generally, if the atomic part of $M^{**}$ is not a rank two Cartan factor.
Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$ denote the unit spheres of the C$^*$-algebras $B(H)$ and $B(K)$ of all bounded linear operators on $H$ and $K$, respectively. We prove that every surjective isometry $f: S(B(K)) \to S(B(H))$ admits an extension to a surjective complex linear or conjugate linear isometry $T: B(K)\to B(H)$. This provides a positive answer to Tingley's problem in the setting of $B(H)$ spaces.
In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute matrix order unit space. As an application, we describe the comparison of order projections. We also discuss finiteness of order projections.
We prove that every surjective isometry from the unit sphere of a rank-2 Cartan factor $C$ onto the unit sphere of a real Banach space $Y$, admits an extension to a surjective real linear isometry from $C$ onto $Y$. The conclusion also covers the case in which $C$ is a spin factor. This result closes an open problem and, combined with the conclusion in a previous paper, allows us to establish that every JBW$^*$-triple $M$ satisfies the Mazur--Ulam property, that is, every surjective isometry from its unit sphere onto the unit sphere of a arbitrary real Banach space $Y$ admits an extension to a…
We prove that for a bijective, unital, linear map between absolute order unit spaces is an isometry if, and only if, it is absolute value preserving. We deduce that, on (unital) $JB$-algebras, such maps are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces and prove that for a bijective, unital, linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is completely absolute value preserving. We obtain that on (unital) C$^*$-algebras such maps are precisely C$^*$-algebra …
We study 2-local reflexivity of the set of all surjective isometries between certain function spaces. We do not assume linearity for isometries. We prove that a 2-local isometry in the group of all surjective isometries on the algebra of all continuously differentiable functions on the closed unit interval with respect to several norms is a surjective isometry. We also prove that a 2-local isometry in the group of all surjective isometries on the Banach algebra of all Lipschitz functions on the closed unit interval with the sum-norm is a surjective isometry.
Surjective isometries on the positive parts of the unit spheres of some function spaces
Graduate School of Science and Technology, Niigata University（新潟大学科学技术研究生院）；National Institute of Technology, Tsuruoka College（国立高冈技术学院）；National Institute of Technology, Yonago College（国立米子技术学院）；Department of Applied Mathematics, National Sun Yat-sen University（国立中山大学应用数学系）；National Sun Yat-sen University（国立中山大学）
We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the group of the invertible elements in a unital semisimple commutative Banach algebra onto an open subgroup of the group of the invertible elements in a unital Banach algebra, then $T(1)^{-1}T$ is an isometrical group isomorphism. In particular, $T(1)^{-1}T$ is extended to an isometrical real algebra isomorphism from $A$ onto $B$.
We obtain a formula that relates the spherical moments of the multiplication tuple on a Dirichlet-type space to a complex moment problem in several variables. This can be seen as the ball-analogue of a formula originally invented by Richter. We capitalize on this formula to study Dirichlet-type spaces on the unit ball and joint $2$-isometries.
We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.
We give some new characterizations of unitaries, isometries, unital operator spaces, unital function spaces, operator systems, C*-algebras, and related objects. These characterizations only employ the vector space and operator space structure.
Let $H(\mathbb{D})$ be the linear space of all analytic functions on the open unit disc $\mathbb{D}$ and $H^p(\mathbb{D})$ the Hardy space on $\mathbb{D}$. The characterization of complex linear isometries on $\mathcal{S}^p=\{f\in H(\mathbb{D}):f'\in H^p(\mathbb{D})\}$ was given for $1\leq p<\infty$ by Novinger and Oberlin in 1985. Here, we characterize surjective, not necessarily linear, isometries on $\mathcal{S}^\infty$.
We obtain the admissible sets on the unit circle to be the spectrum of a strict $m$-isometry on an $n$-finite dimensional Hilbert space. This property gives a better picture of the correct spectrum of an $m$-isometry. We determine that the only $m$-isometries on $\mathbb{R}^2$ are $3$-isometries and isometries giving by $\pm I+Q$, where $Q$ is a nilpotent operator. Moreover, on real Hilbert space, we obtain that $m$-isometries preserve volumes. Also we present a way to construct a strict $(m+1)$-isometry with an $m$-isometry given, using ideas of Aleman and Suciu \cite[Proposition 5.2]{AS} on …
We examine the surjectivity of isometries between weighted spaces of holomorphic functions. We show that for certain classical weights on the open unit disc all isometries of the weighted space of holomorphic functions, ${ \mathcal H}_{v_o}( Δ)$, are surjective. Criteria for surjectivity of isometries of ${ \mathcal H}_v(U)$ in terms of a separation condition on points in the image of ${ \mathcal H}_{v_o}(U)$ are also given for $U$ a bounded open set in $\mathbb{C}$. Considering the weight $v(z)= 1-|z|^2$ and the isomorphism $f\mapsto f'$ we are able to show that all isometries of the little B…
A known general program, designed to endow the quotient space ${\cal U}_{\cal A} / {\cal U}_{\cal B}$ of the unitary groups ${\cal U}_{\cal A}$, ${\cal U}_{\cal B}$ of the C$^*$ algebras ${\cal B}\subset{\cal A}$ with an invariant Finsler metric, is applied to obtain a metric for the space ${\cal I}({\cal H})$ of partial isometries of a Hilbert space ${\cal H}$. ${\cal I}({\cal H})$ is a quotient of the unitary group of ${\cal B}({\cal H})\times{\cal B}({\cal H})$, where ${\cal B}({\cal H})$ is the algebra of bounded linear operators in ${\cal H}$. Under this program, the solution of a linear …
This note is concerned with isometries on the spaces of self-adjoint traceless matrices. We compute the group of isometries with respect to any unitary similarity invariant norm. This completes and extends the result of Nagy on Schatten $p$-norm isometries. Furthermore, we point out that our proof techniques could be applied to obtain an old result concerning isometries on skew-symmetric matrices.
In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identify the isometries and spectra of a particular class of weighted composition operators on the Bloch space.
We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the invertible group $A^{-1}$ of a unital Banach algebra $A$ onto an open subgroup of the invertible group $B^{-1}$ of a unital Banach algebra $B$, then $T$ is extended to a real-linear isometry up to translation between these Banach algebras. We consider multiplicativity or unti-multiplicativity of the isometry. Note that a unital linear isometry between unital semisimple commutative Banach algebra need be multiplicative. On the other hand, we show that if $A$ is commutative and $A$ or $B$ are semisimple, then $(T(…
Globalization of local sign structures for phase-isometries on uniform algebras
Graduate School of Science and Technology, Niigata University（新潟大学科学技术研究科）；National Institute of Technology, Tsuruoka College（国立大学法人鹤岗工业高等专门学校）；Department of Mathematics, Faculty of Science, Niigata University（新潟大学理学部数学科）
Based on the characterization of surjective $L^p$-isometries of unitary groups in finite factors, we describe all surjective $L^p$-isometries between Grassmann spaces of projections with the same trace value in semifinite factors.
This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBranges-Rovnyak spaces. This characterization is applied to investig…
Corresponding to any $(m-1)$-tuple of semi-spectral measures on the unit circle, a weighted Dirichlet-type space is introduced and studied. We prove that the operator of multiplication by the coordinate function on these weighted Dirichlet-type spaces acts as an analytic $m$-isometry and satisfies a certain set of operator inequalities. Moreover, it is shown that an analytic $m$-isometry which satisfies this set of operator inequalities can be represented as an operator of multiplication by the coordinate function on a weighted Dirichlet-type space induced from an $(m-1)$-tuple of semi-spectra…
Let $(S_1, S_2)$ be a bi-isometry, that is, a pair of commuting isometries $S_1$ and $S_2$ on a complex Hilbert space $\mathscr H.$ By the von Neumann-Wold decomposition, the hyper-range $\mathscr H_\infty(S_1):=\cap_{n=0}^\infty S^n_1\mathscr H$ of $S_1$ reduces $S_1$ to a unitary operator. Although $\mathscr H_\infty(S_1)$ is an invariant subspace for $S_2,$ in general, $\mathscr H_\infty(S_1)$ is not a reducing subspace for $S_2.$ We show that $\mathscr H_\infty(S_1)$ reduces $S_2$ to an isometry if and only if the subspaces $S_2(\ker S^*_1)$ and $\mathscr H_\infty(S_1)$ of $\mathscr H$ are…
We examine the group of isometries of the open unit ball of a complex Banach space of certain bounded linear operators equipped with the Carathéodory metric. Therein we obtain a charactrization of the normal isometries in terms of their special type of fixed points.
We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of "generalized weighted composition" operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is irreducible and contains a compact operator then the underlying measure space is p…
We extend existing results that characterize isometries on the Tsirelson-type spaces $T\big[\frac{1}{n}, \mathcal{S}_1\big]$ ($n\in \mathbb{N}, n\geq 2$) to the class $T[θ, \mathcal{S}_α]$ ($θ\in \big(0, \frac{1}{2}\big]$, $1\leqslant α< ω_1$), where $\mathcal{S}_α$ denote the Schreier families of order $α$. We prove that every isometry on $T[θ, \mathcal{S}_1]$ ($θ\in \big(0, \frac{1}{2}\big]$) is determined by a permutation of the first $\lceil θ^{-1} \rceil$ elements of the canonical unit basis followed by a possible sign-change of the corresponding coordinates together with a sign-change of…
This paper is, in a first stage, devoted to establish a topological--algebraic characterization of the principal component, $\mathcal{U}^0 (M)$, of the set of unitary elements, $\mathcal{U} (M)$, in a unital JB$^*$-algebra $M$. We arrive to the conclusion that, as in the case of unital C$^*$-algebras, $$\begin{aligned}\mathcal{U}^0(M) &= M^{-1}_{\textbf{1}}\cap\mathcal{U} (M) =\left\lbrace U_{e^{i h_n}}\cdots U_{e^{i h_1}}(\textbf{1}) \colon \begin{array}{c} n\in \mathbb{N}, \ h_j\in M_{sa} \forall\ 1\leq j \leq n \end{array} \right\rbrace \end{aligned}$$ is analytically arcwise connected. Our…
We prove some noncommutative analogues of a theorem by Plotkin and Rudin about isometries between subspaces of Lp-spaces. Let 0<p<\infty, p not an even integer. The main result of this paper states that in the category of unital subspaces of noncommutative probability Lp-spaces, under some boundedness condition, the unital completely isometric maps come from *-isomorphisms of the underlying von Neumann algebras. Some applications are given, including to non commutative H^p spaces.
An operator T on Hilbert space is a 3-isometry if there exists operators B and D such that (T*)^n T^n = I+nB +n^2 D. An operator J is a Jordan operator if it the sum of a unitary U and nilpotent N of order two which commute. If T is a 3-isometry and c>0, then I-c^{-2} D + sB + s^2D is positive semidefinite for all real s if and only if T is the restriction to an invariant subspace of a Jordan operator J=U+N with the norm of N at most c. As a corollary, an analogous result for 3-symmetric operators, due to Helton and Agler, is recovered.
We study the structure of isometries defined on the algebra $\mathcal{A}$ of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative isometry $\mathcal{A}\to M_n$ must be of the form either $A\mapsto UAU^*$ or $A\mapsto U\overline AU^*$, where $\overline A$ is the complex conjugation and $U$ is a unitary matrix. In our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry $\mathcal{A}\to M_n(\mathbb{C})$ is of the form $A\mapsto UAV$ where $U$ and $V$ are two unitary matrices. This implies, in particular,…
Isometries between C$^*$-algebras with finite corank
Faculty of Engineering, Niigata University（新潟大学工学部）；Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所交叉理论数学科学中心(iTHEMS)）
In this paper we characterize surjective isometries on certain classes of non-commutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$, as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$. The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.
Let $\{T(t)\}_{t\ge 0}$ be a $C_0$-semigroup on a separable Hilbert space $H$. We characterize that $T(t)$ is an $m$-isometry for every $t$ in terms that the mapping $t\in \Bbb R^+ \rightarrow \|T(t)x\|^2$ is a polynomial of degree less than $m$ for each $x\in H$. This fact is used to study $m$-isometric right translation semigroup on weighted $L^p$-spaces. We characterize the above property in terms of conditions on the infinitesimal generator operator or in terms of the cogenerator operator of $\{ T(t)\}_{t\geq 0}$. Moreover, we prove that a non-unitary $2$-isometry on a Hilbert space satisf…
We prove that if X is an infinite dimensional Banach lattice with a weak unit then there exists a probability space (Omega, Sigma,mu) so that the unit sphere S(L_1(Omega, Sigma, mu) is uniformly homeomorphic to the unit sphere S(X) if and only if X does not contain l_{infty}^n's uniformly.
A pair of Hilbert space linear operators $(V_1,V_2)$ is said to be $q$-commutative, for a unimodular complex number $q$, if $V_1V_2=qV_2V_1$. A concrete functional model for $q$-commutative pairs of isometries is obtained. The functional model is parametrized by a collection of Hilbert spaces and operators acting on them. As a consequence, the collection serves as a complete unitary invariance for $q$-commutative pairs of isometries. A $q$-commutative operator pair $(V_1,V_2)$ is said to be doubly $q$-commutative, if in addition, it satisfies $V_2V_1^*=qV_1^*V_2$. Doubly $q$-commutative pairs …
Given a category of objects, it is both useful and important to know if all the objects in the category may be realised as sub-objects -- via morphisms in the given category -- of a single object in that category enjoying some nice properties. In the category of separable Banach spaces with morphisms consisting of linear isometries, such an example of (a universal) object is provided by the well-known Banach Mazur theorem: the space C[0,1] of continuous functions on the unit interval contains each separable Banach spaces as a closed subspace via a linear isometry. Here the question also arises…
Let $n > 1$. Let $\{U_{ij}\}_{1 \leq i < j \leq n}$ be $\binom{n}{2}$ commuting unitaries on some Hilbert space $\mathcal{H}$, and suppose $U_{ji} := U_{ij}^*$, $1 \leq i < j \leq n$. An $n$-tuple of isometries $V = (V_1, \ldots ,V_n)$ on $\mathcal{H}$ is called $\mathcal{U}_n$-twisted isometry with respect to $\{U_{ij}\}_{i<j}$ (or simply $\mathcal{U}_n$-twisted isometry if $\{U_{ij}\}_{i<j}$ is clear from the context) if $V_i$'s are in the commutator $\{U_{st}: s \neq t\}'$, and $V_i^*V_j=U_{ij}^*V_jV_i^*$, $i \neq j$ We prove that each $\mathcal{U}_n$-twisted isometry admits a von Neumann-W…
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…
We prove that, if the closed unit ball of a normed space $X$ has sufficiently many extreme points, then every mapping $Φ$ from $X$ into itself with the following property is affine: For any pair of points in $X$, there exists a (not necessarily linear) surjective isometry on $X$ that coincides with $Φ$ at the two points. We also consider surjectivity of such a mapping in some special cases including C$^*$-algebras.
