Product Of Symmetries Research Articles

Length problems for automorphisms of modules over local rings

Let R be a local ring and M a free module of a finite rank over R. An element τ ∈ Aut R M is said to be simple if τ ≠ 1 fixes a hyperplane of M. We shall show that for any σ ∈ Aut R M there exist a basis X for M and ρ ∈ Aut R M such that ρ acts as a permutation on X and ρ −1 σ is a product of m or less than m simple elements in Aut R M, where m is the order of the invariant factors of σ modulo the maximal ideal of R. Also we shall investigate the problem treated by E.W. Ellers and H. Ishibashi [Factorizations of transformations over a valuation ring, Linear Algebra Appl. 85 (1987) 17–27], in which they showed that σ is a product of simple elements and gave an upper bound of the smallest number of such factors of σ, whereas in the present paper we will give lower bounds of σ in case that R is a local domain. Moreover we will factorize θσ as a product of symmetries and transvections for some θ the matrix of which is diagonal.

Products of Symmetries in Unitary Groups

Products of Symmetries in Unitary Groups

Products of symmetries in unitary groups

Given a regular −-hermitian form on a finite-dimensional vector space V over a commutative field K of characteristic ≠2 such that the norm on K is surjective onto the fixed field of − (this is true whenever K is finite). Call an element σ of the unitary group a symmetry if σ 2 = 1 and the negative space of σ is 1-dimensional. If π is unitary and det π ∈ {1, − 1}, we prove that π is a product of symmetries (with a few exceptions when K = GF9 and dim V = 2) and we find the minimal number of factors in such a product.

A Unitary as a Product of Symmetries

It was proved by Fillmore that a unitary of a properly infinite von Neumann algebra A can be expressed as a product of at most four symmetries. In this paper we introduce an axiom (ENCP) for Baer $^ \ast$-rings and prove that Fillmore’s result is true if A is a properly infinite Baer $^ \ast$-ring satisfying (ENCP) and $LP \sim RP$. This also affirmatively answers the open problem on $A{W^ \ast }$-algebras posed by Berberian.

A unitary as a product of symmetries

It was proved by Fillmore that a unitary of a properly infinite von Neumann algebra A can be expressed as a product of at most four symmetries. In this paper we introduce an axiom (ENCP) for Baer *-rings and prove that Fillmore's result is true if A is a properly infinite Baer *-ring satisfying (ENCP) and LP RP. This also affirmatively answers the open problem on AW*algebras posed by Berberian.

Generators of Orthogonal Groups over Valuation Rings

Let be a valuation ring with unit element, i.e., is a commutative ring such that for any a and b in , either a divides b or b divides a. We assume 2 is a unit of . V is an n-ary nonsingular quadratic module over , O(V) or On(V) is the orthogonal group on V, and S is the set of symmetries in O(V). We define l(σ) to be the minimal number of factors in the expression of a of O(V) as a product of symmetries on V. For the case where is a field, l(σ) has been determined by P. Scherk [6] and J. Dieudonné [1]. In [3] I have generalized the results of Scherk to orthogonal groups over valuation domains. In the present paper I generalize my results of [3] to orthogonal groups over valuation rings.Since is a valuation ring, it is a local ring with the maximal ideal A which consists of all nonunits of .

On Products of Symmetries

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebraA(i.e., one with no finite non-zero central projections), then the symmetries may be chosen fromA.A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

Products of symmetries

Products of symmetries