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 .

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: