2,200 research outputs found
Representation of non-commutative topological algebras
The well known Gelfland-Naimark theorem enables us to represent a complex commutative C*-algebra as a full algebra of complex valued functions defined on its set of primitive ideals which is called the structure space of the algebra. In is thesis we are concerned with the generalization of this type of representation theorem to non-commutative rings and algebras. In order to prove the Gelfand-Naimark theorem, we needed the Stone-Weierstrass theorem to enable us to show that a subalgebra is actually equal to a full algebra of functions. We shall see that in order to represent a non-commutative algebra as a set of functions taking values in a variable range, we shall need a suitable type of Stone-Weierstrass theorem. This thesis can therefore be considered as an illustration of the application of Stone-Weierstrass type argunents to the theory of C*-algebra representations
The Weierstrass Approximation Theorem
In this thesis we will consider the work began by Weierstrass in 1855 and several generalization of his approximation theorem since. Weierstrass began by proving the density of algebraic polynomials in the space of continuous real-valued functions on a finite interval in the uniform norm. His theorem has been generalized to an arbitrary compact Hausdorff space and the approximation with elements from more general algebras of continuous real-valued functions. We will consider proofs that use brute force and proofs based on convolutions and approximate identities, trudge through probability and the use of the Bernstein polynomials, and become intimately close to what is meant by an algebra. We will also see what inspired Weierstrass and Stone throughout their life as we take a sneak peak into their Biographies
Finite Rank Bargmann-Toeplitz Operators with Non-Compactly Supported Symbols
Theorems about characterization of finite rank Toeplitz operators in
Fock-Segal-Bargmann spaces, known previously only for symbols with compact
support, are carried over to symbols without that restriction, however with a
rather rapid decay at infinity. The proof is based upon a new version of the
Stone-Weierstrass approximation theorem
Elementary approach to homogeneous C*-algebras
A C*-algebra is n-homogeneous (where n is finite) if every its nonzero
irreducible representation acts on an n-dimensional Hilbert space. An
elementary proof of Fell's characterization of n-homogeneous C*-algebras (by
means of their spectra) is presented. A spectral theorem and a functional
calculus for finite systems of elements which generate n-homogeneous
C*-algebras are proposed.Comment: 22 page
Extremalis problémák többváltozós és súlyozott polinomokra = Extremal problems for multivariate and weighted polynomials
JĂłl ismert hogy a többváltozĂłs polinomok sűrűek a d-dimenziĂłs kompakt halmazokon folytonos fĂĽggvĂ©nyek terĂ©ben. A többváltozĂłs polinomok egy fontos rĂ©szhalmaza a homogĂ©n polinomok osztálya. Igy termĂ©szetesen felmerĂĽl az a kĂ©rdĂ©s, hogy igaz-e a sűrĂĽsĂ©g a homogĂ©n polinomokra? Egy ismert sejtĂ©s szerint a konvex felĂĽleteken folytonos fĂĽggvĂ©nyek megközelĂthetĹ‘ek kĂ©t homogĂ©n polinom összegĂ©vel. A pályázat keretĂ©ben kĂ©t fontos Ăşj eredmĂ©ny szĂĽletett 1) igazoltuk a sejtĂ©st tetszĹ‘leges sima ( egyĂ©rtelmĂĽ támasz sikkal rendelkezĹ‘) konvex testeken egyenletes normában 2) igazoltuk a sejtĂ©st teljes általánosságban Lp normában Ezen kivĂĽl általánosĂtott Freud sĂşlyokra vonatkozĂł polinom-approximáciĂłs problĂ©mákat vizsgáltunk. Itt az általánosĂtás azt jelenti, hogy az eredeti Freud sĂşlyokat megszorozzuk olyan un. általánosĂtott polinomokkal, amelyeknek csak valĂłs gyökeik vannak. A klasszikus polinom-egyenlotlensĂ©gek analogonjait, valamint direkt Ă©s fordĂtott approximáciĂłs tĂ©teleket bizonyĂtottunk. HibabecslĂ©seket adtunk fĂĽggvĂ©nyek sĂşlyozott approximáciĂłjára Freud sĂşlyok esetĂ©n, olyan egĂ©sz fĂĽggvĂ©nyekkel törtĂ©no approximáciĂł esetĂ©n, amelyek vĂ©ges, ill. vĂ©gtelen sok pontban interpolálják a fĂĽggvĂ©nyt. Ezek a hibabecslĂ©sek olyan sĂşlyozott folytonossági modulusokat tartalmaznak, amelyeknĂ©l a polinom-surusĂ©g nem mindig garantált | It is well known that multivariate polynomials are dense in the space of continuous functions on compact subsets of the d-dimensional space. An important family of multivariate polynomials is the space of all homogeneous polynomials. Thus it is natural to ask if the density holds for homogeneous polynomials. It has been conjectured that any function continuous on a convex surface can be approximated by sums of two homogeneous polynomials. In the framework of the present project the above conjecture was verified in two new important cases: 1) the conjecture was verified for uniform norm on arbitrary regular convex bodies, i.e., in case when the body possesses a unique tangent plane at each point of its boundary 2) the conjecture was verified in full generality in the Lp norm We also considered polynomial approximation problems on the real line with generalized Freud weights. The generalization means multiplying these weights by so-called generalized polynomials which have real roots only. Analogues of classical polynomial inequalities, as well as direct and converse approximation theorems were proved. We gave error estimates for the weighted approximation of functions with Freud-type weights, by entire functions interpolating at finitely or infinitely many points on the real line. The error estimates involve weighted moduli of continuity corresponding to general Freud-type weights for which the density of polynomials is not always guaranteed
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