Algebra Of Real Functions Research Articles

Generalization of a Theorem on the Equivalence of the Coordinate and Algebraic Definitions of a Smooth Manifold

In this paper, we generalize the theorem on the equivalence of the coordinate and algebraic definitions of a smooth manifold. Within the framework of the algebraic approach, a point is considered as a homomorphism from the algebra of smooth real functions defined on a manifold into the field of real numbers. We consider a generalization for the case where the field of real numbers is replaced by an arbitrary associative normalized algebra, generally speaking, noncommutative.

Hilbert series of symplectic quotients by the 2-torus

We compute the Hilbert series of the graded algebra of real regular functions on a linear symplectic quotient by the 2-torus as well as the first four coefficients of the Laurent expansion of this Hilbert series at \(t = 1\). We describe an algorithm to compute the Hilbert series as well as the Laurent coefficients in explicit examples.

Hilbert series associated to symplectic quotients by SU2

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an [Formula: see text]-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at [Formula: see text]. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most [Formula: see text]. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary [Formula: see text]- or [Formula: see text]-module as well as its first three Laurent coefficients.

Constructing symplectomorphisms between symplectic torus quotients

We identify a family of torus representations such that the corresponding singular symplectic quotients at the 0-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may have graded isomorphic algebras of real regular functions and graded Poisson isomorphic complex symplectic quotients yet not be graded regularly diffeomorphic nor graded regularly symplectomorphic.

Boundaries for the Cartesian product of real function algebras

The concepts of Choquet sets and Silov boundary for a real function algebra and real function space has been introduced and studied mainly by Kulkarni and Limaye. Here we study these concepts for the Cartesian product of real function algebras and function spaces.

Multiplication of periodic hyperfunctions via harmonic regularization and applications

We build a locally convex algebra of real analytic functions defined in a strip of the Poincaré half-plane in which a class of periodic hyperfunctions on the real line is topologically embedded. This is accomplished via a harmonic regularization method. In this algebra, we can give a sense to differential problems involving products of hyperfunctions which are a priori not defined in the classical setting. Some examples and an application are given.

Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions

Real numbers are divided into fictitious (non-computable) and essential (computable). Fictitious numbers do not have numerical values, essential numbers have algorithms for constructing these numbers with any exactness. The set of fictitious numbers is continual, the set of essential numbers is countable. Functions are also divided into fictitious, defined over the set of fictitious numbers, and essential, defined over the set of essential numbers. Essential functions have an algorithm for calculating any value with any exactness. All functions of applied mathematics and some functions of abstract mathematics are essential The set these functions is countable. The four upper levels of classification of real functions are constructed. This classification uses superpositions of functions and diagonal sets borrowed from the algebra of finite-valued functions.

Non-radial solutions for some semilinear elliptic equations on the disk

Starting with approximate solutions of the equation −Δu=wu3 on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a computer, using a Banach algebra of real analytic functions, based on Zernike polynomials. Besides proving existence, and symmetry properties, we also determine the Morse index of the solutions.

Orthogonal forms and orthogonality preservers on real function algebras revisited

In [Garcés J, Peralta AM. Orthogonal forms and orthogonality preservers on real function algebras. Linear Multilinear Algebra. 2014;62:275–296.] we determine the precise form of a continuous orthogonal form on a commutative real C-algebra. We also describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between commutative unital real C-algebras. Among the consequences, we show that every orthogonality preserving linear bijection between commutative unital real C-algebras is continuous. In this note we revisit these results and their proofs with the aim of filling a gap in the arguments, and to extend the original conclusions.

Exceptional quantum geometry and particle physics

Based on an interpretation of the quark–lepton symmetry in terms of the unimodularity of the color group SU(3) and on the existence of 3 generations, we develop an argumentation suggesting that the “finite quantum space” corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space C⊕C3 is associated to the quark–lepton symmetry (one complex for the lepton and 3 for the corresponding quark). More generally it is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of “the algebra of real functions” on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.

Multiplication of convex sets in $C(K)$ spaces

Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. We study this for a special class of closed convex sets, namely closed intervals. Our main interest is in correlations between properties of the product of order intervals in C(K) and properties of the underlying space K. When K is finite, the product of two intervals in C(K) is always an interval. Surprisingly, the converse of this result is true for a wide class of compacta. We show that a first-countable space K is finite whenever it has the property that the product of two nonnegative intervals is closed, or the property that the product of an interval with itself is convex. That some assumption on K is needed, can be seen from the fact that, if K is the Stone–Cech compactification of N, then the product of two intervals in C(K) with continuous boundary functions is always an interval. For any K, it is proved that the product of two positive intervals is always an interval, and that the product of two nonnegative intervals is always convex. Finally, square roots of intervals are investigated, with results of similar type. Mathematics Subject Classification: primary 52A05; secondary 46J10, 46E15.

Characterizations of peripherally multiplicative mappings between real function algebras

Let X be a compact Hausdorff space; let τ : X → X be a topological involution; and let A ⊂ C(X, τ) be a real function algebra. Given an f ∈ A, the peripheral spectrum of f is the set σπ(f) of spectral values of f of maximum modulus. We demonstrate that if T1, T2 : A → B and S1, S2 : A → A are surjective mappings between real function algebras A ⊂ C(X, τ) and B ⊂ C(Y, φ) that satisfy σπ(T1(f)T2(g)) = σπ(S1(f)S2(g)) for all f, g ∈ A, then there exists a homeomorphism ψ : Ch(B) → Ch(A) between the Choquet boundaries such that (ψ ◦ φ)(y) = (τ ◦ ψ)(y) for all y ∈ Ch(B), and there exist functions κ1, κ2 ∈ B, with κ−1 1 = κ2, such that Tj(f)(y) = κj(y)Sj(f)(ψ(y)) for all f ∈ A, all y ∈ Ch(B), and j = 1, 2. As a corollary, it is shown that if either Ch(A) or Ch(B) is a minimal boundary (with respect to inclusion) for its corresponding algebra, then the same result holds for surjective mappings T1, T2 : A → B and S1, S2 : A → A that satisfy σπ(T1(f)T2(g)) ∩ σπ(S1(f)S2(g)) 6= ∅ for all f, g ∈ A.

Inequalities connected with averaging operators. Part II

We continue our previous studies of inequalities stemming from the notions of averaging operators. We focus our attention on the following system of functional inequalities: {T(f+T(g))≥T(f)+T(g),T(f⋅T(g))≥T(f)⋅T(g), postulated for all f,g from some algebra of real functions.

Linear Isometries between Real Banach Algebras of Continuous Complex-Valued Functions

Let and be compact Hausdorff spaces, and let and be topological involutions on and , respectively. In 1991, Kulkarni and Arundhathi characterized linear isometries from a real uniform function algebra on (, ) onto a real uniform function algebra on (, ) applying their Choquet boundaries and showed that these mappings are weighted composition operators. In this paper, we characterize all onto linear isometries and certain into linear isometries between and applying the extreme points in the unit balls of and .

On the Generality of Assuming that a Family of Continuous Functions Separates Points

For an algebra $${\mathcal{A}}$$ of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that $${\mathcal{A}}$$ separates points in the sense that for each distinct pair $${x, y \in X}$$ , there exists an $${f \in \mathcal{A}}$$ such that $${f(x) \neq f(y)}$$ . If $${\mathcal{A}}$$ does not separate points, it is known that there exists an algebra $${\widehat{\mathcal{A}}}$$ on a compact Hausdorff space $${(\widehat{X}, \widehat{\tau})}$$ that does separate points such that the map $${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$$ is a uniform norm isometric algebra isomorphism. So it is, to a degree, without loss of generality that we assume $${\mathcal{A}}$$ separates points. The construction of $${{\widehat{\mathcal{A}}}}$$ and $${(\widehat{X}, \widehat{\tau})}$$ does not require that $${\mathcal{A}}$$ has any algebraic structure nor that $${(X, \tau)}$$ has any properties, other than being a topological space. In this work we develop a framework for determining the degree to which separation of points may be assumed without loss of generality for any family $${\mathcal{A}}$$ of bounded, complex-valued, continuous functions on any topological space $${(X, \tau)}$$ . We also demonstrate that further structures may be preserved by the mapping $${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$$ , such as boundaries of weak peak points, the Lipschitz constant when the functions are Lipschitz on a compact metric space, and the involutive structure of real function algebras on compact Hausdorff spaces.

Orthogonal forms and orthogonality preservers on real function algebras

We initiate the study of orthogonal forms on a real C-algebra. Motivated by previous contributions, due to Ylinen, Jajte, Paszkiewicz and Goldstein, we prove that for every continuous orthogonal form on a commutative real C-algebra,, there exist functionals and in satisfying for every in. We describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between unital commutative real C-algebras. As a consequence, we show that every orthogonality preserving linear bijection between unital commutative real C-algebras is continuous.

Large free linear algebras of real and complex functions

Let X be a set of cardinality κ such that κω=κ. We prove that the linear algebra RX (or CX) contains a free linear algebra with 2κ generators. Using this, we prove several algebrability results for spaces CC and RR. In particular, we show that the set of all perfectly everywhere surjective functions f:C→C is strongly 2c-algebrable. We also show that the set of all functions f:R→R whose sets of continuity points equals some fixed Gδ set G is strongly 2c-algebrable if and only if R⧹G is c-dense in itself.

Choquet and Shilov Boundaries, Peak Sets, and Peak Points for Real Banach Function Algebras

Let be a compact Hausdorff space and let be a topological involution on . In 1988, Kulkarni and Arundhathi studied Choquet and Shilov boundaries for real uniform function algebras on . Then in 2000, Kulkarni and Limaye studied the concept of boundaries and Choquet sets for uniformly closed real subspaces and subalgebras of or . In 1971, Dales obtained some properties of peak sets and p-sets for complex Banach function algebras on . Later in 1990, Arundhathi presented some results on peak sets for real uniform function algebras on . In this paper, while we present a brief account of the work of others, we extend some of their results, either to real subspaces of or to real Banach function algebras on .

The Technique of Measures of Noncompactness in Banach Algebras and Its Applications to Integral Equations

We study the solvability of some nonlinear functional integral equations in the Banach algebra of real functions defined, continuous, and bounded on the real half axis. We apply the technique of measures of noncompactness in order to obtain existence results for equations in question. Additionally, that technique allows us to obtain some characterization of considered integral equations. An example illustrating the obtained results is also included.

Projective Freeness of Algebras of Real Symmetric Functions

Let D-n := {z = (z(1),...,z(n)) is an element of C-n : vertical bar z(j)vertical bar (n) denote its closure in C-n. Consider the ring C-r((D) over bar (n); C) = {f : (D) over bar (n) -> C : f is ...