Almost multiplicative functionals

Jarosz, Krzysztof

Studia Mathematica, Tome 122 (1997), p. 37-58 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1997-01-01

Zbl 0897.46043

EUDML-ID : urn:eudml:doc:216396

Bibliographie

[00000] [1] E. Bishop, A generalization of the Stone-Weierstrass theorem, Pacific J. Math. 11 (1961), 777-783. | Zbl 0104.09002

[00001] [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. | Zbl 0533.32002

[00002] [3] P. Colwell, Blaschke Products, The University of Michigan Press, 1985.

[00003] [4] J. B. Conway, Functions of One Complex Variable, Grad. Texts in Math. 11, Springer, 1986.

[00004] [5] J. B. Conway, Functions of One Complex Variable II, Grad. Texts in Math. 159, Springer, 1995.

[00005] [6] R. Frankfurt, Weak* generators of quotient algebras of $H^∞$, J. Math. Anal. Appl. 73 (1980), 52-64. | Zbl 0482.46036

[00006] [7] T. W. Gamelin, Uniform Algebras, Chelsea, New York, 1984.

[00007] [8] J. B. Garnett, Bounded Analytic Functions, Academic Press, 1981. | Zbl 0469.30024

[00008] [9] P. Gorkin, Decompositions of the maximal ideal space of $L^∞$, Trans. Amer. Math. Soc. 282 (1984), 33-44. | Zbl 0545.30040

[00009] [10] C. Guillory and K. Izuchi, Interpolating Blaschke products and nonanalytic sets, Complex Variables Theory Appl. 23 (1993), 163-175. | Zbl 0795.30031

[00010] [11] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72-104. | Zbl 0412.46017

[00011] [12] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, 1962. | Zbl 0117.34001

[00012] [13] K. Jarosz, Into isomorphisms of spaces of continuous functions, Proc. Amer. Math. Soc. 90 (1984), 373-377. | Zbl 0535.46013

[00013] [14] K. Jarosz, Perturbations of Banach Algebras, Lecture Notes in Math. 1120, Springer, 1985. | Zbl 0557.46029

[00014] [15] K. Jarosz, Small perturbations of algebras of analytic functions on polydiscs, in: K. Jarosz (ed.), Function Spaces, Marcel Dekker, 1991, 223-240. | Zbl 0780.46034

[00015] [16] K. Jarosz, Ultraproducts and small bound perturbations, Pacific J. Math. 148 (1991), 81-88. | Zbl 0755.46005

[00016] [17] B. E. Johnson, Approximately multiplicative functionals, J. London Math. Soc. 34 (1986), 489-510. | Zbl 0625.46059

[00017] [18] A. Kerr-Lawson, A filter description of the homeomorphisms of $H^∞$, Canad. J. Math. 17 (1965), 734-757. | Zbl 0128.34702

[00018] [19] A. Kerr-Lawson, Some lemmas on interpolating Blaschke products and a correction, ibid. 21 (1969), 531-534. | Zbl 0206.08702

[00019] [20] S. G. Krantz, Function Theory of Several Complex Variables, Wiley, 1982. | Zbl 0471.32008

[00020] [21] G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), 595-611. | Zbl 0439.47022

[00021] [22] P. McKenna, Discrete Carleson measures and some interpolating problems, Michigan Math. J. 24 (1977), 311-319. | Zbl 0391.30023

[00022] [23] A. Nicolau, Finite products of interpolating Blaschke products, J. London Math. Soc. 50 (1994), 520-531. | Zbl 0819.30019

[00023] [24] R. Rochberg, Deformation of uniform algebras on Riemann surfaces, Pacific J. Math. 121 (1986), 135-181. | Zbl 0591.46046

[00024] [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.

[00025] [26] G. E. Shilov, On rings of functions with uniform convergence, Ukrain. Mat. Zh. 3 (1951), 404-411 (in Russian). | Zbl 0045.21204

[00026] [27] S. J. Sidney, Are all uniform algebras AMNM?, preprint, Institut Fourier, 1995.

[00027] [28] E. L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, Belmont, Calif., 1971. | Zbl 0286.46049

[00028] [29] I. Suciu, Function Algebras, Noordhoff, Leyden, 1975.

[00029] [30] V. Tolokonnikov, Extremal functions of the Nevanlinna-Pick problem and Douglas algebras, Studia Math. 105 (1993), 151-158. | Zbl 0816.30037