×

Pseudo-characters and almost multiplicative functionals. (English) Zbl 0958.43001

The main subject studied in the paper under review is the so called stable approximability of the set of characters or continuous characters on a group, i.e. the question whether they can uniformly on \(G\) approximate every almost character. Theorem 1 asserts that on an amenable locally compact group every measurable \(\varepsilon\)-character can uniformly be approximated on \(G\) by continuous characters. The logarithms of multiplicative pseudo-characters can be chosen to be some real additive almost characters (Theorem 2) and in Theorem 3 the author shows that the involutive Banach group algebra \(\ell_1(G)\) is an AMNM (i.e. “algebra on which almost multiplicative functionals are near to multiplicative functionals”) if and only if the set of characters on \(G\) is stable.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Banach, S.; Tarski, A., Sur la décomposition des ensembles de points en parties respectivement congruentes, Fund. math., 6, 244-277, (1924) · JFM 50.0370.02
[2] Cenzer, D., The stability problem for transformations of the circle, Proc. roy. soc. Edinburgh sect. A, 84, 279-281, (1979) · Zbl 0439.39004
[3] Cenzer, D., The stability problem: new results and counterexamples, Lett. math. phys., 10, 155-160, (1985) · Zbl 0595.39010
[4] Forti, G.L., The stability of homomorphisms and amenability, with applications to functional equations, Abh. math. sem. univ. Hamburg, 57, 215-226, (1987) · Zbl 0619.39012
[5] Forti, G.L., Hyers – ulam stability of functional equations in several variables, Aequationes math., 50, 143-190, (1995) · Zbl 0836.39007
[6] Gelbaum, B.R.; Olmstead, J.M.H., Theorems and counterexamples in mathematics, Problem books in math., (1990), Springer-Verlag New York
[7] Greenleaf, F.P., Invariant means on topological groups, Van nostrand mathematical studies, 16, (1969), Van Nostrand New York · Zbl 0174.19001
[8] Grove, K.; Karcher, H.; Roh, E.A., Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problems, Math. ann., 211, 7-21, (1974) · Zbl 0273.53051
[9] de la Harpe, P.; Karoubi, M., Representations approchées d’un groupe dans une algèbre de Banach, Manuscripta math., 22, 293-310, (1977) · Zbl 0371.22007
[10] Hyers, D.H.; Rassias, Th.M., Approximate homomorphisms, Aequationes math., 44, 125-153, (1992) · Zbl 0806.47056
[11] Jarosz, K., Perturbations of Banach algebras, Springer lecture notes in mathematics, 1120, (1985), Springer-Verlag Berlin
[12] Johnson, B.E., Approximately multiplicative functionals, J. London math. soc., 34, 489-510, (1986) · Zbl 0625.46059
[13] Johnson, B.E., Approximately multiplicative maps between Banach algebras, J. London math. soc., 37, 294-316, (1988) · Zbl 0652.46031
[14] Johnson, B.E., Cohomology in Banach algebras, Memoirs AMS, 127, (1972), Am. Math. Soc Providence · Zbl 0246.46040
[15] Kazhdan, D., On ε-representations, Israel J. math., 43, 315-323, (1982) · Zbl 0518.22008
[16] Ore, O., Some remarks on commutators, Proc. amer. math. soc., 2, 307-314, (1951) · Zbl 0043.02402
[17] Pełczyński, A., Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes math., LVIII, (1968) · Zbl 0165.14603
[18] Rudin, W., Análisis real y complejo, (1979), Alhambra Madrid
[19] Shtern, A.I., Quasirepresentations and pseudorepresentations, Funkt. anal. prilozhen., 25, 70-73, (1991) · Zbl 0737.22003
[20] Székelyhidi, L., Note on a stability theorem, Canad. math. bull., 25, 500-501, (1982) · Zbl 0505.39002
[21] Ulam, S.M., A collection of mathematical problems, (1960), Interscience New York · Zbl 0086.24101
[22] Ulam, S.M., Problems in modern mathematics, (1964), Wiley New York · Zbl 0137.24201
[23] Ulam, S.M., Set, numbers and universes, (1974), MIT Press Cambridge
[24] Ulam, S.M., An anecdotal history of the scottish book, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.