\medskip\begin{defn}\label{defn:q-comm}
Let $\CL$ be an ortholattice and $p,q\in \CL$.
\smnoind
(a) We say that $p$ \emph{q-commutes} with $q$ if $p\wedge(p\wedge q)' \leq (q\wedge (p\wedge q)')'$.
\smnoind
(b) $p\in \CL$ is said to be \emph{q-central} if it q-commutes with all other elements in $\CL$.
\end{defn}\medskip\begin{defn}\label{defn:quantum top}
Let $\CL$ be an ortholattice.
A \emph{quantum topology} on $\CL$ is a subcollection $\CC\subseteq \CL$ satisfying:
\begin{enumerate}[\ \ S1).]
\item $0, 1\in \CC$;
\item if $\{{p_\lambda}\}_{\lambda\in \Lambda}$ is a family in $\CC$, then $\bigwedge_{\lambda\in \Lambda}p_\lambda$ exists and belongs to $\CC$;
\item if $p$ and $q$ are q-commuting elements in $\CC$ then $p\vee q\in \CC$.
\end{enumerate}
In this case, elements in $\CC$ are said to be \emph{quantum closed}, while elements of the form $p'$ for some $p\in \CC$ are said to be \emph{quantum open}.
\end{defn}\medskip

For a quantum system modeled on the self-adjoint part
$B_\sa$ of a (complex)
$C^*$-algebra
$B$, we introduce the
semi-classical object of Gelfand spectrum for this system as follows.
Let $P(B)$ be the set of all
pure states on
$B$. For any
$\phi,\psi \in P(B)$, we
denote
$\phi\neq_\mo \psi$ if
$\phi$ and
$\psi$ have orthogonal
support projections; i.e.,
$\phi$ and
$\psi$ has zero
transition probability. For any left closed ideal
$L\subseteq B$, we set
\begin{align*}\hull(L) & :=\{\phi\in P(B): \phi(x^*x) = 0, \text{ for every }x\in L \}\\ &= \{\phi\in P(B): L\subseteq \ker \phi \}\end{align*}
(see e.g., (Murphy 1990,
Theorem 5.3.4)). Then
$\hull(L)$ is a q-subset
of $(P(B), \neq_\mo)$, and
the collection of all such q-subsets form a quantum topology
$\CC^B$ on
$(P(B), \neq_\mo)$.

\medskip

The good point for the Gelfand spectrum of a
$C^*$-algebra is that it
captures the self-adjoint part of original algebra up to a Jordan
isomorphism (under a mild assumption), which is good enough for the
consideration of physical structure modeled on the self-adjoint parts of
$C^*$-algebras. Let us
recall that a linear map
$\Gamma$ from the
self-adjoint part of a
$C^*$-algebra
$A$ to that of another
$C^*$-algebra is a
Jordan isomorphism if it preserves the Jordan product; i.e.
$\Gamma(ab + ba)= \Gamma(a)\Gamma(b) + \Gamma(b)\Gamma(a)$
($a,b\in A_\sa$). The
following can be seen as a non-commutative generalization of the Gelfand
theorem:

\medskip\begin{thm2}\label{thm:quantum spec}
Let $A$ and $B$ be two $C^*$-algebras.
Suppose that there is a bijection $\Psi:P(A) \to P(B)$ preserving the q-distinctness relations such that $\CC^B = \big\{\Psi(C): C\in \CC^A \big\}$.
\smnoind
(a) If $A$ has no 2-dimensional irreducible $^*$-representation, then there is a Jordan isomorphism $\Gamma: B_\sa\to A_\sa$ such that $\Psi(\omega) = \omega\circ \Gamma$ ($\omega\in P(A)$).
\smnoind
(b) If $A$ is simple (including the case when $A=\BM_2$), then $A$ and $B$ are either $^*$-isomorphic or $^*$-anti-isomorphic.
\end{thm2}\medskip \section*{References}

Murphy, GJ. 1990. “C* Algebras and Operator Theory.”