Compact operators in type III\(_{\lambda}\) and type III\(_0\) factors
For semifinite von Neumann algebras N, the two-sided ideal I(N) generated by the finite projections and its norm closure J(N) can be viewed as the generalization of the ideals of finite rank and compact operators on a Hilbert space. Analogous notions of finiteness and compactness are defined by the authors in the purely infinite case. The approach consists in using the semifiniteness of the crossed product of the algebra by the modular action of a given weight. More precisely, if M is a type \(III_{\lambda}\) factor \((0\leq \lambda <1)\), \(M=N\otimes Z\) where N is of type \(II_{\infty}\). If E is the conditional expectation \(M\to N\), then \[ I(M)=span\{x\in M^+;\quad E(x)\in I(N)\}\quad and\quad J(M)=\overline{I(M)}. \] They are respectively hereditary \({}^*\)- subalgebra and hereditary \(C^*\)-subalgebra of M, and J(M) is the hereditary class defined by J(N). Then the elements of J(M) verify the classical conditions verified by compacts operators on a Hilbert space. Similarly, the \(C_ p(M)\) spaces are introduced, generalizing the trace class operators when \(p=1\), with \(I(M)\subset C_ p(M)\subset J(M)\), and their relation with the \(L_ p(M)\) spaces defined by Haagerup is described.
ideals of finite rank and compact operators on a Hilbert space
semifiniteness of the crossed product of the algebra by the
semifiniteness of the crossed product of the algebra by the modular action of a given weight
scientific article; zbMATH DE number 3949044
scientific article