Automorphisms of the UHF algebra that do not extend to the Cuntz algebra

Automorphisms of the canonical core UHF-subalgebra F_n of the Cuntz algebra O_n do not necessarily extend to automorphisms of O_n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras M_n. In that case, necessary and sufficient conditions for the extension property are presented. It is also addressed the problem of extending to O_n the automorphisms of the diagonal D_n, which is a regular MASA with Cantor spectrum. In particular, it is shown the existence of product-type automorphisms of D_n that are not extensible to (possibly proper) endomorphisms of O_n

Conjugacy classes of affine automorphisms of K^n and linear automorphisms of P^n in the Cremona groups

We describe the conjugacy classes of affine automorphisms in the group Aut(n,\K) (respectively Bir(\K^n)) of automorphisms (respectively of birational maps) of \K^n. From this we deduce also the classification of conjugacy classes of automorphisms of \Pn in the Cremona group Bir(\K^n).Comment: 17 pages, no figure

On the Cone conjecture for Calabi-Yau manifolds with Picard number two

Following a recent work of Oguiso, we calculate explicitly the groups of automorphisms and birational automorphisms on a Calabi-Yau manifold with Picard number two. When the group of birational automorphisms is infinite, we prove that the Cone conjecture of Morrison and Kawamata holds.Comment: to appear in Math. Res. Let

Directions of automorphisms of Lie groups over local fields compared to the directions of Lie algebra automorphisms

To each totally disconnected, locally compact topological group G and each group A of automorphisms of G, a pseudo-metric space of ``directions'' has been associated by U. Baumgartner and the second author. Given a Lie group G over a local field, it is a natural idea to try to define a map from the space of directions of analytic automorphisms of G to the space of directions of automorphisms of the Lie algebra L(G) of G, which takes the direction of an analytic automorphism of G to the direction of the associated Lie algebra automorphism. We show that, in general, this map is not well-defined. However, the pathology cannot occur for a large class of linear algebraic groups (called ``generalized Cayley groups'' here). For such groups, the assignment just proposed defines a well-defined isometric embedding from the space of directions of inner automorphisms of G to the space of directions of automorphisms of L(G). Some counterexamples concerning the existence of small joint tidy subgroups for flat groups of automorphisms are also provided.Comment: 20 pages, LaTe