arXiv subjects

Chi-Keung Ng

Publications and source records attributed to Chi-Keung Ng.

At least 37 records · Page 2Linked to original sources

arXiv2015

A property for locally convex $^*$-algebras related to Property $(T)$ and character amenability

For a locally convex $^*$-algebra $A$ equipped with a fixed continuous $^*$-character $\varepsilon$, we define a cohomological property, called property $(FH)$, which is similar to character amenability. Let $C_c(G)$ be the space of continuous functions on a second countable locally compact group $G$ with compact supports, equipped with the convolution $^*$-algebra structure and a certain inductive topology. We show that $(C_c(G), \varepsilon_G)$ has property $(FH)$ if and only if $G$ has property $(T)$. On the other hand, many Banach algebras equipped with canonical characters have property $(FH)$ (e.g., those defined by a nice locally compact quantum group). Furthermore, through our studies on both property $(FH)$ and character amenablility, we obtain characterizations of property $(T)$, amenability and compactness of $G$ in terms of the vanishing of one-sided cohomology of certain topological algebras, as well as in terms of fixed point properties. These characterizations can be regarded as analogues of one another. Moreover, we show that $G$ is compact if and only if the normed algebra $\big\{f\in C_c(G): \int_G f(t)dt =0\big\}$ (under $\|\cdot\|_{L^1(G)}$) admits a bounded approximate identity with the supports of all its elements being contained in a common compact set.

math.FA↗

arXiv2014

Berkovich spectra of elements in Banach Rings

Adapting the notion of the spectrum $Σ_a$ for an element $a$ in an ultrametric Banach algebra (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum $σ^{Ber}_R(u)$ of an element $u$ in a Banach ring $R$. This spectrum is a compact subset of the affine analytic space $A_Z^1$ over $Z$, and the later can be identified with the "equivalence classes" of all elements in all complete valuation fields. If $R$ is generated by $u$ as a unital Banach ring, then $σ^{Ber}_R(u)$ coincides with the spectrum of $R$ (as defined by Berkovich). If $R$ is a unital complex Banach algebra, then $σ^{Ber}_R(u)$ is the "folding up" of the usual spectrum $σ_B(u)$ alone the real axis. For a non-Archimedean complete valuation field $k$ and an infinite dimensional ultrametric $k$-Banach space $E$ with an orthogonal base, if $u\in L(E)$ is a completely continuous operator, we show that many different ways to define the spectrum of $u$ give the same compact set $σ^{Ber}_{L(E)}(u)$. As an application, we give a lower bound for the valuations of the zeros of the Fredholm determinant $\det(1- t\cdot u)$ (as defined by Serre) in complete valuation field extensions of $k$. Using this, we give a concrete example of a completely continuous operator whose Fredholm determinant does not have any zero in any complete valuation field extension of $k$.

math.FA↗

arXiv2013

Spectral gap actions and invariant states

We define spectral gap actions of discrete groups on von Neumann algebras and study their relations with invariant states. We will show that a finitely generated ICC group $Γ$ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra $L(Γ)$. Moreover, a countable discrete group $Γ$ has property $(T)$ if and only if for any action $α$ of $Γ$ on a von Neumann algebra $N$, every $α$-invariant state on $N$ is a weak-$^*$-limit of a net of normal $α$-invariant states.

math.OA↗

arXiv2011

On genuine infinite algebraic tensor products

A genuine infinite tensor product of complex vector spaces is a vector space ${\bigotimes}_{i\in I} X_i$ whose linear maps coincide with multilinear maps on an infinite family $\{X_i\}_{i\in I}$ of vector spaces. We give a direct sum decomposition of ${\bigotimes}_{i\in I} X_i$ over a set $Ω_{I;X}$, through which we obtain a more concrete description and some properties of ${\bigotimes}_{i\in I} X_i$. If $\{A_i\}_{i\in I}$ is a family of unital $^*$-algebras, we define, through a subgroup $Ω^{\rm ut}_{I;A}\subseteq Ω_{I;A}$, an interesting subalgebra ${\bigotimes}_{i\in I}^{\rm ut} A_i$. Moreover, it is shown that ${\bigotimes}_{i\in I}^{\rm ut} \mathbb{C}$ is the group algebra of $Ω^{\rm ut}_{I;\mathbb{C}}$. In general, ${\bigotimes}_{i\in I}^{\rm ut} A_i$ can be identified with the algebraic crossed product of a cocycle twisted action of $Ω^{\rm ut}_{I;A}$. If $\{H_i\}_{i\in I}$ is a family of inner-product spaces, we define a Hilbert $C^*(Ω^{\rm ut}_{I;\mathbb{C}})$-module $\bar\bigotimes^{\rm mod}_{i\in I} H_i$, which is the completion of a subspace ${\bigotimes}_{i\in I}^{\rm unit} H_i$ of ${\bigotimes}_{i\in I} H_i$. If $χ_{Ω^{\rm ut}_{I;\mathbb{C}}}$ is the canonical tracial state on $C^*(Ω^{\rm ut}_{I;\mathbb{C}})$, then $\bar\bigotimes^{\rm mod}_{i\in I} H_i\otimes_{χ_{Ω^{\rm ut}_{I;\mathbb{C}}}}\mathbb{C}$ is a natural dilation of the infinite direct product $\prod {{\otimes}_{i\in I}} H_i$ as defined by J. von Neumann. We will show that the canonical representation of ${\bigotimes}_{i\in I}^{\rm ut} \mathcal{L}(H_i)$ on $\bar\bigotimes^{ϕ_1}_{i\in I} H_i$ is injective. We will also show that if $\{A_i\}_{i\in I}$ is a family of unital Hilbert algebras, then so is ${\bigotimes}_{i\in I}^{\rm ut} A_i$.

math.RA↗

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↗

arXiv2009

Property $(T)$ and strong Property $(T)$ for unital $C^*$-algebras

In this paper, we will give a thorough study of the notion of Property $(T)$ for $C^*$-algebras (as introduced by M.B. Bekka in \cite{Bek-T}) as well as a slight stronger version of it, called "strong property $(T)$" (which is also an analogue of the corresponding concept in the case of discrete groups and type $\rm II_1$-factors). More precisely, we will give some interesting equivalent formulations as well as some permanence properties for both property $(T)$ and strong property $(T)$. We will also relate them to certain $(T)$-type properties of the unitary group of the underlying $C^*$-algebra.

math.OA↗

arXiv2008

An abstract characterization of unital operator spaces

In this article, we give an abstract characterization of the ``identity'' of an operator space $V$ by looking at a quantity $n_{cb}(V,u)$ which is defined in analogue to a well-known quantity in Banach space theory. More precisely, we show that there exists a complete isometry from $V$ to some $\mathcal{L}(H)$ sending $u$ to ${\rm id}_H$ if and only if $n_{cb}(V,u) =1$. We will use it to give an abstract characterization of operator systems. Moreover, we will show that if $V$ is a unital operator space and $W$ is a proper complete $M$-ideal, then $V/W$ is also a unital operator space. As a consequece, the quotient of an operator system by a proper complete $M$-ideal is again an operator system. In the appendix, we will also give an abstract characterisation of ``non-unital operator systems'' using an idea arose from the definition of $n_{cb}(V,u)$.

math.OA↗

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

Some remarks on groupoids and small categories

This unpublished note contains some materials taken from my old study note on groupoids and small categories. It contains a proof for the fact that any groupoid is a group bundle over an equivalence relation. Moreover, the action of a category $G$ on a category $H$ as well as the resulting semi-direct product category $H\times_αG$ will be defined (when either $G$ is a groupoid or $H^{(0)} = G^{(0)}$). If both $G$ and $H$ are groupoids, then $H\times_αG$ is also a groupoid. The reason of producing this note is for people who want to check some details in a recent work of Li.

math.CT↗

arXiv2006

Regular normed bimodules

In this article, we will give a characterization of Banach bimodules over $C^*$-algebras of compact operators that arises from operator spaces as well as a characterization of (F)-Banach bundles amongst all (H)-Banach bundles over a hyper-Stonian space. These two characterizations are concerned with whether certain natural map from a Banach bimodule to its canonical bidual is isometric (we call such bimodule \emph{regular}).

math.OA↗

arXiv2006

On quaternionic functional analysis

In this article, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion $B^*$-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real $C^*$-algebras respectively. We will also give a Riesz representation theorem for quaternion Hilbert spaces and will extend two results of Kulkarni (namely, we will give the full versions of the Gelfand-Naimark theorem and the Gelfand theorem for quaternion $B^*$-algebras). On our way to these results, we compare, clarify and unify the term "quaternion Hilbert spaces" in the literatures.

math.OA↗

arXiv2003

Cohomology of Hopf $C^*$-algebras and Hopf von Neumann algebras

In this article, we will define two canonical cohomology theories for Hopf $C^*$-algebras and for Hopf von Neumann algebras (with coefficients in their bicomodules). We will then study the situations when these cohomologies vanish. The cases of locally compact groups and compact quantum groups will be considered in more details.

math.OA↗

arXiv2002

Invariant ideals of twisted crossed products

We prove a result concerning the inclusion of non-trivial invariant ideals inside non-trivial ideals of a twisted crossed product. We will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on $C^*$-algebras.

math.OA↗

arXiv2002

Duality of Hopf $C^*$-algebras

In this paper, we study the duality theory of Hopf $C^*$-algebras in a general ``Hilbert-space-free'' framework. Our particular interests are the ``full duality'' and the ``reduced duality''. In order to study the reduced duality, we define the interesting notion of Fourier algebra of a general Hopf $C^*$-algebra.

math.OA↗

arXiv2000

Examples of Amenable Kac system

By giving an interesting characterisation of amenable multiplicative unitaries in term of one dimensional representations, we show in a simple way that bicrossproducts of amenable locally compact groups is both amenable and coamenable.

math.OA↗

arXiv1999

Amenability of Hopf C^*-algebras

Three natural definitions for amenability of general Hopf C^*-algebras (all of them being generalizations of the case of locally compact groups) were given and the relations between them were studied. Moreover, amenability in the situation of duality of Hopf C^*-algebras was also studied.

math.OA↗