A mean ergodic theorem of an amenable group action

We consider a sequence of weak Kadison–Schwarz maps τn on a von-Neumann algebra ℳ with a faithful normal state ϕ sub-invariant for each (τn, n ≥ 1) and use a duality argument to prove strong convergence of their pre-dual maps when their induced contractive maps (Tn, n ≥ 1) on the GNS space of (ℳ, ϕ) are strongly convergent. The result is applied to deduce improvements of some known ergodic theorems and Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space ℳ*.

On characterization of integrable sesquilinear forms

We give a necessary and sufficient condition for a sesquilinear form to be integrable with respect to a faithful normal state on a von Neumann algebra.

INTERSECTING JONES PROJECTIONS

Let M be a von Neumann algebra on a Hilbert space ℋ with a cyclic and separating unit vector Ω and let ω be the faithful normal state on M given by ω(·) = (Ω,·Ω). Moreover, let {Ni : i ∈ I} be a family of von Neumann subalgebras of M with faithful normal conditional expectations Ei of M onto Ni satisfying ω = ω ◦ Ei for all i ∈ I and let N = ∩i∈I Ni. We show that the projections ei, e of ℋ onto the closed subspaces [Formula: see text] and [Formula: see text] respectively satisfy e = ∧i∈I ei. This proves a conjecture of Jones and Xu in [1].

Conditional expectation and stochastic integrals in non-commutative Lp spaces

The aim of the paper is to propose a general scheme for the consideration of non-commutative stochastic integrals. The role of a probability space is played by a couple (, φ0), where is a von Neumann algebra and φ0 is a faithful normal state on . Our processes live in the algebra of all measurable operators associated with the crossed product of by the modular automorphism group The algebra contains all the (Haagerup's) Lp spaces over . The measure topology of the algebra has the nice feature of inducing the Lp norm topology on the Lp spaces, which makes it particularly suitable for defining stochastic integrals. The commutative theory fits smoothly into the scheme, although there exists no canonical way of embedding the algebra of (commutative) random variables into . In fact, for any commutative stochastic process we have a family of different non-commutative stochastic processes corresponding to the process. This arbitrariness seems to be quite natural in the non-commutative context. An appropriate example can be found at the end of the paper (Section 6, C4).