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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 $\Phi(f)(y) = f(\tau(y))$ ($f\in C(X)_+^\mathrm{sp}; y\in Y$).

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

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

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

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