Searcharxiv⌕ Search

arXiv subjects

Antonio M. Peralta

Publications and source records attributed to Antonio M. Peralta.

At least 73 records · Page 4Linked to original sources
arXiv2018

Contractive linear preservers of absolutely compatible pairs between C*-algebras

Let $a$ and $b$ be elements in the closed ball of a unital C$^*$-algebra $A$ (if $A$ is not unital we consider its natural unitization). We shall say that $a$ and $b$ are domain (respectively, range) absolutely compatible ($a\triangle_d b$, respectively, $a\triangle_r b$, in short) if $\Big| |a| -|b| \Big| + \Big| 1-|a|-|b| \Big| =1$ (resp., $\Big| |a^*| -|b^*| \Big| + \Big| 1-|a^*|-|b^*| \Big| =1$), where $|a|^2= a^* a$. We shall say that $a$ and $b$ are absolutely compatible ($a\triangle b$ in short) if they are both range and domain absolutely compatible. In general, $a\triangle_d b$ (respectively, $a\triangle_r b$ and $a\triangle b$) is strictly weaker than $ab^*=0 $ (respectively, $a^* b =0$ and $a\perp b$). Let $T: A\to B$ be a contractive linear mapping between C$^*$-algebras. We prove that if $T$ preserves domain absolutely compatible elements (i.e., $a\triangle_d b\Rightarrow T(a)\triangle_d T(b)$) then $T$ is a triple homomorphism. A similar statement is proved when $T$ preserves range absolutely compatible elements. It is finally shown that $T$ is a triple homomorphism if, and only if, $T$ preserves absolutely compatible elements.

math.OA↗
arXiv2018

Tingley's problem for $p$-Schatten von Neumann classes

Let $H$ and $H'$ be a complex Hilbert spaces. For $p\in(1, \infty)\backslash\{2\}$ we consider the Banach space $C_p(H)$ of all $p$-Schatten von Neumann operators, whose unit sphere is denoted by $S(C_p(H))$. We prove that every surjective isometry $Δ: S(C_p(H))\to S(C_p(H'))$ can be extended to a complex linear or to a conjugate linear surjective isometry $T:C_p(H)\to C_p(H')$.

math.FA↗
arXiv2018

On the unit sphere of positive operators

Given a C$^*$-algebra $A$, let $S(A^+)$ denote the set of those 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 note we prove a variant of Tingley's problem by showing that every surjective isometry $Δ: S(B(H_1)^+)\to S(B(H_2)^+)$ or (respectively, $Δ: S(K(H_3)^+)\to S(K(H_4)^+)$) admits a unique extension to a surjective complex linear isometry from $B(H_1)$ onto $B(H_2))$ (respectively, from $K(H_3)$ onto $B(H_4)$). This provides a positive answer to a conjecture posed by G. Nagy [\emph{Publ. Math. Debrecen}, 2018].

math.FA↗
arXiv2018

Characterizing projections among positive operators in the unit sphere

Let $E$ and $P$ be subsets of a Banach space $X$, and let us define the unit sphere around $E$ in $P$ as the set $$Sph(E;P) :=\left\{ x\in P : \|x-b\|=1 \hbox{ for all } b\in E \right\}.$$ Given a C$^*$-algebra $A$, and a subset $E\subset A,$ we shall write $Sph^+ (E)$ or $Sph_A^+ (E)$ for the set $Sph(E;S(A^+)),$ where $S(A^+)$ stands for the set of all positive operators in the unit sphere of $A$. We prove that, for an arbitrary complex Hilbert space $H$, then a positive element $a$ in the unit sphere of $B(H)$ is a projection if and only if $Sph^+_{B(H)} \left( Sph^+_{B(H)}(\{a\}) \right) =\{a\}$. We also prove that the equivalence remains true when $B(H)$ is replaced with an atomic von Neumann algebra or with $K(H_2)$, where $H_2$ is an infinite-dimensional and separable complex Hilbert space. In the setting of compact operators we prove a stronger conclusion by showing that the identity $$Sph^+_{K(H_2)} \left( Sph^+_{K(H_2)}(a) \right) =\left\{ b\in S(K(H_2)^+) : \!\! \begin{array}{c} s_{_{K(H_2)}} (a) \leq s_{_{K(H_2)}} (b), \hbox{ and } \textbf{1}-r_{_{B(H_2)}}(a)\leq \textbf{1}-r_{_{B(H_2)}}(b) \end{array}\!\! \right\},$$ holds for every $a$ in the unit sphere of $K(H_2)^+$, where $r_{_{B(H_2)}}(a)$ and $s_{_{K(H_2)}} (a)$ stand for the range and support projections of $a$ in $B(H_2)$ and $K(H_2)$, respectively.

math.OA↗
arXiv2018

The $λ$-function in the space of trace class operators

Let $C_1(H)$ denote the space of all trace class operators on an arbitrary complex Hilbert space $H$. We prove that $C_1(H)$ satisfies the $λ$-property, and we determine the form of the $λ$-function of Aron and Lohman on the closed unit ball of $C_1(H)$ by showing that $$λ(a) = \frac{1 - \|a\|_1 + 2 \|a\|_{\infty}}{2},$$ for every $a$ in ${C_1(H)}$ with $\|a\|_1 \leq 1$. This is a non-commutative extension of the formula established by Aron and Lohman for $\ell_1$.

math.OA↗
arXiv2018

The Mazur-Ulam property for commutative von Neumann algebras

Let $(Ω,μ)$ be a $σ$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (Ω,μ)$ 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 $Δ: S(L^{\infty} (Ω,μ))\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\infty} (Ω,μ)\to X$. This conclusion is derived from a more general statement which assures that every surjective isometry $Δ: S(C(K))\to S(X),$ where $K$ is a Stonean space, admits an extension to a surjective real linear isometry from $C(K)$ onto $X$.

math.FA↗
arXiv2018

A survey on Tingley's problem for operator algebras

We survey the most recent results on extension of isometries between special subsets of the unit spheres of C$^*$-algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras, and $p$-Schatten-von Neumann spaces, with special interest on Tingley's problem.

math.OA↗
arXiv2017

Preservers of $λ$-Aluthge transforms

Let $M$ and $N$ be arbitrary von Neumann algebras. For any $a$ in $M$ or in $N$, let $Δ_λ(a)$ denote the $λ$-Aluthge transform of $a$. Suppose that $M$ has no abelian direct summand. We prove that every bijective map $Φ:M\to N$ satisfying $$Φ(Δ_λ(a\circ b^*))=Δ_λ(Φ(a) \circ Φ(b)^*), \hbox{ for all } a,\;b\in M,$$ (for a fixed $λ\in [0,1]$), maps the hermitian part of $M$ onto the hermitian part of $N$ (i.e. $Φ(M_{sa}) = N_{sa}$) and its restriction $Φ|_{M_{sa}} : M_{sa}\to N_{sa}$ is a Jordan isomorphism. If we also assume that $Φ(x +i y ) = Φ(x) +Φ(i y)$ for all $x,y\in M_{sa}$, then there exists a central projection $p_c$ in $M$ such that $Φ|_{p_c M}$ is a complex linear Jordan $^*$-isomorphism and $Φ|_{(\textbf{1}-p_c) M}$ is a conjugate linear Jordan $^*$-isomorphism. Given two complex Hilbert spaces $H$ and $K$ with dim$(H)\geq 2$, we also establish that every bijection $Φ: \mathcal{B}(H)\to \mathcal{B}(K)$ satisfying $$Φ(Δ_λ(a b^*))=Δ_λ(Φ(a) Φ(b)^*), \hbox{ for all } a,\;b\in \mathcal{B}(H),$$ must be a complex linear or a conjugate linear $^*$-isomorphism.

math.OA↗
arXiv2017

Extending surjective isometries defined on the unit sphere of $\ell_\infty(Γ)$

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 $X$.

math.FA↗
arXiv2017

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

Given an infinite set $Γ$, we prove that the space of complex null sequences $c_0(Γ)$ satisfies the Mazur-Ulam property, that is, for each Banach space $X$, every surjective isometry from the unit sphere of $c_0(Γ)$ onto the unit sphere of $X$ admits a (unique) extension to a surjective real linear isometry from $c_0(Γ)$ to $X$. We also prove that the same conclusion holds for the finite dimensional space $\ell_{\infty}^m$.

math.FA↗
arXiv2017

2-local standard isometries on vector-valued Lipschitz function spaces

Under the right conditions on a compact metric space $X$ and on a Banach space $E$, we give a description of the $2$-local (standard) isometries on the Banach space $\hbox{Lip}(X,E)$ of vector-valued Lipschitz functions from $X$ to $E$ in terms of a generalized composition operator, and we study when every $2$-local (standard) isometry on $\hbox{Lip}(X,E)$ is both linear and surjective.

math.FA↗
arXiv2017

Linear maps on C*-algebras which are derivations or triple derivations at a point

We prove new results on generalized derivations on C$^*$-algebras. By considering the triple product $\{a,b,c\} =2^{-1} (a b^* c + c b^* a)$, we introduce the study of linear maps which are triple derivations or triple homomorphisms at a point. We prove that a continuous linear $T$ map on a unital C$^*$-algebra is a generalized derivation whenever it is a triple derivation at the unit element. If we additionally assume $T(1)=0,$ then $T$ is a $^*$-derivation and a triple derivation. Furthermore, a continuous linear map on a unital C$^*$-algebra which is a triple derivation at the unit element is a triple derivation. Similar conclusions are obtained for continuous linear maps which are derivations or triple derivations at zero. We also give an automatic continuity result, that is, we show that generalized derivations on a von Neumann algebra and linear maps on a von Neumann algebra which are derivations or triple derivations at zero are all continuous even if not assumed a priori to be so.

math.OA↗
arXiv2017

Weak-2-local isometries on uniform algebras and Lipschitz algebras

We establish spherical variants of the Gleason-Kahane-Zelazko and Kowalski-Słodkowski theorems, and we apply them to prove that every weak-2-local isometry between two uniform algebras is a linear map. Among the consequences, we solve a couple of problems posed by O. Hatori, T. Miura, H. Oka and H. Takagi in 2007. Another application is given in the setting of weak-2-local isometries between Lipschitz algebras by showing that given two metric spaces $E$ and $F$ such that the set Iso$((\hbox{Lip}(E),\|.\|),(\hbox{Lip}(F),\|.\|))$ is canonical, then every\hyphenation{every} weak-2-local Iso$((\hbox{Lip}(E),\|.\|),(\hbox{Lip}(F),\|.\|))$-map $Δ$ from $\hbox{Lip}(E)$ to $\hbox{Lip}(F)$ is a linear map, where $\|.\|$ can indistinctly stand for $\|f\|_{_L} := \max\{L(f), \|f\|_{\infty} \}$ or $ \|f\|_{_s} := L(f) + \|f\|_{\infty}.$

math.FA↗
arXiv2017

Tingley's problem for spaces of trace class operators

We prove that every surjective isometry between the unit spheres of two trace class spaces admits a unique extension to a surjective complex linear or conjugate linear isometry between the spaces. This provides a positive solution to Tingley's problem in a new class of operator algebras.

math.FA↗
arXiv2017

Pointwise-generalized-inverses of linear maps between C$^*$-algebras and JB$^*$-triples

We study pointwise-generalized-inverses of linear maps between C$^*$-algebras. Let $Φ$ and $Ψ$ be linear maps between complex Banach algebras $A$ and $B$. We say that $Ψ$ is a pointwise-generalized-inverse of $Φ$ if $Φ(aba)=Φ(a)Ψ(b)Φ(a),$ for every $a,b\in A$. The pair $(Φ,Ψ)$ is Jordan-triple multiplicative if $Φ$ is a pointwise-generalized-inverse of $Ψ$ and the latter is a pointwise-generalized-inverse of $Φ$. We study the basic properties of this maps in connection with Jordan homomorphism, triple homomorphisms and strongly preservers. We also determine conditions to guarantee the automatic continuity of the pointwise-generalized-inverse of continuous operator between C$^*$-algebras. An appropriate generalization is introduced in the setting of JB$^*$-triples.

math.OA↗
arXiv2017

On the extension of isometries between the unit spheres of a C*-algebra and B(H)

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.

math.FA↗