Automorphisms of G-Azumaya Algebras

1985 ◽  
Vol 37 (6) ◽  
pp. 1047-1058 ◽  
Author(s):  
Margaret Beattie
Keyword(s):  
Finite Abelian Group ◽  
Equivalence Classes ◽  
Usual Sense ◽  
Smash Product ◽  
Azumaya Algebras ◽  
Azumaya Algebra ◽  
Linear Automorphism ◽  
Inner Automorphism ◽  
Image Position ◽  
Inner Automorphisms

Let R be a commutative ring, G a finite abelian group of order n and exponent m, and assume n is a unit in R. In [10], F. W. Long defined a generalized Brauer group, BD(R, G), of algebras with a G-action and G-grading, whose elements are equivalence classes of G-Azumaya algebras. In this paper we investigate the automorphisms of a G-Azumaya algebra A and prove that if Picm(R) is trivial, then these automorphisms are all, in some sense, inner.In fact, each of these “inner” automorphisms can be written as the composition of an inner automorphism in the usual sense and a “linear“ automorphism, i.e., an automorphism of the typewith r(σ) a unit in R. We then use these results to show that the group of gradings of the centre of a G-Azumaya algebra A is a direct summand of G, and thus if G is cyclic of order pr, A is the (smash) product of a commutative and a central G-Azumaya algebra.

The finite inner automorphism groups of division rings

1995 ◽  
Vol 118 (2) ◽  
pp. 207-213 ◽  
Author(s):  
M. Shirvani
Keyword(s):  
Finite Groups ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Commutative Field ◽  
Division Rings ◽  
Inner Automorphism ◽  
Projective Representations ◽  
Inner Automorphisms ◽  
A Finite Group

Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

On the solution of certain integral equations by generalized functions

1961 ◽  
Vol 57 (4) ◽  
pp. 767-777 ◽  
Author(s):  
J. B. Miller
Keyword(s):  
Integral Equations ◽  
Generalized Functions ◽  
Generalized Solutions ◽  
Generalized Function ◽  
Usual Sense ◽  
Inverse Transformation ◽  
Extended Sense ◽  
Image Position

This note is concerned with a method by which generalized solutions can be shown to exist for certain types of integral equationswhere f(x) ∈ L2(0, ∞). The method is briefly this. An extended meaning is given to such equations by using generalized functions of a particular type. Then, if (1) denotes a transformation which has no everywhere-defined inverse in the usual sense, it may be possible to define in the extended sense an inverse transformationso that if f is given, a generalized function g = can be determined as a solution of (1).

Finite bang-bang controllability for certain non-linear systems

1977 ◽  
Vol 77 (1-2) ◽  
pp. 137-144 ◽  
Author(s):  
Gunnar Aronsson
Keyword(s):  
Control System ◽  
Sufficient Conditions ◽  
Linear Control ◽  
Usual Sense ◽  
Unperturbed System ◽  
Control Functions ◽  
Bounded Controls ◽  
Non Linear ◽  
Non Linear Systems ◽  
Image Position

SynopsisThis paper gives sufficient conditions ensuring that a non-linear control system of the formis controllable by means of control functions u(t), such that each ui(t) only takes two values, with a finite number of switches. It is assumed that the ‘unperturbed’ system ẋ = A(t)x + B(t)u is controllable in the usual sense, i.e. by measurable and bounded controls.

ON AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE UNIVERSAL ALGEBRA

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1085-1106 ◽  
Author(s):  
G. MASHEVITZKY ◽  
B. I. PLOTKIN
Keyword(s):  
Lie Algebras ◽  
Universal Algebra ◽  
Automorphism Groups ◽  
Endomorphism Semigroup ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Inner Automorphism ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

Open induction and the true theory of rationals

1987 ◽  
Vol 52 (3) ◽  
pp. 793-801
Author(s):  
Zofia Adamowicz
Keyword(s):  
Linear Space ◽  
Usual Sense ◽  
Saturated Model ◽  
The Real ◽  
Maximal Subset ◽  
True Theory ◽  
Real Closure ◽  
Open Induction ◽  
Infinite Sets ◽  
Image Position

In the paper we prove the following theorem:Theorem. There is a model N of open induction in which the set of primes is bounded and N is such that its field of fractions 〈N*, +, ·, <〉 is elementarily equivalent to 〈Q, +, ·, <〉 (the standard rationals).We fix an ω1-saturated model 〈M, +, ·, <〉 of PA. Let 〈M*, +, ·, <〉 denote the field of fractions of M. The model N that we are looking for will be a substructure of 〈M*, +, ·, <〉.If A ⊆ M* then let Ā denote the ring generated by A within M*, Ậ the real closure of A, and A* the field of fractions generated by A. We haveLet J ⊆ M. Then 〈M*, +, ·〉 is a linear space over J*. If x1,…,xk ∈ M*, we shall say that x1,…,xk are J-independent if 〈1, x1,…, xk〉 are J*-independent in the usual sense. As usual, we extend the notion of J-independence to the case of infinite sets.If A ⊆ M* and X ⊆ A, then we say that X is a J-basis of A if X is a maximal subset of A which is J-independent.Definition 1.1. By a J-form ρ we mean a function from (M*)k into M*, of the formwhere q0,…, qk ∈ J*If υ ∈ M, we say that ρ is a υ-form if the numerators and denominators of the qi's have absolute values ≤ υ.

scholarly journals Ext and OrderExt Classes of Certain Automorphisms of C*-Algebras Arising from Cantor Minimal Systems

2001 ◽  
Vol 53 (2) ◽  
pp. 325-354 ◽  
Author(s):  
Hiroki Matui
Keyword(s):  
Transformation Group ◽  
Minimal System ◽  
Full Group ◽  
C Algebras ◽  
Inner Automorphism ◽  
Inner Automorphisms ◽  
Cantor Minimal Systems

AbstractGiordano, Putnam and Skau showed that the transformation group C*-algebra arising from a Cantor minimal system is an AT-algebra, and classified it by its K-theory. For approximately inner automorphisms that preserve C(X), we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

Minimal Operators for Schrodinger-Type Differential Expressions with Discontinuous Principal Coefficients

1980 ◽  
Vol 32 (6) ◽  
pp. 1423-1437 ◽  
Author(s):  
M. Faierman ◽  
I. Knowles
Keyword(s):  
Basic Problem ◽  
Adjoint Operator ◽  
General Setting ◽  
Equivalence Classes ◽  
The Self ◽  
Singular Potentials ◽  
Image Position ◽  
Schrödinger Type Operators ◽  
Complex Valued

The objective of this paper is to extend the recent results [7, 8, 9] concerning the self-adjointness of Schrödinger-type operators with singular potentials to a more general setting. We shall be concerned here with formally symmetric elliptic differential expressions of the form1.1where x = (x1, …, xm) ∈ Rm (and m ≧ 1), i = (–1)1/2, ∂j = ∂/∂xj, and the coefficients ajk, bj and q are real-valued and measurable on Rm.The basic problem that we consider is that of deciding whether or not the formal operator defined by (1.1) determines a unique self-adjoint operator in the space L2(Rm) of (equivalence classes of) square integrable complex-valued functions on Rm.

Totally Integrally Closed Azumaya Algebras

1990 ◽  
Vol 33 (4) ◽  
pp. 398-403
Author(s):  
R. Macoosh ◽  
R. Raphael
Keyword(s):  
Semiprime Ring ◽  
Commutative Rings ◽  
Azumaya Algebras ◽  
Azumaya Algebra ◽  
Integral Extension ◽  
Integrally Closed ◽  
Carry Over

AbstractEnochs introduced and studied totally integrally closed rings in the class of commutative rings. This article studies the same question for Azumaya algebras, a study made possible by Atterton's notion of integral extensions for non-commutative rings.The main results are that Azumaya algebras are totally integrally closed precisely when their centres are, and that an Azumaya algebra over a commutative semiprime ring has a tight integral extension that is totally integrally closed. Atterton's integrality differs from that often studied but is very natural in the context of Azumaya algebras. Examples show that the results do not carry over to free normalizing or excellent extensions.

scholarly journals ENTROPY OF SOME INNER AUTOMORPHISMS OF THE HYPERFINITE II1-FACTOR

1993 ◽  
Vol 04 (02) ◽  
pp. 319-322 ◽  
Author(s):  
ERLING STØRMER
Keyword(s):  
Cartan Subalgebra ◽  
Unitary Operator ◽  
Inner Automorphism ◽  
Inner Automorphisms

It is shown that the entropy of an inner automorphism Ad u of the hyperfinite II1-factor is zero if the unitary operator u belongs to a Cartan subalgebra.

scholarly journals On the number of eigenvalues in the spectral gap of a Dirac system

1986 ◽  
Vol 29 (3) ◽  
pp. 367-378 ◽  
Author(s):  
D. B. Hinton ◽  
A. B. Mingarelli ◽  
T. T. Read ◽  
J. K. Shaw
Keyword(s):  
Differential Operators ◽  
Spectral Gap ◽  
Equivalence Classes ◽  
Value Functions ◽  
Complex Vector ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Integrable Functions ◽  
The One ◽  
Image Position

We consider the one-dimensional operator,on 0<x<∞ with. The coefficientsp,V1andV2are assumed to be real, locally Lebesgue integrable functions;c1andc2are positive numbers. The operatorLacts in the Hilbert spaceHof all equivalence classes of complex vector-value functionssuch that.Lhas domainD(L)consisting of ally∈Hsuch thatyis locally absolutely continuous andLy∈H; thus in the language of differential operatorsLis a maximal operator. Associated withLis the minimal operatorL0defined as the closure ofwhereis the restriction ofLto the functions with compact support in (0,∞).

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