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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., It is the preprint (before peer review) version of a paper to appear in J. Phys. A. The names for some terminologies are changed in the publised version; in particular, "quantum sets" are now called "ortho-sets" to avoid confusion with existing terminologies

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

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 B\'edos, 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., Comment: 9 pages; a new co-author; added to Theorem 3.2 a second part about the case of a trivial scaling group, and gave an alternative proof of one of the implications; to appear in the Bulletin of the London Mathematical Society

Let $1\leq p \leq +\infty$. We show that the positive part of the closed unit ball of a non-commmutative $L^p$-space, as a metric space, is a complete Jordan $^*$-invariant for the underlying von Neumann algebra., 8 pages

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., Comment: 12 pages; accepted for publication in Int. J. Math

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., Comment: 24 pages, accepted for publication in Studia Mathematica

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., 41 pages; we added a lot of details and some new results

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 $\mathfrak{A}$., Comment: 28 pages. The appendix in the original version of this paper was removed, and was then expanded to another paper (in Chinese). The current version of this paper is appeared in the proceedings of the conference: "Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics", took place in Herrnhut, Germany, in honor of Charles Batty for his 60th birthday

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 $\Phi : E\to F$ is an $A$-module map, not assumed to be bounded but satisfying $$ \langle \Phi(x),\Phi(y)\rangle_A\ =\ 0\quad\text{whenever}\quad\langle x,y\rangle_A\ =\ 0, $$ then there exists a unique central positive multiplier $u\in M(I_E)$ such that $$ \langle \Phi(x), \Phi(y)\rangle_A\ =\ u \langle x, y\rangle_A\qquad (x,y\in E). $$ As a consequence, $\Phi$ is automatically bounded, the induced map $\Phi_0: E\to \overline{\Phi(E)}$ is adjointable, and $\overline{Eu^{1/2}}$ is isomorphic to $\overline{\Phi(E)}$ as Hilbert $A$-modules. If, in addition, $\Phi$ is bijective, then $E$ is isomorphic to $F$., Comment: 15 pages

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

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 $n_{cb}(V,u)$., Comment: Some remarks were added

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 $H\times_\alpha G$ will be defined (when either $G$ is a groupoid or $H^{(0)} = G^{(0)}$). If both $G$ and $H$ are groupoids, then $H\times_\alpha G$ 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., Comment: 6 pages, unpublished note

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})., to appear in Journal of Operator Theory

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., Comment: The first statement of Remark 1.9(a) in the original published article (in Proc. LMS) does not follow directly from Lemma 1.7(a) but it does follow very easily from the argument of Proposition 1.10. Therefore, we change the presentations of Remark 1.9 and Proposition 1.10 (one line is added in the proof of Proposition 1.10)

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.

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., some changes in major terminology to avoid confusion

In this paper, we will give a natural definition for morphisms between multiplicative unitaries. We will then discuss some equivalences of this definition and some interesting properties of them. Moreover, we will define normal sub-multiplicative unitaries for multiplicative unitaries of discrete type and prove an imprimitivity type theorem for discrete multiplicative unitaries., Comment: 21 pages, Latex; to appear in the Journal of Operator Theory

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 $\sigma$-unital C*-algebras do not have property (T_e)., Comment: 7 pages; to appear in J. Math. Anal. Appl