Showing papers in "Mathematische Annalen in 1976"

Markovian master equations. II

Abstract: We prove some theorems on the behaviour of solutions of master equations in the weak coupling limit, obtaining an exponential decay law under more general conditions than in an earlier paper. As well as applying the theory to a new type of example, we analyse some previously unstudied aspects of the dissipative behaviour.

A new compactification of the Siegel space and degeneration of Abelian varieties. II

Abstract: This is a continuation of the article with the same title appeared in this journal, to the introduction of which we referr the reader for the summary of contents of this article.

Strong Morita Equivalence of Certain Transformation Group C*-Algebras

Abstract: In the course of extending certain results of Takesaki [11], Nielsen obtained [6] a generalization of a theorem of Mackey [5] and Blattner [1] concerning intertwining operators for induced representations. Nielsen's generalization concerns intertwining operators for the restrictions of induced representations to a subgroup. Now in [7] it was shown that the theorem of Mackey and Blattner is a large part of the statement that the inducing process establishes a Morita equivalence, that is, an equivalence between the representation theories [8], for certain C*-algebras related to induced representations. In fact, the equivalence is a strong Morita equivalence in the sense that it is implemented by what is called in [7] an "imprimitivity bimodule". All this suggests that perhaps Nielsen's theorem is a reflection of the fact that certain algebras associated with the situation he considers are strongly Morita equivalent. We show here that this is in fact the case. Specifically, let G be a locally compact group, and let H and K be closed subgroups of G. Let K act on the left on G/H and let C*(K, G/H) be the corresponding transformation group C*-algebra [4]. Similarly, let H act on the right on K\G and let C*(H,K\G) be the corresponding C*-algebra. We show that there is a natural imprimitivity bimodule establishing a strong Morita equivalence between these two C*-algebras. tn particular, it follows that these two algebras have equivalent representation theories. We then show that Nielsen's theorem follows from this. Our proofs are essentially algebraic -- requiring little more measure theory than those facts about Haar measure needed to define transformation group algebras and their representations. In particular, our results give a proof of Nielsen's theorem which avoids the measure-theoretic details involving liftings of measures in which his proof becomes embroiled. Some of the formulas developed here can be used to motivate a considerably more elementary proof of Takesaki's results [11] concerning generalized commutation relations (as extended by Nielsen). But the theorems obtained here are not needed for that purpose, and so this matter will be treated elsewhere [9].

A partial solution of the Pompeiu problem

Abstract: A nonempty bounded open subset D of ℝ n is said to have the Pompeiu property if and only if for every continuous complex-valued function f on ℝ n which does not vanish identically there is a rigid motion σ of ℝ n onto itself — taking D onto σ(D) — such that the integral of f over σ(D) is not zero. This article gives a partial solution of the Pompeiu problem, the problem of finding all sets D with the Pompeiu property. In the special case that D is the interior of a homeomorphic image of an(n−1)-dimensional sphere, the main result states that if D has a portion of an(n−1)-dimensional real analytic surface on its boundary, then either D has the Pompeiu property or any connected real analytic extension of the surface also lies on the boundary of D. Thus, for example, any such region D having a portion of a hyperplane as part of its boundary must have the Pompeiu property, since the entire hyperplane cannot lie in the boundary of the bounded set D.

An increasing sequence of stein manifolds whose limit is not Stein

Abstract: This question was raised in 1933 by Behnke-Thullen [2] in the case when M is an open subset of complex Euclidean space. In the same paper they solved this problem for various special domains M. The problem was solved affirmatively for arbitrary open subsets M in IF" by Behnke-Stein [1, 1938]. K. Stein [4, 1956] proved that M is Stein if each M i is relatively Runge in Mi+ 1. Docquier-Grauert [3, 1960] showed that M is a Stein manifold if there exists a family {Mt},t~[O, 1), of Stein open subsets such that M = U M t and such O ' o_-

Isoperimetric inequalities and the problem of Plateau

Abstract: In this work we treat the following generalized problem of Plateau: Given a rectifiable, closed Jordan curve F in IR 3 and a function H:IR3-,IR we want to find a surface x of the type of the disc which spans F and has mean curvature H(~) in every point ~ on the surface. Representing x as a mapping x:B-~IR 3 of the unit disc B := {(u, v)e IR2:u 2 + v 2 < 1} into IR 3 one can formulate the problem as follows: Find x~ C°(/~, IR3)~C2(B, p3) such that