arXiv subjects

Ngai-Ching Wong

Publications and source records attributed to Ngai-Ching Wong.

35 records · Page 2Linked to original sources

arXiv2015

Transition probabilities of normal states determine the Jordan structure of a quantum system

Let $Φ:\mathfrak{S}(M_1)\to \mathfrak{S}(M_2)$ be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras $M_1$ and $M_2$, respectively. This paper concerns with the situation when $Φ$ preserves (or partially preserves) one of the following three notions of "transition probability" on the normal state spaces: the Uhlmann transition probability $P_U$, the Raggio transition probability $P_B$ and an "asymmetric transition probability" $P_0$ as defined in this article. It is shown that the two systems are isomorphic, i.e. $M_1$ and $M_2$ are Jordan $^*$-isomorphic, if $Φ$ preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, i.e., for any normal states $μ$ and $ν$, we have $$ P\big(Φ(μ),Φ(ν)\big) = 0 \quad \text{if and only if} \quad P(μ,ν)=0, $$ where $P$ stands for $P_U$ (respectively, $P_R$ or $P_0$). Furthermore, as an extension of Wigner's theorem, it is shown that there is a Jordan $^*$-isomorphism $Θ:M_2\to M_1$ with $$Φ= Θ^*|_{\mathfrak{S}(M_1)}$$ if and only if $Φ$ preserves the "asymmetric transition probability". This is also equivalent to $Φ$ preserving the Raggio transition probability. Consequently, if $Φ$ preserves the Raggio transition probability, it will preserve the Uhlmann transition probability as well. As another application, the sets of normal states equipped with either the usual metric, the Bures metric or "the metric induced by the self-dual cone" are complete Jordan $^*$-invariants for the underlying von Neumann algebras.

math-ph↗

arXiv2014

The spectrum of the product of operators, and the product of their numerical ranges

We show that a compact operator $A$ is a multiple of a positive semi-definite operator if and only if $$ σ(AB) \subseteq \overline{W(A)W(B)}, \quad\text{for all (rank one) operators $B$}. $$ An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators $A$ so that equivalence conditions hold.

math.FA↗

arXiv2014

Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices

Let $H:M_m\to M_m$ be a holomorphic function of the algebra $M_m$ of complex $m\times m$ matrices. Suppose that $H$ is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of $H$ consists of zero trace elements, or there is a scalar sequence $\{λ_n\}$ and an invertible $S$ in $M_m$ such that $$ H(x) =\sum_{n\geq 1} λ_n S^{-1}x^nS, \quad\forall x \in M_m,%\eqno{(‡)} $$ or $$ H(x) =\sum_{n\geq 1} λ_n S^{-1}(x^t)^nS, \quad\forall x \in M_m. $$ Here, $x^t$ is the transpose of the matrix $x$. In the latter case, we always have the first representation form when $H$ also preserves zero products. We also discuss the cases where the domain and the range carry different dimensions.

math.FA↗

arXiv2014

Inner products and module maps of Hilbert C*-modules

Let $E$ and $F$ be two Hilbert $C^*$-modules over $C^*$-algebras $A$ and $B$, respectively. Let $T$ be a surjective linear isometry from $E$ onto $F$ and $φ$ a map from $A$ into $B$. We will prove in this paper that if the $C^*$-algebras $A$ and $B$ are commutative, then $T$ preserves the inner products and $T$ is a module map, i.e., there exists a $*$-isomorphism $φ$ between the $C^*$-algebras such that $$ \langle Tx,Ty\rangle=φ(\langle x,y\rangle), $$ and $$ T(xa)=T(x)φ(a). $$ In case $A$ or $B$ is noncommutative $C^*$-algebra, $T$ may not satisfy the equations above in general. We will also give some condition such that $T$ preserves the inner products and $T$ is a module map.

math.OA↗

arXiv2014

Isometric embeddings of Banach bundles

We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an \emph{into} isometry from a continuous section space into an other. However, we demonstrate by an example that a non-surjective linear isometry can be far away from a subbundle embedding.

math.FA↗

arXiv2011

Operator Ideals arising from Generating Sequences

In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks operators will be demonstrated.

math.FA↗

arXiv2010

Linear orthogonality preservers of Hilbert $C^$-modules over general $C^$-algebras

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 there exists a unique central positive multiplier $u\in M(I_E)$ such that $$ \langle Φ(x), Φ(y)\rangle_A\ =\ u \langle x, y\rangle_A\qquad (x,y\in E). $$ As a consequence, $Φ$ is automatically bounded, the induced map $Φ_0: E\to \overline{Φ(E)}$ is adjointable, and $\overline{Eu^{1/2}}$ is isomorphic to $\overline{Φ(E)}$ as Hilbert $A$-modules. If, in addition, $Φ$ is bijective, then $E$ is isomorphic to $F$.

math.OA↗

arXiv2010

Linear orthogonality preservers of Hilbert bundles

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 $\left =0$. We prove that if $θ$ is an orthogonality preserving map from a full Hilbert $C_0(Ω)$-module $E$ into another Hilbert $C_0(Ω)$-module $F$ that satisfies a weaker notion of $C_0(Ω)$-linearity (known as "localness"), then $θ$ is bounded and there exists $ϕ\in C_b(Ω)_+$ such that $$ \left<θ(x),θ(y)\right>\ =\ ϕ\cdot\left , \quad \forall x,y \in E. $$ On the other hand, if $F$ is a full Hilbert $C^*$-module over another commutative $C^*$-algebra $C_0(Δ)$, we show that a "bi-orthogonality preserving" bijective map $θ$ with some "local-type property" will be bounded and satisfy $$ \left<θ(x),θ(y)\right>\ =\ ϕ\cdot\left \circσ, \quad \forall x,y \in E $$ where $ϕ\in C_b(Ω)_+$ and $σ: Δ\rightarrow Ω$ is a homeomorphism.

math.OA↗

arXiv2010

Automatic continuity and $C_0(Ω)$-linearity of linear maps between $C_0(Ω)$-modules

Let $Ω$ be a locally compact Hausdorff space. We show that any local $\mathbb{C}$-linear map (where "local" is a weaker notion than $C_0(Ω)$-linearity) between Banach $C_0(Ω)$-modules are "nearly $C_0(Ω)$-linear" and "nearly bounded". As an application, a local $\mathbb{C}$-linear map $θ$ between Hilbert $C_0(Ω)$-modules is automatically $C_0(Ω)$-linear. If, in addition, $Ω$ contains no isolated point, then any $C_0(Ω)$-linear map between Hilbert $C_0(Ω)$-modules is automatically bounded. Another application is that if a sequence of maps $\{θ_n\}$ between two Banach spaces "preserve $c_0$-sequences" (or "preserve ultra-$c_0$-sequences"), then $θ_n$ is bounded for large enough $n$ and they have a common bound. Moreover, we will show that if $θ$ is a bijective "biseparating" linear map from a "full" essential Banach $C_0(Ω)$-module $E$ into a "full" Hilbert $C_0(Δ)$-module $F$ (where $Δ$ is another locally compact Hausdorff space), then $θ$ is "nearly bounded" (in fact, it is automatically bounded if $Δ$ or $Ω$ contains no isolated point) and there exists a homeomorphism $σ: Δ\rightarrow Ω$ such that $θ(e\cdot φ) = θ(e)\cdot φ\circ σ$ ($e\in E, φ\in C_0(Ω)$).

math.OA↗

arXiv2010

"Maps preserving the spectrum of generalized Jordan product of operators", and its "Addendum"

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 $Φ$ contains all operators of rank at most three, then $Φ$ must be a Jordan isomorphism multiplied by an $m$th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained. After our paper "Maps preserving the spectrum of generalized Jordan product of operators" was published in Linear Algebra Appl. 432 (2010), 1049-1069, Jianlian Cui pointed out that some arguments in the proof of Theorem 3.1 are not entirely clear and accurate. Here we supply some details in the "Addendum".

math.FA↗

arXiv2009

A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

Let $X$ and $Y$ be compact Hausdorff spaces, and $E$, $F$ be Banach lattices. Let $C(X,E)$ denote the Banach lattice of all continuous $E$-valued functions on $X$ equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism $\mathnormalΦ: C(X,E)\to C(Y,F)$ such that $\mathnormalΦf$ is non-vanishing on $Y$ if and only if $f$ is non-vanishing on $X$, then $X$ is homeomorphic to $Y$, and $E$ is Riesz isomorphic to $F$. In this case, $\mathnormalΦ$ can be written as a weighted composition operator: $\mathnormalΦ f(y)=\mathnormalΠ(y)(f(φ(y)))$, where $φ$ is a homeomorphism from $Y$ onto $X$, and $\mathnormalΠ(y)$ is a Riesz isomorphism from $E$ onto $F$ for every $y$ in $Y$. This generalizes some known results obtained recently.

math.FA↗

arXiv2007

Property (T) for non-unital C*-algebras

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T) for not necessarily unital C*-algebras. The stronger one of the two is called ``property (T)'' and the weaker one is called ``property (T_{e})''. It is shown that all non-unital C*-algebras do not have property (T) (neither do their unitalizations). Moreover, all non-unital $σ$-unital C*-algebras do not have property (T_e).

math.OA↗

arXiv2007

Zero product preservers of C*-algebras

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.

math.OA↗

arXiv2005

The triangle of operators, topologies, bornologies

This paper discusses two common techniques in functional analysis: the topological method and the bornological method. In terms of Pietsch's operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of Grothendieck (via Banach space operators), Randtke (via continuous seminorms) and Hogbe-Nlend (via convex bounded sets) are compared.

math.FA↗

arXiv2005

Triple homomorphisms of C*-algebras

In this note, we will discuss what kind of operators between C*-algebras preserves Jordan triple products {a,b,c}= (ab*c + cb*a)/2. These include especially isometries and disjointness preserving operators.

math.FA↗

arXiv1996

Into isometries of $C_0(X,E)$'s

Suppose $X$ and $Y$ are locally compact Hausdorff spaces, $E$ and $F$ are Banach spaces and $F$ is strictly convex. We show that every linear isometry $T$ from $C_0(X,E)$ {\em into} $C_0(Y,F)$ is essentially a weighted composition operator $Tf(y) = h(y) (f(φ(y)))$. This supplements results of Jerison (when $T$ is onto) and Cambern (when $X,Y$ are compact).

math.FA↗

arXiv1996

Weighted composition operators of $C_0(X)$'s

In this paper, we prove that into isometries and disjointness preserving linear maps from $C_0(X)$ into $C_0(Y)$ are essentially weighted composition operators $Tf = h\cdot f\circφ$ for some continuous map $φ$ and some continuous scalar-valued function $h$.

math.FA↗