caa a22 4500 388023791 CHVBK 20180307124931.0 cr unu---uuuuu 161130e199802 xx s 000 0 eng 10.1017/S0013091500019489 doi S0013091500019489 pii (NATIONALLICENCE)cambridge-10.1017/S0013091500019489 Thomas Marc P. Mathematics Department, California State University, Bakersfield, 9001 Stockdale Highway, Bakersfield, California 93311-1099, USA Closed ideals in the Banach algebra ℓ1(ω) when ω is -star shaped [Elektronische Daten] [Marc P. Thomas] Provided that the weight function ω satisfies certain submultiplicative and decay conditions, the discrete convolution algebra ℓ1(ω) becomes a commutative radical Banach algebra with identity adjoined. There are obvious closed ideals in ℓ1(ω) and these are denoted standard ideals. Earlier results of Thomas, strengthened by Yakubovich and Domar, showed that if the weight ω is star-shaped then all closed ideals are standard. Consequently, the closed ideal generated by any element f in ℓ1(ω) must be standard. The requirement that ω be star-shaped (essentially that ω(n)1/n must decrease to zero) is somewhat restrictive in that no local maxima of ω(n)1/n are allowed. We generalize this previous result to apply to the larger class of ε-star shaped weights (0 < ε ≤ 1) which allow such local maxima. If f is a non-zero element on ℓ1(ω) we let the integer α(f) = k0 denote the index of its first non-zero term. We introduce the concept of an ε-peak point for k0. If ε = 1 then ω is star-shaped in the usual sense and there are an infinite number of 1-peak points for any k0. Although this latter fact may fail if 0 < ε < 1, if ω(n)1/n tends to zero sufficiently quickly (dependent on k0 and ε) there will always be an infinite number of ε-peak points for k0. Our main result is that if ω is an ε-star shaped weight, if f is an non-zero element of ℓ1(ω), if α(f) = k0, and if the number of ε-peak points for k0 is infinite, then the closed ideal generated by f is standard. Copyright © Edinburgh Mathematical Society 1998 Proceedings of the Edinburgh Mathematical Society Cambridge University Press 41/1(1998-02), 161-176 0013-0915 41:1<161 1998 41 PEM https://doi.org/10.1017/S0013091500019489 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0013091500019489 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Thomas Marc P. Mathematics Department, California State University, Bakersfield, 9001 Stockdale Highway, Bakersfield, California 93311-1099, USA NATIONALLICENCE 773 0- Proceedings of the Edinburgh Mathematical Society Cambridge University Press 41/1(1998-02), 161-176 0013-0915 41:1<161 1998 41 PEM CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge