Eigenvalues and eigenspaces of quantum dynamical systems and their tensor products (Q1269557)

The author studies here a dynamical system \(({\mathfrak m},\alpha,\omega)\). \({\mathfrak m}\): a von Neumann algebra, \(\alpha\): a representation of an Abelian semigroup \(G\) into the set of linear normal positive identity-preserving mappings on \({\mathfrak m}\), and \(\omega\): a normal faithful \(\alpha\)-invariant state on \({\mathfrak m}\). A complex-valued function \(f(g)\) with \(| f(g)|= 1\), \(g\in G\) is called an eigenvalue of \(\alpha\), if there exists a nonzero \(x\in{\mathfrak m}\) such that \(\alpha_gx= f(g)x\), \(\forall g\in G\). The spectrum \(\sigma(\alpha)\) is the set of all eigenvalues of \(\alpha\). The eigenspace associated to \(f\) is \({\mathfrak m}_f= \{x\in{\mathfrak m}:\alpha_gx= f(g)x\}\). (I) For each \(f\in\sigma(\alpha)\), there exists a normal norm one projection \(\varepsilon_f\) from \({\mathfrak m}\) onto \({\mathfrak m}_f\). \(\varepsilon_f\) is the unique normal projection in \(\overline{[\text{Conv}\{\overline{f(g)}\cdot \alpha_g: g\in G\}]}\) satisfying \(\varepsilon_f\alpha_g= \alpha_g\varepsilon_f= f(g)\varepsilon_f\). (II) Let \(({\mathfrak m}^{(i)}, \alpha^{(i)}, \omega^{(i)})\), \(i= 1,2\) be two completely positive dynamical systems on Abelian semigroups \(G^{(i)}\), \(i= 1,2\). Then \(\sigma(\alpha^{(1)}\otimes \alpha^{(2)})= \sigma(\alpha^{(1)})\otimes \sigma(\alpha^{(2)})\), and \(\sigma(\alpha')= \{f^{(1)}\cdot f^{(2)}: f^{(i)}\in \sigma(\alpha^{(i)}), i=1,2\}\), \(\alpha_g'= \alpha_g^{(1)}\otimes \alpha_g^{(2)}\) for \(\forall g\in G^{(1)}= G^{(2)}\), hold.