Geometric Invariants of Surjective Isometries between Unit Spheres

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

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

Li 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.

ON THE EXTENSION OF ISOMETRIES BETWEEN THE UNIT SPHERES OF A -TRIPLE AND A BANACH SPACE

We prove that if $M$ is a $\text{JBW}^{\ast }$ -triple and not a Cartan factor of rank two, then $M$ satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another real Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$ .

Единственность симметричной структуры в идеалах компактных операторов

Let Hbe a separable infinite-dimensional complex Hilbert space, let L(H) be the C∗-algebra of bounded linear operators acting in H, and let K(H) be the two-sided ideal of compact linear operators in L(H). Let (E,∥⋅∥E) be a symmetric sequence space, and let CE:={x∈K(H):{sn(x)}∞n=1∈E} be the proper two-sided ideal in L(H), where {sn(x)}∞n=1 are the singular values of a compact operator x. It is known that CE is a Banach symmetric ideal with respect to the norm ∥x∥CE=∥{sn(x)}∞n=1∥E. A symmetric ideal CE is said to have a unique symmetric structure if CE=CF, that is E=F, modulo norm equivalence, whenever (CE,∥⋅∥CE) is isomorphic to another symmetric ideal (CF,∥⋅∥CF). At the Kent international conference on Banach space theory and its applications (Kent, Ohio, August 1979), A. Pelczynsky posted the following problem: (P) Does every symmetric ideal have a unique symmetric structure? This problem has positive solution for Schatten ideals Cp, 1≤p<∞ (J. Arazy and J. Lindenstrauss, 1975). For arbitrary symmetric ideals problem (P) has not yet been solved. We consider a version of problem (P) replacing an isomorphism U:(CE,∥⋅∥CE)→(CF,∥⋅∥CF) by a positive linear surjective isometry. We show that if F is a strongly symmetric sequence space, then every positive linear surjective isometry U:(CE,∥⋅∥CE)→(CF,∥⋅∥CF) is of the form U(x)=u∗xu, x∈CE, where u∈L(H) is a unitary or antiunitary operator. Using this description of positive linear surjective isometries, it is established that existence of such an isometry U:CE→CF implies that (E,∥⋅∥E)=(F,∥⋅∥F).

Nonlinear Isometries on Schatten-pClass in Atomic Nest Algebras

LetHbe a complex Hilbert space; denote by Alg 𝒩and𝒞p(H)the atomic nest algebra associated with the atomic nest𝒩onHand the space of Schatten-pclass operators on,Hrespectively. Let𝒞p(H)∩Alg 𝒩be the space of Schatten-pclass operators in Alg 𝒩. When1≤p<+∞andp≠2, we give a complete characterization of nonlinear surjective isometries on𝒞p(H)∩Alg 𝒩. Ifp=2, we also prove that a nonlinear surjective isometry on𝒞2(H)∩Alg 𝒩is the translation of an orthogonality preserving map.

ISOMETRIES BETWEEN UNIT SPHERES OF THE -SUM OF STRICTLY CONVEX NORMED SPACES

We prove that any surjective isometry between unit spheres of the ${\ell }^{\infty } $-sum of strictly convex normed spaces can be extended to a linear isometry on the whole space, and we solve the isometric extension problem affirmatively in this case.

New criteria for equivalence of locally compact abelian groups

Suppose that there exists a surjective isometry between open subgroups of the groups of invertible elements in the measure algebras on LCA groups. We show that the LCA groups are topologically isomorphic to each other.

2-Local Isometries on Spaces of Lipschitz Functions

Let (X, d) be a metric space, and let Lip(X) denote the Banach space of all scalar-valued bounded Lipschitz functions ƒ on X endowed with one of the natural normswhere L(ƒ) is the Lipschitz constant of ƒ. It is said that the isometry group of Lip(X) is canonical if every surjective linear isometry of Lip(X) is induced by a surjective isometry of X. In this paper we prove that if X is bounded separable and the isometry group of Lip(X) is canonical, then every 2-local isometry of Lip(X) is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of Lip(X) when X is bounded.

Isometries of spaces of unbounded continuous functions

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

Isometries of Hilbert space valued function spaces

Let X be a (real or complex) rearrangement-invariant function space on Ω (where Ω = [0, 1] or Ω ⊆ N) whose norm is not proportional to the L2-norm. Let H be a separable Hilbert space. We characterize surjective isometries of X (H). We prove that if T is such an isometry then there exist Borel maps a: Ω → + K and σ: Ω → Ω and a strongly measurable operator map S of Ω into B (H) so that for almost all ω, S(ω) is a surjective isometry of H, and for any f ∈ X(H), T f(ω) = a(ω)S(ω)(f(σ(ω))) a.e. As a consequence we obtain a new proof of the characterization of surjective isometries in complex rearrangement-invariant function spaces.