Let A and A' be two Cstar-algebras with identities I_A and I_A', respectively, and P_1 and P_2 = I_A - P_1 nontrivial projections in A. In this paper we study the characterization of multiplicative star-Lie-Jordan-type maps, where the notion of these maps arise here. In particular, if M_A is a von Neumann algebra relative Cstar-algebra A without central summands of type I_1 then every bijective unital multiplicative star-Lie-Jordan-type maps are star-ring isomorphisms.
Let $\mathfrak{A}$ and $\mathfrak{A}'$ be two $C^*$-algebras with identities $I_{\mathfrak{A}}$ and $I_{\mathfrak{A}'}$, respectively, and $P_1$ and $P_2 = I_{\mathfrak{A}} - P_1$ nontrivial projections in $\mathfrak{A}$. In this paper we study the characterization of multiplicative $*$-Jordan-type maps. In particular, if $\mathcal{M}$ is a factor von Neumann algebra then every bijective unital multiplicative $*$-Jordan-type maps are $*$-ring isomorphisms.
Let $η\neq -1$ be a non-zero complex number, and let $ϕ$ be a not necessarily linear bijection between two von Neumann algebras, one of which has no central abelian projections preserving the Jordan $η$-$\ast$-$n$-product. It is showed that $ϕ$ is a linear $\ast$-isomorphism if $η$ is not real and $ϕ$ is the sum of a linear $\ast$-isomorphism and a conjugate linear $\ast$-isomorphism if $η$ is real.
We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of purity for quantum systems. We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category (which sets it apart from other possible definitions.) In fact, from the results presented in this paper, it follows that the category of EJAs with positive contractive linear maps is a dagger-effectus, a type of structure originally defined to study von Neumann algebras in an abstract categorical setti…
Without the faithful assumption, we prove that every Jordan derivation on a class of path algebras of quivers without oriented cycles is a derivation and that every Lie derivation on such kinds of algebras is of the standard form.
We develop a cohomology theory for Jordan triples, including the infinite dimensional ones, by means of the cohomology of TKK Lie algebras. This enables us to apply Lie cohomological results to the setting of Jordan triples. Some preliminary results for von Neumann algebras are obtained.
Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras and $η$ be a non-zero complex number. A nonlinear bijective map $ϕ:\mathcal A\rightarrow\mathcal B$ has been demonstrated to satisfy $$ϕ([A,B]_{*}^η\diamond_η C)=[ϕ(A),ϕ(B)]_{*}^η\diamond_ηϕ(C)$$ for all $A,B,C\in\mathcal A.$ If $η=1,$ then $ϕ$ is a linear $*$-isomorphism, a conjugate linear $*$-isomorphism, the negative of a linear $*$-isomorphism, or the negative of a conjugate linear $*$-isomorphism. If $η\neq 1$ and satisfies $ϕ(I)=1,$ then $ϕ$ is either a linear $*$-isomorphism or a conjugate linear $*$-isomorphism.
Let $\mathcal{A}$ and $\mathcal{B}$ be two $C^{*}$-algebras such that $\mathcal{B}$ is prime. In this paper, we investigate the additivity of map $Φ$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$Φ(AP+ηPA^{*})=Φ(A)Φ(P)+ηΦ(P)Φ(A)^{*},$$ for all $A\in\mathcal{A}$ and $P\in\{P_{1},I_{\mathcal{A}}-P_{1}\}$ where $P_{1}$ is a nontrivial projection in $\mathcal{A}$. Let $η$ be a non-zero complex number such that $|η|\neq1$, then $Φ$ is additive. Moreover, if $η$ is rational then $Φ$ is $\ast$-additive.
Let H and K be infinite dimensional Hilbert spaces, while B(H) and B(K) denote the algebras of all linear bounded operators on H and K, respectively. We characterize the forms of additive mappings from B(H) into B(K) that preserve the nonzero idempotency of either Jordan products of operators or usual products of operators in both directions.
Let $\A$ be a Banach algebra with unity $\textbf{1}$ and $ \M $ be a unital Banach left $ \A $-module. let $ δ: \A \rightarrow \M$ be a continuous linear map with the property that \[ a,b\in \A, \quad ab+ba=z \Rightarrow 2aδ(b)+2bδ(a)=δ(z), \] where $z\in \A$. In this article, first we characterize $δ$ for $z=\textbf{1}$. Then we consider the case $\A=\M=Alg \mathcal{L}$, where $Alg \mathcal{L}$ is areflexive algebra on a Hilbert space $ \Hh $ and $z=P$ is a non-triavial idempotent in $\A$ with $P(\Hh) \in \mathcal{L}$ and describe $δ$. Finally we apply the main results to $CSL$-algebras, irre…
Let $\mathcal{A}$ be a unital $C^{*}$-algebra. We consider Jordan $*$-homomorphisms on $C(X, \mathcal{A})$ and Jordan $*$-homomorphisms on $\operatorname{Lip}(X,\mathcal{A})$. More precisely, for any unital $C^{*}$-algebra $\mathcal{A}$, we prove that every Jordan $*$-homomorphism on $C(X, \mathcal{A})$ and every Jordan $*$-homomorphism on $\operatorname{Lip}(X,\mathcal{A})$ is represented as a weighted composition operator by using the irreducible representations of $\mathcal{A}$. In addition, when $\mathcal{A}_1$ and $\mathcal{A}_2$ are primitive $C^{*}$-algebras, we characterize the Jordan …
In the paper "Maps preserving the spectrum of generalized Jordan product of operators", we define a generalized Jordan products on standard operator algebras $A_1, A_2$ on complex Banach spaces $X_1, X_2$, respectively. This includes the usual Jordan product $A_1 \circ A_2 = A_1 A_2 + A_2 A_1$, and the triple $\{A_1,A_2,A_3\} = A_1 A_2 A_3 + A_3 A_2 A_1$. Let a map $Φ: A_1 \to A_2$ prserving the spectra of the products $$ σ(Φ(A_1) \circ ... \circ Φ(A_k)) = σ(A_1\circ ... \circ A_k) $$ whenever any one of $A_1, ..., A_k$ has rank at most one. It is shown in this paper that if the range of $Φ$ c…
Let $X_1$ and $X_2$ be complex Banach spaces with dimension at least three, $\mathcal{A}_1$ and $\mathcal{A}_2$ be standard operator algebras on $X_1$ and $X_2$, respectively. For $k\geq2$, let $(i_1,...,i_m)$ be a sequence with terms chosen from $\{1,\ldots,k\}$ and assume that at least one of the terms in $(i_1,\ldots,i_m)$ appears exactly once. Define the generalized Jordan product $T_1\circ T_2\circ\cdots\circ T_k=T_{i_1} T_{i_2}\cdots T_{i_m}+T_{i_m}\cdots T_{i_2} T_{i_1}$ on elements in $\mathcal{A}_i$. This includes the usual Jordan product $A_1A_2+A_2A_1$, and the Jordan triple $A_1A_2…
We investigate linear operators between C$^\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 prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized derivations from $\mathcal{J}$ into $\mathcal{J}^*$. We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all anti-symmetric orthogonal forms on $\mathcal{J}$, and of all Lie Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$.
We study contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space. For example we find generalizations and variants of certain classical results on contractive projections on C*-algebras and JB-algebras due to Choi, Effros, Størmer, Friedman and Russo, and others. In fact most of our arguments generalize to contractive `real positive' projections on Jordan operator algebras, that is on a norm-closed space A of operators on a Hilbert space which are closedunder the Jordan product. We also prove many new general results on real positive maps which a…
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Institute for Quantum Computing（量子计算研究所）；Leibniz Universität Hannover, Institut für Theoretische Physik（汉诺威莱布尼茨大学理论物理研究所）
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 $T, S \in \mathcal{B} (X)$.
