Linear Isometry Research Articles

Fundamental aspects of vector-valued Banach limits

This paper is divided into four parts. In the first we study the existence of vector- valued Banach limits and show that a real Banach space with a monotone Schauder basis admits vector- valued Banach limits if and only if it is -complemented in its bidual. In the second we prove two vector- valued versions of Lorentz' intrinsic characterization of almost convergence. In the third we show that the unit sphere in the space of all continuous linear operators from to which are invariant under the shift operator on cannot be obtained via compositions of surjective linear isometries with vector- valued Banach limits. In the final part we show that if enjoys the Krein–Milman property, then the set of vector- valued Banach limits is a face of the unit ball in the space of all continuous linear operators from to which are invariant under the shift operator on .

Isometries of real Hilbert C⁎-modules

Let T:V→W be a surjective real linear isometry between full real Hilbert C⁎-modules over real C⁎-algebras A and B, respectively. We show that the following conditions are equivalent: (a) T is a 2-isometry; (b) T is a complete isometry; (c) T preserves ternary products; (d) T preserves inner products; (e) T is a module map. When A and B are commutative, we give a full description of the structure of T.

LINEAR SURJECTIVE ISOMETRIES BETWEEN VECTOR-VALUED FUNCTION SPACES

We prove some Banach–Stone type theorems for linear isometries of vector-valued continuous function spaces, by making use of the extreme point method.

Dynamics of Biholomorphic Self-Maps on Bounded Symmetric Domains

Let $g$ be a fixed-point free biholomorphic self-map of a bounded symmetric domain $B$. It is known that the sequence of iterates $(g^n)$ may not always converge locally uniformly on $B$ even, for example, if $B$ is an infinite dimensional Hilbert ball. However, $g=g_a\circ T$, for a linear isometry $T$, $a=g(0)$ and a transvection $g_a$, and we show that it is possible to determine the dynamics of $g_a$. We prove that the sequence of iterates $(g_a^n)$ converges locally uniformly on $B$ if, and only if, $a$ is regular, in which case, the limit is a holomorphic map of $B$ onto a boundary component (surprisingly though, generally not the boundary component of $\frac{a}{\|a\|}$). We prove $(g_a^n)$ converges to a constant for all non-zero $a$ if, and only if, $B$ is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.

ON ISOMETRIC REPRESENTATION SUBSETS OF BANACH SPACES

Let $X,Y$ be two Banach spaces and $B_{X}$ the closed unit ball of $X$. We prove that if there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists an isometry $F:X\rightarrow Y^{\ast \ast }$. If, in addition, $Y$ is weakly nearly strictly convex, then there is an isometry $F:X\rightarrow Y$. Making use of these results, we show that if $Y$ is weakly nearly strictly convex and there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists a linear isometry $S:X\rightarrow Y$.

On linear isometries and ε-isometries between Banach spaces

Let X,Y be two Banach spaces, and f:X→Y be a standard ε-isometry for some ε≥0. Recently, Cheng et al. showed that if co‾[f(X)∪−f(X)]=Y, then there exists a surjective linear operator T:Y→X with ‖T‖=1 such that the following sharp inequality holds:‖Tf(x)−x‖≤2ε for all x∈X. Making use of the above result, we prove the following results: Suppose that co‾[f(X)∪−f(X)]=Y. Then(1)if there is a linear isometry S:X→Y such that TS=IdX, then T⁎S⁎:Y⁎→T⁎(X⁎) is a w⁎-to-w⁎ continuous linear projection with ‖T⁎S⁎‖=1,(2)if there exists a w⁎-to-w⁎ continuous linear projection P:Y⁎→T⁎(X⁎) with ‖P‖=1, then there is an unique linear isometry S(P):X→Y such that TS(P)=IdX and P=T⁎S(P)⁎. Furthermore, if P1≠P2 are two w⁎-to-w⁎ continuous linear projection from Y⁎ onto T⁎(X⁎) with ‖P1‖=‖P2‖=1, then S(P1)≠S(P2). We apply these results to provide an alternative proof of a recent theorem, which gives an affirmative answer of a question proposed by Vestfrid. We also unify several known theorems concerning the stability of ε-isometries.

Isometries of the Toeplitz matrix algebra

We study the structure of isometries defined on the algebra A of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative norm preserving map A→Mn(C) must be of the form either A↦UAU⁎ or A↦UA‾U⁎, where 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 A→Mn(C) is of the form A↦UAV where U and V are two unitary matrices. This implies, in particular, that every such an isometry is a complete isometry and that a unital linear isometry A→Mn(C) is necessarily an algebra homomorphism.

Operadic multiplications in equivariant spectra, norms, and transfers

We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N∞ operads, equivariant generalizations of E∞ operads. Algebras in equivariant spectra over an N∞ operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill–Hopkins–Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N∞ operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant infinite loop spaces.To explain the relationship between norms, transfers, and N∞ operads, we discuss the general features of these operads, linking their properties to families of finite sets with group actions and analyzing their behavior under norms and geometric fixed points. A surprising consequence of our study is that in stark contract to the classical setting, equivariantly the little disks and linear isometries operads for a general incomplete universe U need not determine the same algebras.Our work is motivated by the need to provide a framework to describe the flavors of commutativity seen in recent work of the second author and Hopkins on localization of equivariant commutative ring spectra.

Real-Linear Isometries and Jointly Norm-Additive Maps on Function Algebras

In this paper, we describe into real-linear isometries defined between (not necessarily unital) function algebras and show, based on an example, that this type of isometries behaves differently from surjective real-linear isometries and from classical linear isometries. Next we introduce jointly norm-additive mappings and apply our results on real-linear isometries to provide a complete description of these mappings when defined between function algebras which are not necessarily unital or uniformly closed.

Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

Non supercyclic subsets of linear isometries on Banach spaces of analytic functions

Let X be a Banach space of analytic functions on the open unit disk and Γ a subset of linear isometries on X. Sufficient conditions are given for non-supercyclicity of Γ. In particular, we show that the semigroup of linear isometries on the spaces S p (p > 1), the little Bloch space, and the group of surjective linear isometries on the big Bloch space are not supercyclic. Also, we observe that the groups of all surjective linear isometries on the Hardy space H p or the Bergman space L (1 < p < ∞, p ≠ 2) are not supercyclic.

Operator space and operator system analogs of Kirchberg's nuclear embedding theorem

The Gurarij operator space NG introduced by Oikhberg is the unique separable 1-exact operator space that is approximately injective in the category of 1-exact operator spaces and completely isometric linear maps. We prove that a separable operator space X is nuclear if and only if there exist a linear complete isometry φ:X→NG and a completely contractive projection from NG onto the range of φ. This can be seen as the operator space analog of Kirchberg's nuclear embedding theorem. With similar methods we also establish the natural operator system analog of Kirchberg's nuclear embedding theorem involving the Gurarij operator system GS.

A universal theorem for stability of ε-isometries of Banach spaces

Let X, Y be two Banach spaces, and f:X→Y be a standard ε-isometry for some ε≥0. In this paper, we show the following sharp weak stability inequality of f: for every x⁎∈X⁎ there exists ϕ∈Y⁎ with ‖ϕ‖=‖x⁎‖≡r such that|〈x⁎,x〉−〈ϕ,f(x)〉|≤2εrfor allx∈X. It is not only a sharp quantitative extension of Figiel's theorem, but it also unifies, generalizes and improves a series of known results about stability of ε-isometries. For example, if the mapping f satisfies C(f)≡co¯[f(X)∪−f(X)]=Y, then it is equivalent to the following sharp stability theorem: There is a linear surjective operator T:Y→X of norm one such that ‖Tf(x)−x‖≤2ε, for all x∈X; When the ε-isometry f is surjective, it is equivalent to Omladič–Šemrl's theorem: There is a surjective linear isometry U:X→Y so that‖f(x)−Ux‖≤2ε,for allx∈X.

On admissible topologies in JB*-triples

Given a complex JB*-triple X, we define and study admissible topologies on X, i.e., locally convex topologies τ on X coarser than the norm topology, invariant under the group of surjective linear isometries of X, and such that the triple product is jointly -continuous on bounded subsets of X. As a consequence of the joint -continuity of the triple product, all holomorphic automorphisms of the open unit ball are homeomorphisms of and the natural action is jointly -continuous on .

On isometrical extension properties of function spaces

In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C(Q)$ and $C(\Delta)$, where $Q$ and $\Delta$ denote the Hilbert cube $[0,1]^{\infty}$ and a Cantor set, respectively.

Surjective isometries on the vector-valued differentiable function spaces

This paper investigates the surjective linear isometries between the differentiable function spaces C0n(Ω,E) and C0m(Σ,F) (where Ω,Σ are open subsets of Euclidean spaces and E,F are reflexive, strictly convex Banach spaces), and show that such isometries can be written as weighted composition operators.

Multiplicative Isometries on ClassesMp(X)of Holomorphic Functions

In (Iida and Kasuga 2013), the authors described multiplicative (but not necessarily linear) isometries ofMp(X)ontoMp(X)in the case of positive integerp∈N, whereMp(X) (p≥1)is included in the Smirnov classN∗(X). In this paper, we will generalize the result to arbitrary (not necessarily positive integer) value of the exponents0&lt;p&lt;∞.

Algebraic and topological properties of the group of isometries on classes of vector valued function spaces

We study algebraic and topological properties of the group of all surjective isometries on several spaces of vector valued analytic functions and vector valued $L^p$ spaces ($1 \leq p \leq \infty$). We also derive the form for the surjective linear isometries on the vector valued little Zygmund space.

Linear isometries on spaces consisting of absolutely continuous functions

Linear isometries on spaces consisting of absolutely continuous functions

On non- L 0 -linear perturbations of random isometries in random normed modules

The purpose of this paper is to study non- L 0 -linear perturbations of random isometries in random normed modules. Let (Ω,F,P) be a probability space, K the scalar field R of real numbers or C of complex numbers, L 0 (F,K) the equivalence classes of K-valued ℱ-measurable random variables on Ω, ( E 1 , ∥ ⋅ ∥ 1 ) and ( E 2 , ∥ ⋅ ∥ 2 ) random normed modules over K with base (Ω,F,P). In this paper, we first establish the Mazur-Ulam theorem in random normed modules. Making use of this theorem and the relations between random normed modules and classical normed spaces, we show that if f: E 1 → E 2 is a surjective random ε-isometry with f(0)=0 and has the local property, where ε∈ L 0 (F,R) and ε≥0, then there is a surjective L 0 -linear random isometry U: E 1 → E 2 such that ∥ f ( x ) − U ( x ) ∥ 2 ≤4ε, for all x∈ E 1 . We do not obtain a sharp estimate as the classical result, since random normed modules have a complicated stratification structure, which is the essential difference between random normed modules and classical normed spaces.MSC:46A22, 46A25, 46H25.