Clustering for algebraic \(K\)-systems
A von Neumann \(K\)-system \(({\mathcal A}_ n, \sigma, \omega)\) is a family of subalgebras \({\mathcal A}_ n\), \(n\in \mathbb{Z}\), of an algebra \({\mathcal A}\) with the properties (i) \({\mathcal A}_{n+1} \supset {\mathcal A}_ n\), (ii) \(\cup_ n ({\mathcal A}_ n)= {\mathcal A}\), (iii) \(\cap_ n ({\mathcal A}_ n)= z1\), and an automorphism \(\sigma\) of \({\mathcal A}\) with \(\sigma ({\mathcal A}_ n)= ({\mathcal A}_{n+1})\) and the invariant state \(\omega\). The main results. Theorem (3.1). Let \(({\mathcal A}_ n, \sigma, \omega)\) be a von Neumann \(K\)-system. Then \(\forall b\in {\mathcal A}\), \(n\in \mathbb{Z}\), \(\varepsilon>0\) \(\exists N(b,n, \varepsilon)\in \mathbb{Z}^ +\) such that \(\forall a\in {\mathcal A}_ n\), \(k<-N\): \(| \omega(b \sigma^ k a)- \omega(b) \omega(a) |< \varepsilon\| a\|\). (This clustering we call \(K\)- clustering.) Theorem (3.4). A von Neumann \(K\) system is \(r\)-clustering, i.e. \(\forall r\in \mathbb{N} \lim_{\inf| n_ 1- n_ k|\to \infty} \omega (\sigma_{n_ 1} \alpha_ 1 \dots \sigma_{n_ r} a_ r)= \omega (a_ 1) \dots \omega (a_ r)\). Theorem (3.1) has a sort of converse: Theorem (3.1). Let \({\mathcal A}\) be a \(C^*\) algebraic \(K\)-system with respect to the automorphism \(\sigma\) and \(\omega\) a \(\sigma\)-invariant separating state, that is \(K\)-clustering with respect to the \(C^*\) system \(\cup_ n ({\mathcal A}_ n)= {\mathcal A}\). Then it is also \(K\)- clustering for \({\mathcal A}''= \cup_ n ({\mathcal A}_ n^{\prime\prime})\) and \(\cap_ n ({\mathcal A}_ n^{\prime\prime})= \lambda1\). Thus \(K\)- clustering ensures that a \(C^*\)-\(K\)-system entends to a von Neumann \(K\) system and is also necessary.
scientific article; zbMATH DE number 554097
scientific article