Compact operators in type \(III_{\lambda}\) and type \(III_ 0\) factors. I
This paper continues the program initiated in Math. Ann. 273, 251-270 (1986; Zbl 0591.46053) of studying notions of compact operators in type \(III_{\lambda}\) \((0\leq \lambda <1)\) factors M represented as crossed products of a type \(II_{\infty}\) algebra M by an automorphism \(\theta\). E being the canonical normal conditional expectation from M onto N, the compact operator ideal J will be the norm closure of the finite rank operators \(I=span \{x\in M^+;E(x)\in\) two sided ideal of N generated by finite projections\(\}\), while \(K=span \{x\in M^+;E(x)\in\) norm closed two sided ideal of N generated by finite projections\(\}\). The algebra K is characterized in terms of essential central range of its elements. In particular \(x\in K^+\) if and only if N' meets the \(\sigma\)-weak closure of the convex hull of \(\{vxv^*\); v unitary in \(N\}\) only in \(\{\) \(0\}\). The projections p in N such that \(p\theta^ n(p)=0\) for all \(n\neq 0\) are studied and it is shown that every non zero projection majorizes such a non zero projection. This allows to introduce an isomorphism \(\rho\) of \(B(\ell^ 2({\mathbb{Z}}))\) onto a type I factor \(D\subset M\) with matrix units and to characterize the positive part of the intersection of D with the different ideals of compact operators introduced earlier. In particular it is shown that \(J\neq K\) and is not an ideal of K.
canonical normal conditional expectation
scientific article; zbMATH DE number 4019793
scientific article