Automorphisms Of Products Research Articles

A structural result of irreducible inclusions of type 𝐼𝐼𝐼_{𝜆} factors, 𝜆∈(0,1)

Given an irreducible inclusion of factors with finite index N ⊂ M N\subset M , where M M is of type I I I λ 1 / m {III}_{\lambda ^{1/m}} , N N of type I I I λ 1 / n {III}_{\lambda ^{1/n}} , 0 > λ > 1 0>\lambda >1 , and m , n m,n are relatively prime positive integers, we will prove that if N ⊂ M N\subset M satisfies a commuting square condition, then its structure can be characterized by using fixed point algebras and crossed products of automorphisms acting on the middle inclusion of factors associated with N ⊂ M N\subset M . Relations between N ⊂ M N\subset M and a certain G G -kernel on subfactors are also discussed.

Formula omitted]n(A) for a Class of Polynomial Automorphisms and Stably Tameness

In this paper we introduce a set, denoted by Dn(A), for every commutative ringAand every positive integern. It is shown that the elements of this set can be used to give an explicit description of the class Hn(A) introduced in6. We deduce that each polynomial map of the formF=X+HwithH∈Hn(A) can be written as a finite product of automorphisms of the form exp(D), where eachDis a locally nilpotent derivation satisfyingD2(Xi)=0 for alli. Furthermore we deduce that all suchFs are stably tame.

Reduced Free Products of Completely Positive Maps and Entropy for Free Product of Automorphisms

The reduced free product of unital completely positive maps is defined. An invariant state on a C *-algebra for an automorphism (based on the free shift) is the composition of a state of a subalgebra with the reduced free product of expectations. Entropies for the reduced free product and the tensor product of an automorphism γ with the free shift coincide with the entropy of γ .

A comparison of length definitions for maps of modules over local rings

LetM be a finitely generated free module over a local ring. An automorphism α ofM can be written as a product of automorphisms that are elements of a given generating groupG of the set of all automorphisms ofM. The minimal number of elements required for such a product is called the length ofα. We study two decompositions using similar generating groups and compare the two resulting lengths.

Smooth Markov partitions and toral automorphisms

AbstractWe show that the only hyperbolic toral automorphisms f for which there exist Markov partitions with piecewise smooth boundary are those for which a power fk is linearly covered by a direct product of automorphisms of the 2-torus. Only a finite number of shapes occur in a certain natural set of cross-sections of the partition boundary. The behavior of the stratified structure of a piecewise smooth boundary under the mapping forces these shapes to be self-similar. This, together with expanding properties of the mapping, means that a piecewise smooth partition is in fact piecewise linear. Orbits of affine disks in the boundary are used to construct a basis of 2-dimensional invariant toral subgroups, and then the product decomposition of a covering follows easily.

Actions of finite groups on graphs and related automorphisms of free groups

The proof will be completed in Section 6. It is geometrical and the geometrical objects which will be used are graphs on which finite groups act (G-graphs). The following lines are intended to describe the structure of the proof with the main intermediate results. The translation into geometrical language is made possible by the realization theorem of Culler [S] which states that every finite subgroup of Out @ is realized by automorphisms of a graph (see Section 3). Now if G is a finite subgroup of Out Cp and r a graph realizing G then the centralizer of G can be identified with the group Out, n,(T, *) of equivariant conjugacy classes of automorphisms of rc,(f, *). We shall turn to fundamental groupoids and ‘prove with some effort (Sections 4 and 5) that if the G-graph r is suitably chosen (“reduced”) then the natural homomorphism Aut, Z7(r) + Out, z,(& *) is surjective. This way the problem reduces to the more tractable group Aut, n(r) of equivariant automorphisms of the free groupoid n(r). That this group is finitely generated will be proved in Section 6 by an extension of the Nielsen method in free groups. A peculiarity which occurs in the study of Aut, IZ(IJ is that we are forced to consider simultaneously a number of graphs related to r and all isomorphisms between fundamental groupoids of these graphs. In the classical situation of automorphisms of a free group (G trivial, r a wedge of loops) one proves that every automorphism of @ (i.e., of n(r)) can be expressed as a product of Nielsen automorphisms. What we have in the general situation is that (under certain circumstances described in Sec-

An elementary proof of the automorphism theorem for the polynomial ring in two variables

If k is a field, then the automorphism theorem for k[ x, y] states that every k-algebra automorphism of k[ x, y] is a finite compositional product of automorphisms of the type: (i) x↦ x, y↦ y+ h( x) with h(x) ∈ k[x] ; or (ii) x↦ a 11 x+ a 12 y+ a 13, y↦ a 21 x+ a 22 y+ a 23 with a 11 a 22≠ a 21 a 12 and a ij∈ k . The proof presented in this paper, in the case of k being the complex numbers, uses the resultant formula for the inverse of an automorphism (McKay and Wang) and a differential equation associated with the fact that the Jacobian of the automorphism is a nonzero constant. Then the required divisibility condition on the degrees of the images of x and y follows from the fact that this differential equation must have a rational function solution.

Automorphisms of a free associative algebra of rank 2. II

Let R R be a commutative domain with 1. R ⟨ x , y ⟩ R\langle x,y\rangle stands for the free associative algebra of rank 2 over R ; R [ x ~ , y ~ ] R;R[\tilde x,\tilde y] is the polynomial algebra over R R in the commuting indeterminates x ~ \tilde x and y ~ \tilde y . We prove that the map Ab : Aut ⁡ ( R ⟨ x , y ⟩ ) → Aut ⁡ ( R [ x ~ , y ~ ] ) \text {Ab}: \operatorname {Aut} (R\langle x,y\rangle ) \to \operatorname {Aut} (R[\tilde x,\tilde y]) induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [5], Nagata [7] and van der Kulk [8]* that describes the automorphisms of F [ x ~ , y ~ ] F[\tilde x,\tilde y] ( F F a field) we are able to conclude that every automorphism of F ⟨ x , y ⟩ F\langle x,y\rangle is tame (i.e. a product of elementary automorphisms).