We establish a more precise description of those surjective or bijective continuous linear operators preserving orthogonality between C$^*$-algebras. The new description is applied to determine all uniformly continuous one-parameter semigroups of orthogonality preserving operators on an arbitrary C$^*$-algebra. We prove that given a family $\{T_t: t\in \mathbb{R}_0^{+}\}$ of orthogonality preserving bounded linear bijections on a general C$^*$-algebra $A$ with $T_0=Id$, if for each $t\geq 0,$ we set $h_t = T_t^{**} (1)$ and we write $r_t$ for the range partial isometry of $h_t$ in $A^{**},$ and $S_t$ stands for the triple isomorphism on $A$ associated with $T_t$ satisfying $h_t^* S_t(x)$ $= S_t(x^*)^* h_t$, $h_t S_t(x^*)^* =$ $ S_t(x) h_t^*$, $h_t r_t^* S_t(x) =$ $S_t(x) r_t^* h_t$, and $T_t(x) = h_t r_t^* S_t(x) = S_t(x) r_t^* h_t, \hbox{ for all } x\in A,$ the following statements are equivalent:
$(a)$ $\{T_t: t\in \mathbb{R}_0^{+}\}$ is a uniformly continuous one-parameter semigroup of orthogonality preserving operators on $A$;
$(b)$ $\{S_t: t\in \mathbb{R}_0^{+}\}$ is a uniformly continuous one-parameter semigroup of surjective linear isometries (i.e. triple isomorphisms) on $A$ (and hence there exists a triple derivation $δ$ on $A$ such that $S_t = e^{t δ}$ for all $t\in \mathbb{R}$), the mapping $t\mapsto h_t $ is continuous at zero, and the identity $ h_{t+s} = h_t r_t^* S_t^{**} (h_s),$ holds for all $s,t\in \mathbb{R}.$
In a first objective we improve our understanding about surjective and bijective bounded linear operators preserving orthogonality from a JB$^*$-algebra $\mathcal{A}$ into a JB$^*$-triple $E$. Among many other conclusions, it is shown that a bounded linear bijection $T: \mathcal{A}\to E$ is orthogonality preserving if, and only if, it is biorthogonality preserving if, and only if, it preserves zero-triple-products in both directions (i.e., $\{a,b,c\}=0 \Leftrightarrow \{T(a),T(b),T(c)\}=0$). In the second main result we establish a complete characterization of all one-parameter groups of ortho…
Applying a result of abstract ring theory we get that bijective additive mappings on standard algebras of unbounded operators preserving zero products are multiples of ring isomorphisms. The structure of additive bijective mappings on certain classes of standard algebras of unbounded operators preserving orthogonality in both directions is also investigated. The results are quite similar to those for algebras of bounded operators
Let $A$ be a $C^*$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $θ: E\to F$ is a linear map preserving orthogonality, i.e., $<θ(x), θ(y) > = 0$ whenever $<x, y > = 0$. We show in this article that if, in addition, $A$ has real rank zero, and $θ$ is an $A$-module map (not assumed to be bounded), then there exists a central positive multiplier $u\in M(A)$ such that $<θ(x), θ(y) > = u < x, y>$ ($x,y\in E$). In the case when $A$ is a standard $C^*$-algebra, or when $A$ is a $W^*$-algebra containing no finite type II direct summand, we also obtain the same concl…
In 2014, we determine the precise form of a continuous orthogonal form on a commutative real C$^*$-algebra. We also describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between commutative unital real C$^*$-algebras. Among the consequences, we show that every orthogonality preserving linear bijection between commutative unital real C$^*$-algebras is continuous. In this note we revisit these results and their proofs with the idea of filling a gap in the arguments, and to extend the original conclusions.
Due to the corresponding fact concerning Hilbert spaces, it is natural to ask if the linearity and the orthogonality structure of a Hilbert $C^*$-module determine its $C^*$-algebra-valued inner product. We verify this in the case when the $C^*$-algebra is commutative (or equivalently, we consider a Hilbert bundle over a locally compact Hausdorff space). More precisely, a $\mathbb{C}$-linear map $θ$ (not assumed to be bounded) between two Hilbert $C^*$-modules is said to be "orthogonality preserving" if $\left<θ(x),θ(y)\right> =0$ whenever $\left<x,y\right> =0$. We prove that if $θ$ is an ortho…
We initiate the study of orthogonal forms on a real C$^*$-algebra. Motivated by previous contributions, due to Ylinen, Jajte, Paszkiewicz and Goldstein, we prove that for every continuous orthogonal form $V$ on a commutative real C$^*$-algebra, $A$, there exist functionals $φ_1$ and $φ_2$ in $A^{*}$ satisfying $$V(x,y) = φ_1 (x y) + φ_2 (x y^*),$$ for every $x,y$ in $A$. We describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between unital commutative real C$^*$-algebras. As a consequence, we show that every orthogonality preserving linear bijection …
As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module structure. In fact, we have a more general result as follows. Let $A$ be a $C^*$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of $A$ generated by $\{\langle x,y\rangle_A: x,y\in E\}$. If $Φ: E\to F$ is an $A$-module map, not assumed to be bounded but satisfying $$ \langle Φ(x),Φ(y)\rangle_A\ =\ 0\quad\text{whenever}\quad\langle x,y\rangle_A\ =\ 0, $$ then…
Let T be be a zero-product preserving bounded linear map between C*-algebras A and B. Here neither A nor B is necessarily unital. In this note, we investigate when T gives rise to a Jordan homomorphism. In particular, we show that A and B are isomorphic as Jordan algebras if T is bijective and sends zero products of self-adjoint elements to zero products. They are isomorphic as C*-algebras if T is bijective and preserves the full zero product structure.
In this paper we investigate the properties of minus partial order in unital rings. We generalize several results well known for matrices and bounded linear operators on Banach spaces. We also study linear maps preserving the minus partial order in unital semisimple Banach algebras with essential socle.
We study new classes of linear preservers between C$^*$-algebras and JB$^*$-triples. Let $E$ and $F$ be JB$^*$-triples with $\partial_{e} (E_1)$. We prove that every linear map $T:E\to F$ strongly preserving Brown-Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital C$^*$-algebras $A$ and $B,$ for each linear map $T$ strongly preserving Brown-Pedersen quasi-invertible elements, then there exists a Jordan $^*$-homomorphism $S: A\to B$ satisfying $T(x) = T(1) S(x)$, for every $x\in A$. We also study the connections between linea…
As a continuation of the work on linear maps between operator algebras which preserve certain subsets of operators with finite rank, or corank, here we consider the problem inbetween, that is, we treat the question of preserving operators with infinite rank and infinite corank. Since, as it turns out, in this generality our preservers cannot be written in a nice form what we have got used to when dealing with linear preserver problems, hence we restrict our attention to certain important classes of operators like idempotents, or projections, or partial isometries. We conclude the paper with a …
Let $\mathcal{B} (X)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space $X$. For an operator $ T \in \mathcal{B} (X)$, $K(T)$ denotes as usual the analytic core of $T$. We determine the form of surjective maps $ ϕ$ on $ \mathcal{B} (X)$ satisfying $$ K(ϕ(T) ϕ(S) + ϕ(S) ϕ(T)) = K(TS + ST) $$ for all $T, S \in \mathcal{B} (X)$.
A linear mapping $T$ on a JB$^*$-triple is called triple derivable at orthogonal pairs if for every $a,b,c\in E$ with $a\perp b$ we have $$0 = \{T(a), b,c\} + \{a,T(b),c\}+\{a,b,T(c)\}.$$ We prove that for each bounded linear mapping $T$ on a JB$^*$-algebra $A$ the following assertions are equivalent: $(a)$ $T$ is triple derivable at zero; $(b)$ $T$ is triple derivable at orthogonal elements; $(c)$ There exists a Jordan $^*$-derivation $D:A\to A^{**}$, a central element $ξ\in A^{**}_{sa},$ and an anti-symmetric element $η$ in the multiplier algebra of $A$, such that $$ T(a) = D(a) + ξ\circ a +…
Let $a$ and $b$ be elements in the closed ball of a unital C$^*$-algebra $A$ (if $A$ is not unital we consider its natural unitization). We shall say that $a$ and $b$ are domain (respectively, range) absolutely compatible ($a\triangle_d b$, respectively, $a\triangle_r b$, in short) if $\Big| |a| -|b| \Big| + \Big| 1-|a|-|b| \Big| =1$ (resp., $\Big| |a^*| -|b^*| \Big| + \Big| 1-|a^*|-|b^*| \Big| =1$), where $|a|^2= a^* a$. We shall say that $a$ and $b$ are absolutely compatible ($a\triangle b$ in short) if they are both range and domain absolutely compatible. In general, $a\triangle_d b$ (respe…
For a unital $C^*$-algebra $\mathcal A$ and a subspace $\mathcal B$ of $\mathcal A$, a characterization for a best approximation to an element of $\mathcal A$ in $\mathcal B$ is obtained. As an application, a formula for the distance of an element of $\mathcal A$ from $\mathcal B$ has been obtained, when a best approximation of that element to $\mathcal B$ exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert $C^*$-module to a subspace is obtained.
We study holomorphic maps between C$^*$-algebras $A$ and $B$. When $f:B_A (0,\varrho) \longrightarrow B$ is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball $U=B_{A}(0,δ)$ and we assume that $f$ is orthogonality preserving on $A_{sa}\cap U$, orthogonally additive on $U$ and $f(U)$ contains an invertible element in $B$, then there exist a sequence $(h_n)$ in $B^{**}$ and Jordan $^*$-homomorphisms $Θ, \widetildeΘ : M(A) \to B^{**}$ such that $$ f(x) = \sum_{n=1}^\infty h_n \widetildeΘ (a^n)= \sum_{n=1}^\infty Θ (a^n) h_n,$$ uniformly in $a\in U$. W…
In this paper we present results concerning orthogonality in Hilbert $C^*$-modules. Moreover, for a $C^*$-algebra $\mathscr{A}$, we prove theorems concerning the multi-$\mathscr{A}$-linearity and its preservation by $\mathscr{A}$-linear operators. New version of solution of the orthogonality equation on Hilbert $C^*$-modules and mappings preserving orthogonality are also investigated.
Let $\mathcal A$ be a unital $C^*$-algebra with the unit $e$. We consider the elements $a\in \mathcal A$ which are Roberts orthogonal to the unit $e$. We obtain a characterization of this orthogonality in terms of the Davis--Wielandt shell of $a$ and show that, for certain classes of elements of $\mathcal A$, the Roberts orthogonality of $a$ and $e$ is equivalent to the symmetry of the numerical range of $a$ with respect to the origin.
We show that a bounded, linear map between C*-algebras is a weighted $\ast$-homomorphism (the central compression of a $\ast$-homomorphism) if and only if it preserves zero-products, range-orthogonality, and domain-orthogonality. It follows that a self-adjoint, bounded, linear map is a weighted $\ast$-homomorphism if and only if it preserves zero-products. As an application we show that a linear map between C*-algebras is completely positive, order zero in the sense of Winter-Zacharias if and only if it is positive and preserves zero-products.
We prove the following: Let $A$ be a C$^{\ast}$-algebra. Then for $a, b \in A^+ \setminus\{ 0 \}$, we have $a b = 0$ if and only is $\Vert \Vert c \Vert^{-1} c + \Vert d \Vert^{-1} d \Vert = 1$ whenever $0 < c \le a$ and $0 < d \le b$ in $A^+$.
Let $M$ and $N$ be arbitrary von Neumann algebras. For any $a$ in $M$ or in $N$, let $Δ_λ(a)$ denote the $λ$-Aluthge transform of $a$. Suppose that $M$ has no abelian direct summand. We prove that every bijective map $Φ:M\to N$ satisfying $$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 ex…
We introduce the Jordan-strict topology on the multipliers algebra of a JB$^*$-algebra, a notion which was missing despite the fourty years passed after the first studies on Jordan multipliers. In case that a C$^*$-algebra $A$ is regarded as a JB$^*$-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C$^*$-strict topology. We prove that every JB$^*$-algebra $\mathfrak{A}$ is J-strict dense in its multipliers algebra $M(\mathfrak{A})$, and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonalit…
In the first part of the paper, we use states on $C^*$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^*$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^*$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^*$-module under the assumption that their inner product has negative real part. We introduce the concept of Pythagoras orth…
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 automati…
Let A be a C*-algebra and d from A into A** be a continuous linear map. We assume that d acts like derivation or anti-derivation at orthogonal elements for several types of orthogonality conditions such as ab=0, ab*=0, ab=ba=0 and ab*=b*a=0. In each case, we characterize the structure of d. Then we apply our results for von Neumann algebras and unital simple C*-algebras.
We investigate linear operators between C$^\ast$-algebras which approximately preserve involution and orthogonality, the latter meaning that for some $\varepsilon>0$ we have $\|ϕ(x)ϕ(y)\|\leq\varepsilon\|x\|\|y\|$ for all positive $x,y$ with $xy=0$. We establish some structural properties of such maps concerning approximate Jordan-like equations and almost commutation relations. In some situations (e.g. when the codomain is finite-dimensional), we show that $ϕ$ can be approximated by an approximate Jordan $^\ast$-homomorphism, with both errors depending only on $\|ϕ\|$ and $\varepsilon$.
We give necessary and sufficient condition that an element of an arbitrary $C^{*}$-algebra is an isolated vertex of the orthograph related to the mutual strong Birkhoff-James orthogonality. Also, we prove that for all $C^{*}$-algebras except $\mathbb{C},\mathbb{C}\oplus\mathbb{C}$ and $M_2(\mathbb{C})$ all non isolated points make a single connected component of the orthograph which diameter is less than or equal to $4$, i.e. any two non isolated points can be connected by a path with at most $4$ edges. Some related results are given.
We show that the approximate quasiorthogonality of two operator algebras is equivalent to the algebras being approximately private relative to their conditional expectation quantum channels. Our analysis is based on a characterization of the measure of orthogonality in terms of Choi matrices and Kraus operators for completely positive maps. We present examples drawn from different areas of quantum information.
Let $\mathcal{A}$ be a $*$-algebra and $\mathcal{M}$ be a $*$-$\mathcal A$-bimodule, we study the local properties of $*$-derivations and $*$-Jordan derivations from $\mathcal{A}$ into $\mathcal{M}$ under the following orthogonality conditions on elements in $\mathcal A$: $ab^*=0$, $ab^*+b^*a=0$ and $ab^*=b^*a=0$. We characterize the mappings on zero product determined algebras and zero Jordan product determined algebras. Moreover, we give some applications on $C^*$-algebras, group algebra, matrix algebras, algebras of locally measurable operators and von Neumann algebras.
We show that any order isomorphism between ordered structures of associative unital JB-subalgebras of JBW algebras is implemented naturally by a Jordan isomorphism. Consequently, JBW algebras are determined by the structure of their associative unital JB subalgebras. Further we show that in a similar way it is possible to reconstruct Jordan structure from the order structure of associative subalgebras endowed with an orthogonality relation. In case of abelian subalgebras of von Neuman algebra it is we shown that order isomorphisms of the structure of abelian C*-subalgebras that are well behave…
