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## Studia Mathematica

1997 | 124 | 1 | 37-58
Tytuł artykułu

### Almost multiplicative functionals

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A linear functional F on a Banach algebra A is almost multiplicative if |F(ab) - F(a)F(b)| ≤ δ∥a∥ · ∥b∥ for a,b ∈ A, for a small constant δ. An algebra is called functionally stable or f-stable if any almost multiplicative functional is close to a multiplicative one. The question whether an algebra is f-stable can be interpreted as a question whether A lacks an almost corona, that is, a set of almost multiplicative functionals far from the set of multiplicative functionals. In this paper we discuss f-stability for general uniform algebras; we prove that any uniform algebra with one generator as well as some algebras of the form R(K), K ⊂ ℂ, and A(Ω), $Ω ⊂ ℂ^{n}$, are f-stable. We show that, for a Blaschke product B, the quotient algebra $H^{∞}/BH^{∞}$ is f-stable if and only if B is a product of finitely many interpolating Blaschke products.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
37-58
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-02-05
poprawiono
1996-10-30
Twórcy
autor
• Department of Mathematics, Bowling Green State University, Bowling Green, Ohio 43403, U.S.A., kjarosz@siue.edu
• Department of Mathematics, Southern Illinois University at Edwardsville, Edwardsville, Illinois 62026, U.S.A.
Bibliografia
• [1] E. Bishop, A generalization of the Stone-Weierstrass theorem, Pacific J. Math. 11 (1961), 777-783.
• [2] J. Bruna and J. M. Ortega, Closed finitely generated ideals in algebras of holomorphic functions and smooth to the boundary in strictly pseudoconvex domains, Math. Ann. 268 (1984), 137-157.
• [3] P. Colwell, Blaschke Products, The University of Michigan Press, 1985.
• [4] J. B. Conway, Functions of One Complex Variable, Grad. Texts in Math. 11, Springer, 1986.
• [5] J. B. Conway, Functions of One Complex Variable II, Grad. Texts in Math. 159, Springer, 1995.
• [6] R. Frankfurt, Weak* generators of quotient algebras of $H^∞$, J. Math. Anal. Appl. 73 (1980), 52-64.
• [7] T. W. Gamelin, Uniform Algebras, Chelsea, New York, 1984.
• [8] J. B. Garnett, Bounded Analytic Functions, Academic Press, 1981.
• [9] P. Gorkin, Decompositions of the maximal ideal space of $L^∞$, Trans. Amer. Math. Soc. 282 (1984), 33-44.
• [10] C. Guillory and K. Izuchi, Interpolating Blaschke products and nonanalytic sets, Complex Variables Theory Appl. 23 (1993), 163-175.
• [11] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72-104.
• [12] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, 1962.
• [13] K. Jarosz, Into isomorphisms of spaces of continuous functions, Proc. Amer. Math. Soc. 90 (1984), 373-377.
• [14] K. Jarosz, Perturbations of Banach Algebras, Lecture Notes in Math. 1120, Springer, 1985.
• [15] K. Jarosz, Small perturbations of algebras of analytic functions on polydiscs, in: K. Jarosz (ed.), Function Spaces, Marcel Dekker, 1991, 223-240.
• [16] K. Jarosz, Ultraproducts and small bound perturbations, Pacific J. Math. 148 (1991), 81-88.
• [17] B. E. Johnson, Approximately multiplicative functionals, J. London Math. Soc. 34 (1986), 489-510.
• [18] A. Kerr-Lawson, A filter description of the homeomorphisms of $H^∞$, Canad. J. Math. 17 (1965), 734-757.
• [19] A. Kerr-Lawson, Some lemmas on interpolating Blaschke products and a correction, ibid. 21 (1969), 531-534.
• [20] S. G. Krantz, Function Theory of Several Complex Variables, Wiley, 1982.
• [21] G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), 595-611.
• [22] P. McKenna, Discrete Carleson measures and some interpolating problems, Michigan Math. J. 24 (1977), 311-319.
• [23] A. Nicolau, Finite products of interpolating Blaschke products, J. London Math. Soc. 50 (1994), 520-531.
• [24] R. Rochberg, Deformation of uniform algebras on Riemann surfaces, Pacific J. Math. 121 (1986), 135-181.
• [25] D. Sarason, The Shilov and Bishop decompositions of $H^∞ + C$, in: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. II, Wadsworth, Belmont, Calif., 1983, 461-474.
• [26] G. E. Shilov, On rings of functions with uniform convergence, Ukrain. Mat. Zh. 3 (1951), 404-411 (in Russian).
• [27] S. J. Sidney, Are all uniform algebras AMNM?, preprint, Institut Fourier, 1995.
• [28] E. L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, Belmont, Calif., 1971.
• [29] I. Suciu, Function Algebras, Noordhoff, Leyden, 1975.
• [30] V. Tolokonnikov, Extremal functions of the Nevanlinna-Pick problem and Douglas algebras, Studia Math. 105 (1993), 151-158.
Typ dokumentu
Bibliografia
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