Surjective Linear Isometries Research Articles

2-Local isometries between Banach algebras of continuous functions with involution

We give a description of 2-local real isometries between $$C(X,\tau )$$ and $$C(Y,\eta )$$ where X and Y are compact Hausdorff spaces, X is also first countable and $$\tau $$ and $$\eta $$ are topological involutions on X and Y, respectively. In particular, we show that every 2-local real isometry T from $$C(X,\tau )$$ to $$ C(Y,\eta )$$ is a surjective real linear isometry whenever X is also separable.

2-Local isometries between spaces of functions of bounded variation

Given two subsets X and Y of the real line with at least two points, we apply results on surjective linear isometries between Banach spaces of all functions of bounded variation BV(X) and BV(Y) to show that every 2-local isometry $$T:BV(X)\longrightarrow BV(Y)$$ is a constant multiple of an isometric linear algebra isomorphism. Moreover, similar results are given for the closed subspaces of BV(X) and BV(Y) consisting of all continuous (resp. absolutely continuous) functions when X and Y are compact.

Hilbert isometries and maximal deviation preserving maps on JB-algebras

In this paper we characterize the surjective linear variation norm isometries on JB-algebras. Variation norm isometries are precisely the maps that preserve the maximal deviation, the quantum analogue of the standard deviation, which plays an important role in quantum statistics. Consequently, we characterize the Hilbert's metric isometries on cones in JB-algebras.

On the Mazur–Ulam property for the space of Hilbert-space-valued continuous functions

Let K be a compact Hausdorff space and let H be a real or complex Hilbert space with dim(HR)≥2. We prove that the space C(K,H), of all H-valued continuous functions on K, equipped with the supremum norm, satisfies the Mazur–Ulam property, that is, if Y is any real Banach space, every surjective isometry Δ from the unit sphere of C(K,H) onto the unit sphere of Y admits a unique extension to a surjective real linear isometry from C(K,H) onto Y. Our strategy relies on the structure of C(K)-module of C(K,H) and several results in JB⁎-triple theory. For this purpose we determine the facial structure of the closed unit ball of a real JB⁎-triple and its dual space.

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

AbstractWe 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 $Y$.

Banach spaces with a large semigroup of contractive automorphisms

Transitivity, almost transitivity, and convex-transitivity are classical notions in the isometric theory of Banach spaces, all of them related to Mazur's problem asking whether transitive separable Banach spaces are Hilbert spaces [24,33]. In this paper we study ‘near’ versions of the above notions. Roughly speaking, ‘nearness’ means that contractive automorphisms play the role played by surjective linear isometries in the corresponding classical notion. We also consider asymptotic transitivity as defined in [35], which is implied by almost transitivity and implies near almost transitivity, and introduce the corresponding notion of asymptotic convex-transitivity. We show that nearly convex-transitive (respectively, asymptotically convex-transitive) Banach spaces are nearly almost-transitive (respectively, asymptotically transitive), uniformly smooth, and uniformly convex, as soon as they be Asplund spaces or have the Radon-Nikodým property. We also deal with a property stronger than the near transitivity, and weaker than the uniform micro-semitransitivity [18]. We prove that Banach spaces with this intermediate property are uniformly smooth and uniformly convex. Finally we show that nearly almost-transitive norm-unital complete normed (possibly non-associative) algebras are Hilbert spaces.

Algebraic reflexivity of isometry groups of algebras of Lipschitz maps

We study groups of surjective linear isometries on Banach algebras of Lipschitz maps with values in some unital C⁎-algebras. In this paper, these spaces are endowed with the sum norm. For the case where the C⁎-algebras are commutative whose groups of all surjective isometries are algebraically reflexive, we prove that the group of all surjective isometries on the corresponding Banach algebra of Lipschitz maps are algebraically reflexive. We also prove that the group of unital surjective isometries between matrix-valued Lipschitz algebras are reflexive.

Extending biholomorphic automorphisms of the open unit ball of a complex Banach space to the bidual

Let X be a complex Banach space. Dineen (1986) proved that each biholomorphic automorphism of the open unit ball of X extends to a biholomorphic automorphism of the open unit ball of the bidual X″ of X. As far as we know, the question of the uniqueness of the extension remained unanswered, however. In this paper we show that the uniqueness of the extension occurs if and only if each surjective linear isometry on X can be uniquely extended to a surjective linear isometry on X″. Since this last condition is fulfilled whenever X is a JB*-triple (thanks to the results by Barton-Timoney (1986)), we rediscover Isidro's result (2018) on the uniqueness of the extension in the JB*-triple case.

On the unit sphere of positive operators

Given a C ∗ -algebra A , let S ( A + ) denote the set of positive elements in the unit sphere of A . Let H 1 , H 2 , H 3 , and H 4 be complex Hilbert spaces, where H 3 and H 4 are infinite-dimensional and separable. In this article, we prove a variant of Tingley’s problem by showing that every surjective isometry Δ : S ( B ( H 1 ) + ) → S ( B ( H 2 ) + ) (resp., Δ : S ( K ( H 3 ) + ) → S ( K ( H 4 ) + ) ) admits a unique extension to a surjective complex linear isometry from B ( H 1 ) onto B ( H 2 ) (resp., from K ( H 3 ) onto K ( H 4 ) ). This provides a positive answer to a conjecture recently posed by Nagy.

Hermitian operators and isometries on algebras of matrix-valued Lipschitz maps

ABSTRACT This paper deals with hermitian operators and surjective linear isometries, between spaces of Lipschitz maps, defined on a compact metric space, with values in a finite dimensional vector space. These spaces are endowed with the sum norm. The first main result formulates that hermitian operators are composition operators (Theorem 2.2) and the second one (Theorem 3.3) gives a characterization for the surjective unital linear isometries between Banach algebras of Lipschitz maps with values in .

Stability Characterizations of ∈-isometries on Certain Banach Spaces

Suppose that X, Y are two real Banach Spaces. We know that for a standard ∈-isometry f: X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of Y*. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ∈-isometry to be stable in assuming that N is w*-closed in Y*. Making use of this result, we improve several known results including Figiel’s theorem in reflexive spaces. We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then $$L(f) \equiv \overline {span} [f(x)]$$ contains a complemented linear isometric copy of X; Moreover, if X = Y, then for every ∈-isometry f : X → X, there exists a surjective linear isometry S : X → X such that f − S is uniformly bounded by 2∈ on X.

Low rank compact operators and Tingley's problem

Let E and B be arbitrary weakly compact JB⁎-triples whose unit spheres are denoted by S(E) and S(B), respectively. We prove that every surjective isometry f:S(E)→S(B) admits an extension to a surjective real linear isometry T:E→B. This is a complete solution to Tingley's problem in the setting of weakly compact JB⁎-triples. Among the consequences, we show that if K(H,K) denotes the space of compact operators between arbitrary complex Hilbert spaces H and K, then every surjective isometry f:S(K(H,K))→S(K(H,K)) admits an extension to a surjective real linear isometry T:K(H,K)→K(H,K).

On linear maps preserving complex symmetry

An operator T on a complex Hilbert space H is said to be complex symmetric if there exists a conjugate-linear, isometric involution C:H→H so that CTC=T⁎. This paper is devoted to describing which linear maps leave the class of complex symmetric operators invariant. Complete characterizations are obtained for several classes of linear maps, including similarity transformations, surjective linear isometries, multiplication operators and certain completely positive maps.

The Mazur–Ulam property for commutative von Neumann algebras

ABSTRACTLet be a σ-finite measure space. Given a Banach space X, let the symbol stand for the unit sphere of X. We prove that the space of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm satisfies the Mazur–Ulam property, that is, if X is any complex Banach space, every surjective isometry admits an extension to a surjective real linear isometry . This conclusion is derived from a more general statement which assures that every surjective isometry where K is a Stonean space, admits an extension to a surjective real linear isometry from onto X.

A solution to Tingley’s problem for isometries between the unit spheres of compact C*-algebras and JB*-triples

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 $A$ into $B$. These results provide new examples of infinite dimensional Banach spaces where Tingley's problem admits a positive answer.

Extending surjective isometries defined on the unit sphere of $$\ell _\infty (\Gamma )$$ ℓ ∞ ( Γ )

Let $$\Gamma $$ be an infinite set equipped with the discrete topology. We prove that the space $$\ell _{\infty }(\Gamma ,{\mathbb {C}}),$$ of all complex-valued bounded functions on $$\Gamma $$ , satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $$\ell _{\infty }(\Gamma , {\mathbb {C}})$$ onto the unit sphere of an arbitrary complex Banach space X admits a unique extension to a surjective real linear isometry from $$\ell _{\infty }(\Gamma ,{\mathbb {C}})$$ to X.

On the extension of isometries between the unit spheres of von Neumann algebras

We prove that every surjective isometry between the unit spheres of two von Neumann algebras admits a unique extension to a surjective real linear isometry between these two algebras.

The Mazur–Ulam property for the space of complex null sequences

ABSTRACTGiven an infinite set , we prove that the space of complex null sequences, , satisfies the Mazur–Ulam property, that is, for each Banach space X, every surjective isometry from the unit sphere of onto the unit sphere of X admits a (unique) extension to a surjective real linear isometry from to X. We also prove that the same conclusion holds for the finite-dimensional space .

Isometries of perfect norm ideals of compact operators

It is proved that for every surjective linear isometry $V$ on a perfect Banach symmetric ideal $\mathcal C_E\neq \mathcal C_2$ of compact operators, acting in a complex separable infnite-dimensional Hilbert space $\mathcal H$ there exist unitary operators $u$ and $v$ on $\mathcal H$ such that $V(x)=uxv$ or $V(x) = ux^tv$ for all $x\in \mathcal C_E$, where $x^t$ is the transpose of an operator $x$ with respect to a fixed orthonormal basis in $\mathcal H$. In addition, it is shown that any surjective 2-local isometry on a perfect Banach symmetric ideal $\mathcal C_E \neq \mathcal C_2$ is a linear isometry on $\mathcal C_E$.

The Ramsey property for Banach spaces and Choquet simplices, and applications

We show that the class of finite-dimensional Banach spaces and the class of finite-dimensional Choquet simplices have the Ramsey property. As an application, we show that the group Aut(G) of surjective linear isometries of the Gurarij space G is extremely amenable, and that the canonical action Aut(P)↷P is the universal minimal flow of the group Aut(P) of affine homeomorphisms of the Poulsen simplex P. This answers questions of Melleray–Tsankov and Conley–Törnquist.