Volume 2007, Issue 605

We provide explicit, simple, geometric formulas for free involutions ρ of Euclidean spheres that are not conjugate to the antipodal involution. Therefore the quotient S n /ρ is a manifold that is homotopically equivalent but not diffeomorphic to . We use these formulas for constructing explicit non-trivial elements in π 1 Diff( S 5 ) and π 1 Diff( S 13 ) and to provide explicit formulas for non-cancellation phenomena in group actions.

We introduce a notion of C *-symbolic dynamical system, that is a finite family of endomorphisms of a C *-algebra with some conditions. The endomorphisms are indexed by symbols and yield both a nontrivial subshift and a Hilbert C *-bimodule. The associated C *-algebra with the Hilbert C *-bimodule is regarded as a crossed product by the subshift. We also introduce a notion of strong shift equivalence of C *-symbolic dynamical systems. We then prove that if two C *-symbolic dynamical systems are strong shift equivalent, the gauge actions of the crossed product C *-algebras by their subshifts are stably outer conjugate.

We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor ϒ representing this reduction is found to be given by a ternary symmetric form with special properties. A 5-dimensional manifold ( M , g , ϒ) with Riemannian metric g and ternary form generated by such a tensor has a corresponding SO(3) structure, whose Gray-Hervella type classification is established using 𝔰𝔬(3)-valued connections with torsion. Structures with antisymmetric torsions, we call them the nearly integrable SO(3) structures, are studied in detail. In particular, it is shown that the integrable models (those with vanishing torsion) are isometric to the symmetric spaces M + = SU(3)/SO(3), M − = SL(3, R )/SO(3), M 0 = ℝ 5 . We also find all nearly integrable SO(3) structures with transitive symmetry groups of dimension d > 5 and some examples for which d = 5. Given an SO(3) structure ( M , g , ϒ), we define its “twistor space” 𝕋 to be the 𝕊 2 -bundle of those unit 2-forms on M which span ℝ 3 = 𝔰𝔬(3). The 7-dimensional twistor manifold 𝕋 is then naturally equipped with several CR and G 2 structures. The ensuing integrability conditions are discussed and interpreted in terms of the Gray-Hervella type classification.

The starting points for this paper are simple descriptions of the norm and strong * closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the type or cardinality of the algebra. We relate this to several known results and derive some consequences, of which we list a few here. Exactly when the ambient von Neumann algebra is a direct sum of σ-finite algebras, any two normal operators have the same norm-closed unitary orbit if and only if they have the same strong * -closed unitary orbit if and only if they have the same strongclosed unitary orbit. But these three closures generally differ from each other and from the unclosed unitary orbit, and we characterize when equality holds between any two of these four sets. We also show that in a properly infinite von Neumann algebra, the strong-closed unitary orbit of any operator, not necessarily normal, meets the center in the (non-void) left essential central spectrum of Halpern. One corollary is a “type III Weyl-von Neumann-Berg theorem” involving containment of essential central spectra.

In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern-Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely related to the Farey map and to the Gauss map.

We prove that the well-known Anick automorphism (see P. M. Cohn , Free rings and their relations, 2nd ed., Academic Press, London 1985., p. 343) of the free associative algebra F〈x, y, z〉 over an arbitrary field F of characteristic 0 is wild.

We prove that a region D outside a convex set C with smooth boundary in a three-dimensional Cartan-Hadamard manifold M satisfies the relative isoperimetric inequality area, with equality if and only if D is isometric to a Euclidean half ball. We also prove similar sharp inequalities when the sectional curvature of M is bounded above by a negative constant.

M. Rost has proved the existence of an exact sequence relating the group SK 1 —kernel of the reduced norm for K 1 —of a biquaternion algebra D whose center is a field F with Galois cohomology groups of F . In this paper, we relate the group SK 2 —kernel of the reduced norm for K 2 —of D with Galois cohomology of F through an exact sequence.