JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 33, No. 4, November 2020 http://dx.doi.org/10.14403/jcms.2020.33.4.497 APPROXIMATE IDENTITY OF CONVOLUTION BANACH ALGEBRAS Hyuk Han Abstract. A weight ! on the positive half real line [0; 1) is a positive continuous function such that !(s + t) ≤ !(s)!(t), for all s; t 2 [0; 1), and !(0) = 1. The weighted convolution Banach algebra L1(!) is the algebra of all equivalence classes of Lebesgue R 1 measurable functions f such that kfk = 0 jf(t)j !(t)dt < 1, under pointwise addition, scalar multiplication of functions, and R t the convolution product (f ∗ g)(t) = 0 f(t − s)g(s)ds. We give a sufficient condition on a weight function !(t) in order that L1(!) has a bounded approximate identity. 1. Radical of Algebras Let A be an algebra with or without unit over the complex field C. Given a 2 A and subsets E; F of A, we denote by RF , E(1−a), (1−a)E respectively the sets fxy : x; y 2 Eg, fx − xa : x 2 Eg, fx − ax : x 2 Eg. A left ideal of A is a linear subspace J of A such that AJ ⊂ J. An element u of A is a right modular unit for a linear subspace E of A if A(1 − u) ⊂ E.A modular left ideal is a left ideal for which there exists a right modular unit. A left ideal of J of A is proper if J 6= A, maximal if it is proper and not contained in any other proper left ideal, maximal modular if it is proper and modular and not contained in any other such left ideal. Similar definitions apply to right ideals, left modular units, etc., in terms of the inclusions JA ⊂ J, (1 − u)A ⊂ E.A two-sided ideal is a linear subspace that is both a left ideal and a right ideal. It is clear that a left ideal of A is a subalgebra of A, and that if A has a unit element 1, then 1 is a right modular unit for every linear subspace of A. Received October 20, 2020; Accepted October 29, 2020. 2010 Mathematics Subject Classification: 46J45, 46E10. Key words and phrases: radical convolution algebra, approximate identity. This work was supported by the research grant of the Kongju National University in 2019. 498 Hyuk Han Definition 1.1. The(Jacobson) radical of A is the intersection of the maximal modular left ideals of A. It is denoted by rad(A), and, by the usual convention, rad(A) = A if there are no maximal modular left ideals of A. A is said to be semi-simple if rad(A) = f0g, and to be a radical algebra if rad(A) = A. It is well known that the radical is the intersection of all maximal right modular ideals of A. And it is easy to see that, if A has a unit e then rad(A) = fa 2 A : e − ba 2 InvA for all b 2 Ag = fa 2 A : e − ab 2 InvA for all b 2 Ag; where InvA is the set of all invertible elements of A. This is found in [2]. Let A be a Banach algebra. Then every maximal modular left or right ideal of A is closed, so the radical of A is closed, in fact, rad(A) is a closed two-sided ideal of the Banach algebra A and A=rad(A) is a semi-simple Banach algebra. Definition 1.2. Let A be a Banach algebra. An element x 2 A is said to be quasi-nilpotent if n 1 lim kx k n = 0: n!1 n 1 Since limn!1 kx k n is the spectral radius of x, an element of a Ba- nach algebra A is quasi-nilpotent if and only if the spectral radius of x is zero. That is, the spectrum of x contains a single element 0. It is well known that if x 2 rad(A) then x is quasi-nilpotent. But, in a non- commutative Banach algebra, a quasi-nilpotent element need not be in the radical. In fact, Kaplansky proved that every noncommutative C*- algebra contains a non-zero nilpotent element. Since every C*-algebra is semi-simple, this nilpotent element does not belong to the radical. But in a commutative Banach algebra the radical coincides with the set of quasi-nilpotent elements [2]. 2. Approximate Identity of Radical Convolution Algebras Throughout this paper, we shall be concerned with complex valued + functions on the positive half real line R = [0; 1). In each case, we shall Approximate identity of convolution Banach algebras 499 suppose that we are considering Lebesgue measure functions, and, as usual when defining spaces of integrable functions, we equate functions equal almost everywhere. 1 + + Let L (R ) denote the set of all locally integrable functions on R . loc Z 1 + That is, f 2 Lloc(R ) if and only if jfj dt < 1 for each compact K + 1 + subset K of R . If f; g 2 Lloc(R ), their convolution product is f ∗ g where Z t + (f ∗ g)(t) = f(t − s)g(s)ds (t 2 R ): 0 Then clearly f ∗ g = g ∗ f. For all natural number n, let Z n pn(f) = jf(t)j dt: 0 1 + If f; g 2 Lloc(R ), we apply Fubini's theorem to show that f ∗ g 2 1 + Lloc(R ) Z n Z t pn(f ∗ g) = f(t − s)g(s) ds dt 0 0 ZZ ≤ jf(t − s)g(s)j ds dt 0≤s≤t≤n ZZ = jf(u)g(v)j du dv u≥0; v≥0; u+v≤n Z n Z n = jf(u)g(v)j du dv 0 0 = pn(f)pn(g): 1 + Then, pn is a semi norm so that Lloc(R ) is a commutative Fr´echet algebra, and, in fact, it is a radical Fr´echet algebra. + + Definition 2.1. Let ! be a continuous function R into R such that + !(0) = 1;!(t) > 0;!(s + t) ≤ !(s)!(t)(s; t 2 R ): Such a function ! is said to be a weight function. Let Z 1 1 1 + L (!) = ff 2 Lloc(R ): kfk = jf(t)j!(t) dt < 1g: 0 500 Hyuk Han Then with the same convolution multiplication as above, each L1(!) is a commutative Banach algebra without identity with respect to the indicated norm. 1 Proposition 2.2. Let !(t) be a weight function. Then, lim !(t) t t!1 always exists and 1 1 lim !(t) t = inf !(t) t : t!1 t>0 Proof. Let f(t) = ln !(t). Then f(t) is a continuous subadditive 1 function, that is, f(s + t) ≤ f(s) + f(t). Let α > inf f(t) be given, t>0 t 1 and choose s > 0 such that f(s) < α. We set s m = sup ff(t): s ≤ t ≤ 2sg; then m < 1 by the continuity of f. For any positive number n and real number t with (n + 1)s ≤ t ≤ (n + 2)s; since s ≤ t − ns ≤ 2s, we have f(t) ≤ f(ns) + f(t − ns) ≤ nf(s) + m: And so 1 n m f(t) < f(s) + t t t ns m ≤ α + : t t Letting t ! 1 and n ! 1 together so that (n + 1)s ≤ t ≤ (n + 2)s, we have 1 lim sup f(t) ≤ α: t!1 t Hence we have 1 1 lim ln(!(t) t ) = inf ln(!(t) t ): t!1 t>0 Since ln(·) is continuous, we have 1 1 lim !(t) t = inf !(t) t : t!1 t>0 1 + For f 2 Lloc(R ) n f0g, let α(f) = inf(supp f), where suppf is the support of f, that is the closure of the set where f is non-zero. Hence, α(f) ≥ a if and only if f = 0 almost everywhere on [0; a]. For zero function 0, we define α(0) = 1. Approximate identity of convolution Banach algebras 501 1 + Theorem 2.3. (Titchmarsh) Let f; g 2 Lloc(R ). Then, α(f ∗ g) = α(f) + α(g): The proof of Titchmarsh convolution theorem is found in [4]. 1 + Corollary 2.4. The algebra Lloc(R ) and its subalgebras are inte- gral domains. Especially L1(!) is an integral domain. If a > 0, we set 1 Ma = ff 2 L (!): α(f) ≥ ag: Then it is easy to show that Ma is a closed ideal in Banach algebra L1(!). These ideals are called the standard ideal of L1(!). + 1 Let !(t) be a weight function on , and set ρ = lim !(t) t . R t!1 Theorem 2.5. Let !(t) be a weight function. If ρ = 0, then L1(!) is a radical Banach algebra. Proof. Let χ be the characteristic function [a; b] where 0 < a < b. Let χ∗(n) denote the n-times convolution product of χ . Then by induction we have, supp χ∗(n) = [na; nb] and ∗(n) n−1 χ (t) ≤ (b − a) for t 2 [na; nb]: Hence we have, Z nb kχ∗(n)k ≤ (b − a)n−1 !(t) dt: na 1 1 Let 0 < " < be given. Choose t so that !(t) t < " for all t ≥ t . If e 0 0 na > t0 Z nb kχ∗(n)k ≤ (b − a)n−1 "t dt na "na ≤ (b − a)n−1 ; jln "j so that ∗(n) 1 a lim kχ k n ≤ (b − a)" : n!1 502 Hyuk Han 1 Since 0 < " < is arbitrary, e ∗(n) 1 lim kχ k n = 0: n!1 Hence χ is a quasi-nilpotent.