Recurrence and ergodicity in unital *-algebras

Results concerning recurrence and ergodicity are proved in an abstract Hilbert space setting based on the proof of Khintchine's recurrence theorem for sets, and on the Hilbert space characterization of ergodicity. These results are carried over to a non-commutative *-algebraic setting using the GNS-construction. This generalizes the corresponding measure theoretic results, in particular a variation of Khintchine's Theorem for ergodic systems, where the image of one set overlaps with another set, instead of with itself.Comment: 16 page

Noncommutative recurrence over locally compact Hausdorff groups

We extend previous results on noncommutative recurrence in unital *-algebras over the integers, to the case where one works over locally compact Hausdorff groups. We derive a generalization of Khintchine's recurrence theorem, as well as a form of multiple recurrence. This is done using the mean ergodic theorem in Hilbert space, via the GNS construction.Comment: 11 page

Minimum Moduli in Von Neumann Algebras

In this paper we answer a question raised in [12] in the affirmative, namely that the essential minimum modulus of an element in a von Neumann algebra, relative to any norm closed two-sided ideal, is equal to the minimum modulus of the element perturbed by an element from the ideal. As a corollary of this result, we extend some basic perturbation results on semi-Fredholm elements to a von Neumann algebra setting. We can characterize the semi-Fredholm elements in terms of the points of continuity of the essential minimum modulus function. Mathematics Subject Classification (2000): 46L Keywords: algebra, selfadjoint operator algebras, Fredholm theory, von Neumann algebra, minimum modulus, semi-Fredholm Quaestiones Mathematicaes 24 (4) 2001, 493–50