Extensions of factorial states of \(C^*\)-algebras
Let B be a \(C^*\)-subalgebra of a \(C^*\)-algebra A and let \(\phi\) be a factorial state of B. Methods due to Sakai and Tsui, involving tensor products or weak expectations, can sometimes be used to extend \(\phi\) to factorial states of A. These two methods are shown to be applicable in precisely the same circumstances and to yield the same factorial extensions. If \(\phi\) is a type I factorial state then the methods give rise to type I factorial states of A which are extremal extensions of \(\phi\). If \(\phi\) is type III then the methods produce type III extensions. However, if \(\phi\) is type II then the methods yield extensions which may in general be either of type II or of type III. Examples are given to show that both possibilities can occur in isolation. The second-named author has recently obtained related results in the context of \(C^*\)-dynamical systems [J. Functional Analysis, to appear]. Longo and Popa have separately shown that a generalization of the tensor product method, suggested by Sakai, can always be applied when B is separable to obtain a factorial extension of \(\phi\). It seems conceivable that this method might sometimes give a type II extension of a type III state \(\phi\).
scientific article; zbMATH DE number 3921284
scientific article