On Certain Extensions of Function Rings

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply that the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Furthermore, we show that, for a $p$-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the eighties of the last century Rennison studied and characterized the set of quasi-central elements of some unital Banach algebras. In this paper, we improve one of Rennison's result assert that for given unital Banach algebra \(A\) the set of quasi-central elements \(Q(A)\), need not be equal to centre of \(A\) and need not be closed under addition or multiplication. Further, we prove that if \(A\) is unital ultraprime Banach algebra, then \(Q(A)\) need not be equal to centre of \(\mathrm{A}\) and need not be closed under addition or multiplication.

A holomorphically fibred space generates locally trivial bundles with positive dimensional fibers. This paper proposes two varieties of fibrations (compact and non-compact) in the non-uniformly scalable quasinormed topological (C, R) space admitting cylindrically symmetric continuous functions. The projective base space is dense, containing a complex plane, and the corresponding surjective fiber projection on the base space can be fixed at any point on real subspace. The contact category fibers support multiple oriented singularities of piecewise continuous functions within the topological space. A composite algebraic operation comprised of continuous linear translation and arithmetic addition generates an associative magma in the non-compact fiber space. The finite translation is continuous on complex planar subspace under non-compact projection. Interestingly, the associative magma resists transforming into a monoid due to the non-commutativity of composite algebraic operation. However, an additive group algebraic structure can be admitted in the fiber space if the fibration is a non-compact variety. Moreover, the projection on base space supports additive group structure, if and only if the planar base space passes through the real origin of the topological (C, R) space. The topological analysis shows that outward deformation retraction is not admissible within the dense topological fiber space. The comparative analysis of the proposed fiber space with respect to Minkowski space and Seifert fiber space illustrates that the group algebraic structures in each fiber spaces are of different varieties. The proposed topological fiber bundles are rigid, preserving sigma-sections as compared to the fiber bundles on manifolds.

Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set K ⊆ X, we study when a continuous real function on K that is strictly monotonic with respect to ≿ can be extended to X without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when X is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space X starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

Vector invariants for the two-dimensional modular representation of a cyclic group of prime order

It is well known that the complete norm topology on a Banach algebra is not unique in general, though semisimplicity is sufficient (but not necessary) for uniqueness. In this note we consider a class of topological algebras of formal power series which have unique Fréchet space topology. The structure of these algebras in the Banach algebra case will be considered in a later paper.

In this paper, we study biprojectivity of generalized module extension Banach algebras and second dual of Banach algebras. As a main result, we prove that if [Formula: see text] is a contractible closed ideal of [Formula: see text] such that [Formula: see text] is biprojective, then [Formula: see text] is biprojective. As a consequence, we give some results on biprojectivity of generalized module extension Banach algebras. Indeed, this is a generalization of results in [A. Ebaddian and A. Jabbari, Biprojectivity and biflatness of amalgamated duplication of Banach algebras, J. Algebra Its Appl. 19(7) (2020) 2050132 (15 pp.)]. Moreover, we show that if [Formula: see text] is biprojective and [Formula: see text] is a two-sided ideal of [Formula: see text] then [Formula: see text] is biprojective. This fact generalizes some known results in [M. S. Moslehian and A. Niknam, Biflatness and biprojectivity of second dual of Banach algebras, Southeast Asian Bull. Math 27(1) (2003) 129–133].

Finite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in the algebraic properties of the extensions and conversely. Of particular interest to us is the class of finite covering maps for which the induced extensions of function rings admit primitive generators. This is exactly the class of polynomial covering maps and the extensions are algebraic extensions defined by the underlying Weierstrass polynomials.The purpose of this paper is to develop a suitable Galois theory for finite extensions of function rings induced by finite covering maps and to apply it in the case of Weierstrass polynomials and polynomial covering maps.

On the universal completion of pointfree function spaces

A density theorem of semisimple analytic groups acting on locally compact groups is presented.Let G and H be locally compact groups with G acting continuously on H as a group of automorphisms.An element h of H is said to be G-bounded if the orbit Gh = \g(h): g G] has compact closure in H.We write F AH) fot the set of all G-bounded elements in H.It is very easy to verify that F AH) is a G-invariant subgroup of H. However in general, EAH) is not closed in H.In this paper, we shall study the group EAH) fot certain topological groups G. Our main result is the following Theorem.Let G be a semisimple analytic group without compact actors acting on a locally compact group H continuously as a group of automorphisms.If the set F AH) is dense in H, then G acts trivially on H.The theorem generalizes some results in [2], [4] and is closely related to the density property of certain subgroups in semisimple analytic groups without compact factors.The result of Corollaries 4.1 and 4.2 is contained in [2], [4].In the sequel, we shall use the term "G acts on H" fot "G acts on H as a group of automorphisms".1. Minimally almost periodic groups.Let G be a locally compact group.We recall that G is minimally almost periodic if there are no nontrivial continuous homomorphisms /: G -* G of locally compact groups such that the closure C1(/(G)) of (G) in G is compact.Minimally almost periodic groups have been widely studied.Yet for our need, we shall establish some lemmas concerning minimally almost periodic groups.Lemma 1.1.Let G be a minimally almost periodic group acting continuously on a locally compact group H, and N be a closed G-invariant normal subgroup of H.

We show that a real Banach algebra $A$ such that $\Vert a^{2}\Vert =\Vert a\Vert^{2}$ for $a\in A$ is a subalgebra of the algebra $C_{\mathbb{H}}( X) $ of continuous quaternion-valued functions on a compact set $X$.

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper, we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case), and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth’s classical construction to the pointfree setting. The second shows that in the bounded case, the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L, and that in the non-bounded case, it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov’s approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.