Anton Stroh

Recurrence and ergodicity in unital ∗-algebras

Abstract Results concerning recurrence and ergodicity are proved in an abstract Hilbert space setting based on the proof of Khintchines 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 Khintchines theorem for ergodic systems, where the image of one set overlaps with another set, instead of with itself.

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 Khintchines 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.

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].

MINIMUM MODULI IN VON NEUMANN ALGEBRAS

Abstract 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 then characterize the semi-Fredholm elements in terms of the points of continuity of the essential minimum modulus function.

An index theorem for Toeplitz operators on totally ordered groups

We show that for every totally ordered group Γ and invertible function f ∈ C(Γ) which does not have a logarithm, there is a representation in which the Toeplitz operator Tf is a Breuer-Fredholm operator with nonzero index; this representation is the GNS-representation associated to a natural unbounded trace on the Toeplitz algebra T (Γ). The Toeplitz algebra T (Γ) of a totally ordered abelian group Γ is the C∗-algebra of operators on the Hardy space H(Γ) generated by the compressions Tf of the multiplication operators Mf for f ∈ C(Γ). When Γ = Z, the Toeplitz operator Tf is Fredholm if and only if f ∈ C(T)−1, and then its Fredholm index is minus the winding number of f about 0. For other Γ, Tf is Fredholm if and only if it is invertible, and to get an interesting index theorem, one has to change one’s concept of Fredholm operator. Coburn, Douglas, Schaeffer and Singer [3] showed that, if Γ is a subgroup of R, there is a representation π of T (Γ) such that π(Tf ) is a Breuer-Fredholm element of the II∞-factor π(T (Γ))′′ whenever f is invertible in C(Γ), and gave a formula for the index. Subsequently Murphy proved a version of this index theorem for more general ordered groups [8], but only a restricted class of Tf with f invertible are Fredholm in his representation. Here we extend Murphy’s result in two directions. First of all, we prove that for each totally ordered abelian group Γ and each invertible f ∈ C(Γ), there is a representation π of T (Γ) in which π(Tf ) is Breuer-Fredholm. Secondly, we show that these representations are the GNS-representations of certain unbounded traces on the Toeplitz algebra T (Γ), thus explaining more clearly how the representations used in [3], [8] are canonically associated to the Toeplitz algebra. Murphy has shown elsewhere [9] that a trace τ on a C∗-algebra B naturally gives rise to an index theory of Fredholm elements of B. This is not obvious: since C∗algebras need not contain projections, the obvious definitions of dimension as the trace of a projection and index as the difference of two dimensions are not available. His elegant solution, which deserves to be much better known, is to declare b ∈ B to be Fredholm if it has a inverse c modulo the ideal Mτ of elements of finite trace, and then define the index of b to be τ(bc− cb). We have couched our index theorem Received by the editors January 13, 1997 and, in revised form, March 11, 1997. 1991 Mathematics Subject Classification. Primary 46L55, 47B35.

Every Banach algebra has the spectral radius property

Nylen and Rodman [NR] introduced the notion of spectral radius property in Banach algebras in order to generalize a classical theorem of Yamamoto on the asymptotic behaviour of the singular values of ann xn matrix. In this paper we prove a conjecture of theirs in the affirmative, namely that any unital Banach algebra has the spectral radius property. In fact a slightly more general spectral property holds. We show that for every element which has spectral points which are not of finite multiplicity, the essential spectral radius is the supremum of the set of absolute values of the spectral points that are not of finite multiplicity.