Independence of automorphism group, center, and state space of quantum logics (Q1200611)

Let \(L\) be a quantum logic and let \({\mathcal B}(L)\), \({\mathcal G}(L)\) and \({\mathcal S}(L)\) denote the centre of \(L\), the automorphism group of \(L\) and the state space of \(L\). The author proves that given a Boolean algebra \(B\), a group \(G\) and a compact convex set \(S\), one can construct a quantum logic \(L(B,G,S)\) such that \({\mathcal B}(L(B,G,S))=B\), \({\mathcal G}(L(B,G,S))=G\) and \({\mathcal S}(L(B,G,S))=S\). Moreover, he shows that \(L(B,G,S)\) may be required to contain an arbitrary preassigned quantum logic. --- The proof technique is very interesting. The result itself is a highly valuable contribution to the theory of quantum logics and orthomodular structures.