The spectra of non-elliptic operators

We discuss the spectrum of the minimal operator corresponding to a constant coefficient partial differential operator onL ^p (E ⁿ ). We then study effects on the spectrum by various perturbations.     

Commutativity of automorphisms of subfactors modulo inner automorphisms

We introduce a new algebraic invariant of a subfactor . We show that this is an abelian group and that if the subfactor is strongly amenable, then the group coincides with the relative Connes invariant introduced by Y. Kawahigashi. We also show that this group is contained in the center of in many interesting examples such as quantum subfactors with level , but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors.

     

On inner automorphisms and certain central automorphisms of groups

Let G be a group, let M and N be two normal subgroups of G. We denote by Aut _N ^M (G), the set of all automorphisms of G which centralize G/M and N. In this paper we investigate the structure of a group G in which one of the Inn(G) = Aut _N ^M (G), Aut _N ^M (G) ≤ Inn(G) or Inn(G) ≤ Aut _N ^M (G) holds. We also discuss the problem: “what conditions on G is sufficient to ensure that G has a non-inner automorphism which centralizes G/M and N”.     

Lambda-rings of automorphisms

Under composition, the set of automorphisms of a single algebra forms a group. The topic of this paper is the full set of all automorphisms of all algebra structures in a given abstract class on a fixed finite set. A lambda-ring, known as the automorphism type ring, is associated with this full set of automorphisms, and its structure is described. For certain classes (such as the variety of lattices), the full set of automorphisms is represented up to conjugation in the symmetric group by the automorphisms of a single, so-called representative algebra. In this case, the automorphism type ring is a subring of the ring of class functions on the automorphism group of the representative algebra.     

A note on local automorphisms

Let H be an infinite-dimensional almost separable Hilbert space. We show that every local automorphism of ℬ(H), the algebra of all bounded linear operators on a Hilbert space H, is an automorphism.     

Symplectic automorphisms on Kummer surfaces

Nikulin proved that the isometries induced on the second cohomology group of a K3 surface X by a finite abelian group G of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of H²(X,\mathbbZ){H^2(X,\mathbb{Z})} which is fixed by the isometries induced by G. However for certain groups these discriminants are not the same as those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.     

Fundamental solutions for a class of non-elliptic homogeneous differential operators

We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be constructed in terms of radial averages and invariant distributions on the unit sphere.     

Orbits on the projective line

Let PGL(2, q) act in the natural way on the four-sets and the five-sets in PG(1, q). We determine the number of orbits of any given size and use this to construct some 3-designs on q + 2 points.     

Invertible composition operators on Hp

The weakly closed algebras generated by certain sets of composition operators are shown to be reflexive. A structure theorem for invertible composition operators on H² is obtained and used to show that such operators are reflexive. The structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts. Another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal. Thus the general invariant subspace problem for Hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators.     

Localization principle of automorphisms on generalized pseudoellipsoids

We deduce global biholomorphic equivalence from local biholomorphic equivalence near boundary points in the class of generalized pseudoellipsoids, extending local biholomorphisms to global ones. We discuss related results on holomorphic proper maps.     

Imaginary automorphisms on real hyperelliptic curves

A real hyperelliptic curve X is said to be Gaussian if there is an automorphism such that , where [-1] denotes the hyperelliptic involution on X. Gaussian curves arise naturally in several contexts, for example when one studies real Jacobians. In the present paper, we study the properties of Gaussian curves and we describe their moduli spaces.