The purpose of this paper is to construct an embedding of the C * -algebra of the Cartan motion group of a real reductive group Ginto the reduced C * -algebra of G itself. The embedding has a number of applications: we shall use it to characterize the Mackey bijection from the tempered dual of G into the unitary dual of the motion group; to characterize the continuous field of reduced group C * -algebras arising from the contraction of G to its Cartan motion group; and to characterize the Connes-Kasparov assembly map in operator K-theory. The results presented here continue and complete a project that was begun several years ago by the last two authors, who considered the case of complex groups. In the real case, detailed information from the theory of R-groups is used in the construction.
We prove several new rigidity results for polynomial automor-phisms ofC(2)with positive entropy. A first result is that a complexslice of the (forward or backward) Julia set is never a smooth, oreven rectifiable, curve. We also show that such an automorphismcannot preserve a global holomorphic foliation, nor a real-analyticfoliation with complex leaves. These results are used to show that under mild assumptions,two real-analytically conjugate automorphisms are polynomiallyconjugate. The properties of multipliers of saddle periodic orbits play animportant role in these results. We also study some arithmeticquestions related to these multipliers.
On a smooth manifold, we associate to any closed differential form a mapping cone complex. The cohomology of this mapping cone complex can vary with the de Rham cohomology class of the closed form. We present a novel Morse theoretical description for the mapping cone cohomology. Specifically, we introduce a Morse complex for the mapping cone complex which is generated by pairs of critical points with the differential defined by gradient flows and an integration of the closed form over spaces of gradient flow lines. We prove that the cohomology of our cone Morse complex is isomorphic to the mapping cone cohomology and hence independent of both the Riemannian metric and the Morse function used to define the complex. We also obtain sharp inequalities that bound the dimension of the mapping cone cohomology in terms of the number of Morse critical points and the properties of the specified closed form. Our results are widely applicable, especially for any manifold equipped with a geometric structure described by a closed differential form. We also obtain a bound on the difference between the number of Morse critical points and the Betti numbers.
We study the algebraic K-theory of the ring of continuous functions on a compact Hausdorff space with values in a local divisionring, e.g., a local field: We compute its negativeK-theory and show its K-regularity. The complex case reproves the results of Rosen-berg, Friedlander-Walker, and Cortinas-Thom. Our consideration in the real case proves two previously unconfirmed claims made by Rosenberg in 1990. The algebraic nature of our methods enables usto deal with the nonarchimedean and noncommutative cases analogously
