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Chi-Keung Ng

Publications and source records attributed to Chi-Keung Ng.

At least 19 recordsLinked to original sources
arXiv2026

Predual and tight operator systems

Given a (not necessarily unital) complete operator system $S$ with a generating cone, there are some studies in literature on the operator system dual $S^\mathrm{d}$ of $S$, i.e., the dual matrix-ordered space $S^*$ equipped with a matrix norm that turns it into a dual operator system satisfying certain universal property. In order to do further study on $S^\mathrm{d}$, one needs to consider the operator system predual construction. More precisely, given a dual operator system $V$ with a generating cone, there is a matrix norm on the predual space $V_*$, that turns it into a complete operator system $V_\#$ satisfying certain universal property. In this article, we show that $V\cong T^\mathrm{d}$ for a complete operator system $T$ if and only if $V$ satisfies a natural property called tightness; in this case, $V\cong (V_\#)^\mathrm{d}$. Furthermore, we establish that if $S$ is tight, then $S\cong W_\#$ for a dual operator system $W$; in fact $S\cong (S^\mathrm{d})_\#$. Since all unital complete operator systems and all $C^*$-algebra are tight, one sees that every unital complete operator system and every $C^*$-algebra is of the form $W_\#$, for a dual operator system $W$.

math.OA↗
arXiv2025

Duality for operator systems with generating cones

Let $S$ be a complete operator system with a generating cone; i.e. $S_\sa = S_+ - S_+$. We show that there is a matrix norm on the dual space $S^*$, under which, and the usual dual matrix cone, $S^*$ becomes a dual operator system with a generating cone, denoted by $S^\rd$. The canonical complete order isomorphism $ι_{S^*}: S^* \to S^\rd$ is a dual Banach space isomorphism. Furthermore, we construct a canonical completely contractive weak$^*$-homeomorphism $β_S: (S^\rd)^\rd\to S^{**}$, and verify that it is a complete order isomorphism. For a complete operator system $T$ with a generating cone and a completely positive complete contraction $φ:S\to T$, there is a weak$^*$-continuous completely positive complete contraction $φ^\rd:T^\rd \to S^\rd$ with $ι_{S^*}\circ φ^* = φ^\rd \circ ι_{T^*}$. This produces a faithful functor from the category of complete operator systems with generating cones (where morphisms are completely positive complete contractions) to the category of dual operator systems with generating cones (where morphisms are weak$^*$-continuous completely positive complete contractions). We define the notion of approximately unital operator systems, and verify that operator systems considered in \cite{CvS} and \cite{CvS2} are approximately unital. If $S$ is approximately unital, then $ι_{S^*}:S^* \to S^\rd$ is an operator space isomorphism and $β_S: (S^\rd)^\rd\to S^{**}$ is a complete isometry. We will also establish that the restriction of the faithful functor $(S,T,φ)\mapsto (T^\rd, S^\rd, φ^\rd)$ to the category of approximately unital complete operator systems is both full and injective on objects.

math.OA↗
arXiv2022

Non-unital operator systems that are dual spaces

We will give an abstract characterization of an arbitrary self-adjoint weak$^*$-closed subspace of $\mathcal{L}(H)$ (equipped with the induced matrix norm, the induced matrix cone and the induced weak$^*$-topology). In order to do this, we obtain a matrix analogues of a result of Bonsall for $^*$-operator spaces equipped with closed matrix cones. On our way, we observe that for a $^*$-vector $X$ equipped with a matrix cone (in particular, when $X$ is an operator system or the dual space of an operator system), a linear map $ϕ:X\to M_n$ is completely positive if and only if linear functional $[x_{i,j}]_{i,j}\mapsto \sum_{i,j=1}^n ϕ(x_{i,j})_{i,j}$ on $M_n(X)$ is positive.

math.FA↗
arXiv2022

Dual spaces of operator systems

This article is to give an infinite dimensional analogue of a result of Choi and Effros. We say that an (not necessarily unital) operator system $T$ is \emph{dualizable} if one can find an equivalent dual matrix norm on the dual space $T^*$ such that under this dual matrix norm and the canonical dual matrix cone, $T^*$ becomes a dual operator system. We show that "a complete" operator system $T$ is dualizable if and only if $M_\infty(T)^\mathrm{sa}$ satisfies a bounded decomposition property. In this case, $$\|f\|^\mathrm{d}:= \sup \big\{\big\|[f_{i,j}(x_{k,l})]\big\|: x\in M_n(T)^+; \|x\|\leq 1; n\in \mathbb{N}\big\},$$ is the largest dual matrix norm that is equivalent to and dominated by the original dual matrix norm on $T^*$ that turns it into a dual operator system, denoted by $T^\mathrm{d}$. $T^\mathrm{d}$ is again dualizable. For every completely positive completely bounded map $ϕ:S\to T$ between dualizable operator systems, there is a unique weak-$^*$-continuous completely positive completely bounded map $ϕ^\mathrm{d}:T^\mathrm{d} \to S^\mathrm{d}$ which is compatible with the dual map $ϕ^*$. This gives a full and faithful functor from the category of dualizable operator systems to that of dualizable dual operator systems. Moreover, we will verify that that if $S$ is either a $C^*$-algebra or a unital operator system, then $S$ is dualizable and the canonical weak-$^*$-homeomorphism from the unital operator system $S^{**}$ to the operator system $(S^\mathrm{d})^\mathrm{d}$ is a completely isometric complete order isomorphism. Furthermore, the category of $C^*$-algebras and that of unital "complete" operator systems can be regarded as full subcategories of the category of dual operator systems.

math.OA↗
arXiv2021

Quantum sets and Gelfand spectra (Ortho-sets and Gelfand spectra)

Motivated by quantum states with zero transition probability, we introduce the notion of ortho-set which is a set equipped with a relation $\neq_\mathrm{q}$ satisfying: $x\neq_\mathrm{q} y$ implies both $x\neq y$ and $y \neq_\mathrm{q} x$. For an ortho-set, a canonical complete ortholattice is constructed. Conversely, every complete ortholattice comes from an ortho-set in this way. Hence, the theory of ortho-sets captures almost everything about quantum logics. For a quantum system modeled by the self-adjoint part $B_\mathrm{sa}$ of a $C^*$-algebra $B$, we also introduce a "semi-classical object" called the Gelfand spectrum. It is the ortho-set, $P(B)$, of pure states of $B$ equipped with an "ortho-topology", which is a collection of subsets of $P(B)$, defined via a hull-kernel construction with respects to closed left ideals of $B$. We establish a generalization of the Gelfand theorem by showing that a bijection between the Gelfand spectra of two quantum systems that preserves the respective ortho-topologies is induced by a Jordan isomorphism between the self-adjoint parts of the underlying $C^*$-algebras (i.e. an isomorphism of the quantum systems), when the underlying $C^*$-algebras satisfy a mild condition.

math-ph↗
arXiv2020

Coarse metric and uniform metric

We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the coarse structure $\mathcal{E}$ on a set $X$ is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set $\mathcal{P}_0(X)$ of non-empty subsets of $X$ and show that it induces the Hausdorff coarse structure on $\mathcal{P}_0(X)$. On the other hand, we define the notion of pseudo uniform metric. Each pseudo uniform metric induces a uniform structure on the underlying space. In the reverse direction, we show that a uniform structure $\mathcal{U}$ on a set $X$ is induced by a map $d$ from $X\times X$ to a partially ordered set (with no requirement on $d$) if and only if $\mathcal{U}$ admits a base $\mathcal{B}$ such that $\mathcal{B}\cup \{\bigcap \mathcal{U}\}$ is closed under arbitrary intersections. In this case, $\mathcal{U}$ is actually defined by a pseudo uniform metric. We also show that a uniform structures $\mathcal{U}$ comes from a pseudo uniform metric that takes values in a totally ordered set if and only if $\mathcal{U}$ admits a totally ordered base. Finally, a valuation ring will produce an example of a coarse and pseudo uniform metric that take values in a totally ordered set.

math.MG↗
arXiv2020

On a variant of Tingley's problem for some function spaces

Let $(Ω, \mathfrak{A}, μ)$ and $(Γ, \mathfrak{B}, ν)$ be two arbitrary measure spaces, and $p\in [1,\infty]$. Set $$L^p(μ)_+^\mathrm{sp}:= \{f\in L^p(μ): \|f\|_p =1; f\geq 0\ μ\text{-a.e.} \}$$ i.e., the positive part of the unit sphere of $L^p(μ)$. We show that every metric preserving bijection $Φ: L^p(μ)_+^\mathrm{sp} \to L^p(ν)_+^\mathrm{sp}$ can be extended (necessarily uniquely) to an isometric order isomorphism from $L^p(μ)$ onto $L^p(ν)$. A Lamperti form, i.e., a weighted composition like form, of $Φ$ is provided, when $(Γ, \mathfrak{B}, ν)$ is localizable (in particular, when it is $σ$-finite). On the other hand, we show that for compact Hausdorff spaces $X$ and $Y$, if $Φ$ is a metric preserving bijection from the positive part of the unit sphere of $C(X)$ to that of $C(Y)$, then there is a homeomorphism $τ:Y\to X$ satisfying $Φ(f)(y) = f(τ(y))$ ($f\in C(X)_+^\mathrm{sp}; y\in Y$).

math.FA↗
arXiv2020

Analytic bundle structure on the idempotent manifold

Let $X$ be a (real or complex) Banach space, and $\mathcal{I}(X)$ be the set of all (non-zero and non-identity) idempotents; i.e., bounded linear operators on $X$ whose squares equal themselves. We show that the Banach submanifold $\mathcal{I}(X)$ of $\mathcal{L}(X)$ is a locally trivial analytic affine-Banach bundle over the Grassmann manifold $\mathscr{G}(X)$, via the map $κ$ that sends $Q\in \mathcal{I}(X)$ to $Q(X)$, such that the affine-Banach space structure on each fiber is the one induced from $\mathcal{L}(X)$ (in particular, every fiber is an affine-Banach subspace of $\mathcal{L}(X)$). Using this, we show that if $K$ is a real Hilbert space, then the assignment $$(E,T)\mapsto T^*\circ P_{E^\bot} + P_{E}, \quad \text{ where } E\in \mathscr{G}(K)\text{ and } T\in \mathcal{L}(E,E^\bot),$$ induces a bi-analytic bijection from the total space of the tangent bundle, $\mathbf{T}(\mathscr{G}(K))$, of $\mathscr{G}(K)$ onto $\mathcal{I}(K)$ (here, $E^\bot$ is the orthogonal complement of $E$, $P_E\in \mathcal{L}(K)$ is the orthogonal projection onto $E$, and $T^*$ is the adjoint of $T$). Notice that this bi-analytic bijection is an affine map on each tangent plane.

math.DG↗
arXiv2018

Property (T) for locally compact groups and C*-algebras

Let $G$ be a locally compact group and let $C^*(G)$ and $C^*_r(G)$ be the full group $C^*$-algebra and the reduced group $C^*$-algebra of $G$. We investigate the relationship between Property $(T)$ for $G$ and Property $(T)$ as well as its strong version for $C^*(G)$ and $C^*_r(G)$. We show that $G$ has Property $(T)$ if (and only if) $C^*(G)$ has Property $(T)$. In the case where $G$ is a locally compact IN-group, we prove that $G$ has Property $(T)$ if and only if $C^*_r(G)$ has strong Property $(T)$. We also show that $C^*_r(G)$ has strong Property $(T)$ for every non-amenable locally compact group $G$ for which $C^*_r(G)$ is nuclear. Some of these groups (as for instance $G=SL_2(\mathbf{R})$) do not have Property $T$.

math.OA↗
arXiv2017

The topology on Berkovich affine lines over complete valuation rings

In this article, we give a full description of the topology of the one dimensional affine analytic space $\mathbb{A}_R^1$ over a complete valuation ring $R$ (i.e. a valuation ring with "real valued valuation" which is complete under the induced metric), when its field of fractions $K$ is algebraically closed. In particular, we show that $\mathbb{A}_R^1$ is both connected and locally path connected. Furthermore, $\mathbb{A}_R^1$ is the completion of $K\times (1,\infty)$ under a canonical uniform structure. As an application, we describe the Berkovich spectrum $\mathfrak{M}(\mathbb{Z}_p[G])$ of the Banach group ring $\mathbb{Z}_p[G]$ of a cyclic $p$-group $G$ over the ring $\mathbb{Z}_p$ of $p$-adic integers.

math.NT↗
arXiv2017

Amenability of locally compact quantum groups and their unitary co-representations

We prove that amenability of a unitary co-representation $U$ of a locally compact quantum group passes to unitary co-representations that weakly contain $U$. This generalizes a result of Bekka, and answers affirmatively a question of Bédos, Conti and Tuset. As a corollary, we extend to locally compact quantum groups a result of the first-named author, which characterizes amenability of a locally compact group $G$ by nuclearity of the reduced group $C^{*}$-algebra $C_{r}^{*}(G)$ and an additional condition.

math.OA↗
arXiv2016

A Murray-von Neumann type classification of $C^*$-algebras

We define type $\mathfrak{A}$, type $\mathfrak{B}$, type $\mathfrak{C}$ as well as C*-semi-finite C*-algebras. It is shown that a von Neumann algebra is a type $\mathfrak{A}$, type $\mathfrak{B}$, type $\mathfrak{C}$ or C*-semi-finite C*-algebra if and only if it is, respectively, a type I, type II, type III or semi-finite von Neumann algebra. Any type I C*-algebra is of type $\mathfrak{A}$ (actually, type $\mathfrak{A}$ coincides with the discreteness as defined by Peligrad and Zsido), and any type II C*-algebra (as defined by Cuntz and Pedersen) is of type $\mathfrak{B}$. Moreover, any type $\mathfrak{C}$ C*-algebra is of type III (in the sense of Cuntz and Pedersen). Furthermore, any purely infinite C*-algebra (in the sense of Kirchberg and Rordam) with real rank zero is of type $\mathfrak{C}$, and any separable purely infinite C*-algebra with stable rank one is also of type $\mathfrak{C}$. We also prove that type $\mathfrak{A}$, type $\mathfrak{B}$, type $\mathfrak{C}$ and C*-semi-finiteness are stable under taking hereditary C*-subalgebras, multiplier algebras and strong Morita equivalence. Furthermore, any C*-algebra $A$ contains a largest type $\mathfrak{A}$ closed ideal $J_\mathfrak{A}$, a largest type $\mathfrak{B}$ closed ideal $J_\mathfrak{B}$, a largest type $\mathfrak{C}$ closed ideal $J_\mathfrak{C}$ as well as a largest C*-semi-finite closed ideal $J_\mathfrak{sf}$. Among them, we have $J_\mathfrak{A} + J_\mathfrak{B}$ being an essential ideal of $J_\mathfrak{sf}$, and $J_\mathfrak{A} + J_\mathfrak{B} + J_\mathfrak{C}$ being an essential ideal of $A$. On the other hand, $A/J_\mathfrak{C}$ is always C*-semi-finite, and if $A$ is C*-semi-finite, then $A/J_\mathfrak{B}$ is of type $\mathfrak{A}$.

math.OA↗
arXiv2016

On the decomposition into Discrete, type II and type III $C^*$-algebras

We obtained a "decomposition scheme" of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsido), type II C*-algebras and type III C*-algebras (both defined by Cuntz and Pedersen) form a good framework to "classify" C*-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C*-subalgebras as well as taking "essential extension" and "normal quotient". Furthermore, there exist the largest discrete finite ideal $A_{d,1}$, the largest discrete essentially infinite ideal $A_{d,\infty}$, the largest type II finite ideal $A_{II,1}$, the largest type II essentially infinite ideal $A_{II,\infty}$, and the largest type III ideal $A_{III}$ of any C*-algebra $A$ such that $A_{d,1} + A_{d,\infty} + A_{II,1} + A_{II,\infty} + A_{III}$ is an essential ideal of $A$. This "decomposition" extends the corresponding one for $W^*$-algebras. We also give a closer look at C*-algebras with Hausdorff primitive spectrum, AW*-algebras as well as local multiplier algebras of C*-algebras. We find that these algebras can be decomposed into continuous fields of prime C*-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above.

math.OA↗
arXiv2015

Property $T$ of reduced $C^*$-crossed products by discrete groups

We generalize the main result of Kamalov and show that if $G$ is an amenable discrete group with an action $α$ on a finite nuclear unital $C^*$-algebra $A$ such that the reduced crossed product $A\rtimes_{α,r} G$ has property $T$, then $G$ is finite and $A$ is finite dimensional. As an application, an infinite discrete group $H$ is non-amenable if and only if the uniform Roe algebra $C^*_u(H)$ has property $T$.

math.OA↗
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↗
arXiv2015

Property $T$ for general locally compact quantum groups

In this short article, we obtained some equivalent formulations of property $T$ for a general locally compact quantum group $\mathbb{G}$, in terms of the full quantum group $C^*$-algebras $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ and the $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if $\mathbb{G}$ is of Kac type, we show that $\mathbb{G}$ has property $T$ if and only if every finite dimensional irreducible $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ is an isolated point in the spectrum of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ (this also generalizes the corresponding locally compact group result). In addition, we give a way to construct property $T$ discrete quantum groups using bicrossed products.

math.QA↗