Mixing and asymptotic properties of Markov semigroups on von Neumann algebras

In the paper, we investigate weak mixing and ‘approach to equilibrium’ of a Markov semigroup, i.e. a semigroup of linear normal positive unital mappings on a von Neumann algebra. In particular, we show that weak mixing is equivalent to ergodictity and triviality of the (point) spectrum of the semigroup, and give conditions assuring that each normal state tends to an equilibrium state under the action of the semigroup. Received May 6, 1999 / in final form October 21, 1999 / Published online July 20, 2000     

Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory

We study a perturbed semilinear problem with Neumann boundary condition where is a bounded smooth domain of , , , if or if and is the unit outward normal at the boundary of . We show that for any fixed positive integer K any “suitable” critical point of the function generates a family of multiple interior spike solutions, whose local maximum points tend to as tends to zero. Received March 7, 1999 / Accepted October 1, 1999 / Published online April 6, 2000     

Capelli identities for classical Lie algebras

We extend the Capelli identity from the Lie algebra to the other classical Lie algebras and . We employ the theory of reductive dual pairs due to Howe. Received: 12 February 1997 / in revised form: 24 July 1998     

Uniqueness of weak solutions of 1+1 dimensional wave maps

In this paper, we prove that finite energy distributional solutions to the Cauchy problem of wave maps from 1+1 dimensional Minkowsky space to any complete Riemannian manifold are unique. Received July 4, 1996; in final form August 25, 1998     

Classification of sequentially weakly continuous $JB^*$-triples

Let D be the open unit ball of a -triple A and let Aut(D) be the group of all biholomorphic automorphisms of D. It is shown that every element of Aut(D) is sequentially weakly continuous if and only if every primitive ideal of A is a maximal closed ideal and is a type I -triple without infinite-spin part. Implications for general structure theory are explored. In particular, it is deduced that every -triple A contains a smallest ideal J for which the sequentially weakly continuous biholomorphic automorphisms of the open unit ball of A/J are all linear. Received August 27, 1998; in final form February 10, 1999     

Spectral synthesis for Banach algebras, II

This paper continues the study of spectral synthesis and the topologies and on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of . Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of . Received: 6 October 1999; in final form: 15 May 2000 / Published online: 25 June 2001     

Unique tensor factorization of algebras

Tensor product decomposition of algebras is known to be non-unique in many cases. But, as will be shown here, a -indecomposable, finite-dimensional -algebra A has an essentially unique tensor factorization into non-trivial, -indecomposable factors . Thus the semiring of isomorphism classes of finite-dimensional -algebras is a polynomial semiring . Moreover, the field of complex numbers can be replaced by an arbitrary field of characteristic zero if one restricts oneself to schurian algebras. Received: 5 October 1998     

A degenerate Neumann problem for quasilinear elliptic integro-differential operators

This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998     

An operator splitting method for nonlinear convection-diffusion equations

Summary. We present a semi-discrete method for constructing approximate solutions to the initial value problem for the -dimensional convection-diffusion equation . The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case , dimensional splitting is used to reduce the -dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions. ReceivedFebruary 1, 1996 / Revised version received June 24, 1996