On Certain Extensions of Function Rings

1959 ◽  
Vol 11 ◽  
pp. 87-96
Author(s):  
Bernhard Banaschewski
Keyword(s):  
Topological Space ◽  
Galois Extension ◽  
Present Note ◽  
Banach Algebras ◽  
Faithful Representation ◽  
Normal Extension ◽  
Invariance Group ◽  
Group Of Automorphisms ◽  
Normed Ring ◽  
Function Ring

The present note is concerned with the existence and properties of certain types of extensions of Banach algebras which allow a faithful representation as the normed ring C(E) of all bounded continuous real functions on some topological space E. These Banach algebras can be characterized intrinsically in various ways (1); they will be called function rings here. A function ring E will be called a normal extension of a function ring G if E is directly indecomposable, contains C as a Banach subalgebra and possesses a group G of automorphisms for which C is the ring of invariants, that is, the set of all elements fixed under G. G will then be called a group of automorphisms of E over C. If E is a normal extension of C with precisely one group of automorphisms over C, which is then the invariance group of C in E, then E will be called a Galois extension of C. Such an extension will be called finite if its group is finite.

Polynomials with Certain Prescribed Conditions on the Galois Group

1969 ◽  
Vol 21 ◽  
pp. 262-273 ◽  
Author(s):  
Elizabeth Rowlinson ◽  
Hans Schwerdtfeger
Keyword(s):  
Nilpotent Group ◽  
Galois Group ◽  
Canonical Form ◽  
Nilpotent Groups ◽  
Faithful Representation ◽  
Splitting Field ◽  
Normal Extension ◽  
Algebraic Equations ◽  
Abstract Group ◽  
Group A

In this paper, some contributions are made to the theory of algebraic equations over the rational field with conditions imposed on the Galois group. In § 1, for a given abstract group G all faithful permutation representations Ḡ are obtained, and it is shown that if one of them is the group of some equation with splitting field K, then any of them is the group of some equation, also with splitting field K. The method of proof enables us to construct an equation having as group a given faithful permutation representation Ḡ of a prescribed group G if we are given an equation having as group some faithful representation of G. In § 2, equations having nilpotent group are considered, non-normal extension fields are discussed, and a canonical form is obtained for the roots of non-normal irreducible equations; this form is used to characterize fields and equations with nilpotent groups.

scholarly journals Note on Galois Cohomology

1953 ◽  
Vol 5 ◽  
pp. 97-104
Author(s):  
Tadasi Nakayama
Keyword(s):  
Galois Extension ◽  
Class Group ◽  
Present Note ◽  
Galois Cohomology ◽  
Group Cohomology ◽  
Ground Field ◽  
General Group ◽  
3 Dimensional ◽  
The Relationship ◽  
American Authors

Let K/k be a Galois extension. Formerly the writer studied, in [3], [4], a certain correlation of factor sets in K/k with the norm class group of K/k, and extended it, in [5.], to 3-dimensional cocycles. The present note is to study the same relationship for general n-cocycles. As a matter of fact, the constructions which underlie the relationship have become common places in cohomology theory, through the works of French and American authors, and indeed the construction to bring certain (non-Galois) cocycles into the ground field k has been discussed by Baer [1, Theorem C] for general dimensions n under the setting of general group cohomology.

A faithful polynomial representation of Out F3

1989 ◽  
Vol 106 (2) ◽  
pp. 207-213 ◽  
Author(s):  
James McCool
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Polynomial Ring ◽  
Outer Automorphism ◽  
Faithful Representation ◽  
Partial Answer ◽  
Polynomial Representation ◽  
Group Of Automorphisms ◽  
Outer Automorphism Group ◽  
Torsion Free

Let Fn be a free group of rank n and let Out Fn be its outer automorphism group. The main result of this paper is that Out F3 has a faithful representation as a group of automorphisms of the polynomial ring in seven variables over the integers. This extends a similar result for n = 2 (see Helling [3], Horowitz [5] and Rosenberger [12]), and provides a partial answer to a conjecture attributed in [5] to W. Magnus. As an application of the special nature of the representing polynomials, we obtain our second result, that Out F3 is virtually residually torsion-free nilpotent.

scholarly journals Commutators in Banach algebras

1979 ◽  
Vol 22 (3) ◽  
pp. 207-211 ◽  
Author(s):  
Vlastimil Pták
Keyword(s):  
Banach Algebra ◽  
Present Author ◽  
Present Note ◽  
Banach Algebras ◽  
Related Condition ◽  
Image Position

In a recent paper (6) the present author has shown that, for an element a of a Banach algebra A, the conditionfor all x∈A and some constant α is equivalent to [x, a]∈Rad a for all x∈A; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition

scholarly journals Maximal ideal spaces of Banach algebras of derivable elements

1970 ◽  
Vol 11 (3) ◽  
pp. 310-312 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Present Note ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Supremum Norm ◽  
Simple Consequence ◽  
Maximal Ideals ◽  
Image Position

Let A be a commutative Banach algebra, D a closed derivation defined on a subalgebra Δ of A, and with range in A. The elements of Δ may be called derivable in the obvious sense. For each integer k ≦.l, denote by Δk the domain of Dk (so that Dgr;1 = Δ); it is a simple consequence of Leibniz's formula that each Δk is an algebra. The classical example of this situation is A = C(O, 1) under the supremum norm with D ordinary differentiation, and here Δk = Ck(0, 1) is a Banach algebra under the norm ∥.∥k: Furthermore, the maximal ideals of Ak are precisely those subsets of Δk of the form M ∩ Δk where M is a maximal ideal of A, and = M, the bar denoting closure in A. In the present note we show how this extends to the general case.

scholarly journals On automorphism group of free quadratic extensions over a ring

1984 ◽  
Vol 7 (1) ◽  
pp. 103-108
Author(s):  
George Szeto
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Galois Extension ◽  
Zero Divisor ◽  
Quadratic Extension ◽  
Normal Extension ◽  
Quadratic Extensions ◽  
The One

LetRbe a ring with1,ρan automorphism ofRof order2. Then a normal extension of the free quadratic extensionR[x,ρ]with a basis{1,x}overRwith anR-automorphism groupGis characterized in terms of the element(x−(x)α)forαinG. It is also shown by a different method from the one given by Nagahara that the order ofGof a Galois extensionR[x,ρ]overRwith Galois groupGis a unit inR. When2is not a zero divisor, more properties ofR[x,ρ]are derived.

Some Multiplicative Functionals

1953 ◽  
Vol 5 ◽  
pp. 174-178 ◽  
Author(s):  
D. G. Bourgin
Keyword(s):  
Continuous Function ◽  
Compact Space ◽  
Banach Algebras ◽  
The Real ◽  
Function Ring ◽  
Multiplicative Functionals

This note concerns itself primarily with the representation of continuous multiplicative functionals on L2 types of rings or Banach algebras to the real or complex fields where convolution is taken as the ring multiplication. In a recent publication [1] such functionals were studied for the continuous function ring C(S) over a compact space S.

scholarly journals A Group of Automorphisms of the Homotopy Groups

1951 ◽  
Vol 2 ◽  
pp. 73-82
Author(s):  
Hiroshi Uehara
Keyword(s):  
Normal Subgroup ◽  
Topological Space ◽  
Fundamental Group ◽  
Homotopy Group ◽  
Factor Group ◽  
Complete Analysis ◽  
Homotopy Groups ◽  
Group Of Automorphisms ◽  
Arcwise Connected

It is well known that the fundamental group π1(X) of an arcwise connected topological space X operates on the n-th homotopy group πn(X) of X as a group of automorphisms. In this paper I intend to construct geometrically a group 𝒰(X) of automorphisms of πn(X), for every integer n ≥ 1, which includes a normal subgroup isomorphic to π1(X) so that the factor group of 𝒰(X) by π1(X) is completely determined by some invariant Σ(X) of the space X. The complete analysis of the operation of the group on πn(X) is given in §3, §4, and §5,

Systems of modal logic which are not unreasonable in the sense of Halldén

1953 ◽  
Vol 18 (2) ◽  
pp. 109-113 ◽  
Author(s):  
J. C. C. McKinsey
Keyword(s):  
Modal Logic ◽  
Present Note ◽  
The Other ◽  
Normal Extension ◽  
Sentential Calculus ◽  
Other Hand ◽  
Negation Sign

Halldén, in [1], has recently pointed out that it is highly undesirable, in a system of sentential calculus, for there to exist two formulas α and β such that: (i) α and β contain no variable in common; (ii) neither α nor β is provable; (iii) α ∨ β is provable. We shall call a system unreasonable (in the sense of Halldén) if there exists a pair of formulas α and β having properties (i), (ii), and (iii). Halldén shows (in [1]) that the Lewis systems S1 and S3 are unreasonable in this sense; and that the same is true of any system which is between S1 and S3, as well as of every system which is stronger than S3 but weaker than both S4 and S7. In the present note we shall show that this defect does not occur in S4, nor in S5, nor in any “quasi-normal” extension of S5; we give an example, on the other hand, of an unreasonable system which lies between S4 and S5.When we speak, in what follows, of a system of modal logic, we shall mean a system having the same class of well-formed formulas as have the various Lewis calculi. Thus the well-formed formulas of a system of modal logic, when written in unabbreviated form, are just those formulas which can be built up from sentential variables by use of the binary connective ‘·’ (conjunction sign), and the two unary connectives ‘˜’ (negation sign) and ‘◇’ (possibility sign). We shall, however, also make use of some of the defined signs of Lewis.

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