Contractions in Von Neumann Algebras

We associate to any contractionT(and, more generally, to any operatorTof classCρ) in a von Neumann algebraMan operator kernelKα(T) (|α|<1) which allows us to define various kinds of functional calculis forT. WhenMis finite, we use this kernel to give a short proof of the Fuglede-Kadison theorem on the location of the trace and to prove that a contractionTinMis unitary if and only if its spectrum is contained in the unit circle. By using a perturbation of the kernelKα(T) we give, for any operatorTof classCρacting on a separable Hilbert spaceH, a short proof of the power inequality for the numerical range and an accurate conjugacy (to a contraction) result forT. We also get a generalized von Neumann inequality which gives a good control of ‖f(rT*)x+g(rT)x‖ (0⩽r<1) forx∈Handf,gin the disc algebra. Finally, we associate to anyC1contraction in a Hilbert space an asymptotic kernel which allows us to describe new kinds of invariant subspaces forT, from the positive solutionsXof the operator equationT*XT=X. In particular, we recover some results of Beauzamy based on the notion of invariant subspace of “functional type.”