Dye’s theorem for C*-algebras

The following is a main result in the paper joint with Chi-Keung Ng Ortho-sets and Gelfand spectra .

\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 $(P(B), \neq_\mo)$.


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.”

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