On two refinements of the bounded weak approximate identities Mohammad Fozouni1∗ and Raziyeh Farrokhzad2 1,2Department of Mathematics, Faculty of Basic Sciences and Engineering, Gonbad Kavous University, P. O. Box 163, Gonbad-e Kavous, Golestan, Iran, E-mails: [email protected]. ir : r_ [email protected]. ir Abstract Let A be a commutative Banach algebra with non-empty character space ∆(A). In this paper, we change the concepts of convergence and boundedness in the classical notion of bounded approximate identity. This work give us a new kind of approximate identity between bounded approx- imate identity and bounded weak approximate identity. More precisely, a net {eα} in A is a c-w approximate identity if for each a ∈ A, the Gel’fand transform of eαa tends to the Gel’fand transform of a in the compact-open topology and we say {eα} is weakly bounded if the image of {eα} under the Gel’fand transform is bounded in C0(∆(A)). MSC 2010: 46H05, 22D05, 43A15. Keywords: Banach algebra, approximate identity, character space, locally compact group. arXiv:1507.05884v3 [math.FA] 18 Dec 2018 1 Introduction In 1977, Jones and Lahr introduced a new notion of an approximate identity for a commutative Banach algebra A called ”bounded weak approximate identity (b.w.a.i.)” and gave an example of a semisimple and commutative semigroup algebra with a b.w.a.i. which has no approximate identity, bounded or unbounded. Indeed, a bounded net {eα} in A is a b.w.a.i., if for each a ∈ A and φ ∈ ∆(A), |aeα(φ)−a(φ)| −→ 0. Since ∆(A) is a locally compact Hausdoroff space, one may ask the following question: d b ∗Corresponding author 1 2 M. Fozouni, R. Farrokhzad What happens if the net {eα} satisfies the following condition(s); For each a ∈ A and compact subset K of ∆(A), supφ∈K |aeα(φ) − a(φ)| −→ 0, and what happens if we borrow the boundedness of {e } from C (∆(A)) in the α d 0 b sense that there exists an M > 0 such that for each α, keαk∆(A) < M? In this paper, we investigate the above conditions forb commutative Banach algebras with non-empty character spaces. This study provide for us a new kind of approximate identity between bounded approximate identity and bounded weak approximate identity. 2 Preliminaries Throughout the paper, suppose that A is a commutative Banach algebra and suppose ∆(A) is the character space of A, that is, the space consisting of all non-zero homomorphisms from A into C. A bounded net {eα} in A is called a bounded approximate identity (b.a.i.) if for all a ∈ A, limα aeα = a. The notion of a bounded approximate identity first arose in Harmonic analysis; see [1, Section 2.9] for a full discussion of approximate identity and its applications. For a ∈ A, we define a : ∆(A) −→ C by a(φ) = φ(a) for all φ ∈ ∆(A). Then a is in C (∆(A)) and a is called the Gel’fand transform of a. Note that 0 b b ∆(A) is equipped with the Gel’fand topology which turns ∆(A) into a locally b b compact Hausdoroff space; see [8, Definition 2.2.1, Theorem 2.2.3(i)]. Since for each φ ∈ ∆(A), kφk ≤ 1; see [8, Lemma 2.1.5], we have kak∆(A) ≤ kak, where k·k denotes the norm of A. b Suppose that φ ∈ ∆(A). We denote by Ac the space of all a ∈ A such that supp a is compact, and by Jφ the space of all a ∈ Ac such that φ∈ / supp a. Also, let M = ker(φ)= {a ∈ A : φ(a)=0}. bφ b Let X be a non-empty locally compact Hausdoroff space. A subalgebra A of C0(X) is called a function algebra if A separates strongly the points of X, that is, for each x, y ∈ X with x =6 y, there exists f ∈ A such that f(x) =6 f(y), and for each x ∈ X, there exists f ∈ A with f(x) =6 0. A function algebra A is called a Banach function algebra if A has a norm k·k such that (A, k·k) is a Banach algebra. A Banach function algebra A is called natural if X = ∆(A), that is, every character of A is an evaluation functional on some x ∈ X, or x −→ φx is a homeomorphism. In the latter case Mφx is denoted by Mx. A uniform algebra on X is a Banach function algebra (A, k·k) for which k·k is equivalent to the uniform norm k·kX of C0(X), where k·kX defined by kfkX = sup{|f(x)| : x ∈ X} for two refinements of bounded weak approximate identities 3 f ∈ C0(X). Let S be a non-empty set. By c00(S), we denote the space of all functions on S of finite support. A Banach sequence algebra on S is a Banach function algebra A on S such that c00(S) ⊆ A. A net {eα} in A is called a bounded weak approximate identity (b.w.a.i.) if there exists a non-negative constant C such that for each α, keαk < C and for all a ∈ A and φ ∈ ∆(A) lim |φ(aeα) − φ(a)| =0, α or equivalently, limα φ(eα) = 1 for each φ ∈ ∆(A). See [7] and [9] for more details. If ∆(A)= ∅, every bounded net in A is a b.w.a.i. for A. So, to avoid trivialities we will always assume that A is a Banach algebra with ∆(A) =6 ∅. In the case that A is a natural Banach function algebra, a bounded weak approximate identity {uα} is called a bounded pointwise approximate identity (BPAI); see [2, Definition 2.11]. Let A be a Banach function algebra. We say that A has bounded relative approximate units (BRAUs) of bound m if for each non-empty compact subset K of ∆(A) and ǫ> 0, there exists f ∈ A with kfk < m and |1 − ϕ(f)| < ǫ for all ϕ ∈ K; see [2, Section 2.2]. In this paper, we introduce two notions of approximate identities of a Ba- nach algebra A depending on its character space and provide some illuminating examples to show the difference of our notions from those previously known. 3 Definitions Suppose K(∆(A)) denotes the collection of all compact subsets of ∆(A), and τco denotes the compact-open topology of C0(∆(A)). Definition 3.1. A c-w approximate identity for A (in which, by ”c-w” we mean ”compact-weak” ) is a net {eα} in A such that for each a ∈ A and K ∈K(∆(A)) lim kaeα − akK =0. α d b If the net {eα} is bounded, we say that it is a bounded c-w approximate identity (b.c-w.a.i.) for Banach algebra A. The Banach algebra A has c-w approximate units (c-w.a.u.), if for each a ∈ A, K ∈ K(∆(A)) and ǫ > 0, there exists e ∈ A such that kae − akK < ǫ. The approximate units have bound m, if e can be chosen with kek < m. Clearly, if A b b has a b.c-w.a.i., then it has b.c-w.a.u. 4 M. Fozouni, R. Farrokhzad Recall that A[τ] denotes the topological algebra A, where τ indicates the topology of the underlying topological vector space A; see [3] for a general theory of topological algebras. Definition 3.2. A net {eλ} in A is a weakly bounded c-w approximate identity (w.b.c-w.a.i.) if the net {eλ} is a b.a.i. for the topological algebra A[τco], that is, for each a ∈ A and K ∈ K(∆(A)), lim kae − ak = 0 and there exists a b α λ K b constant M > 0 such that for all λ and K ∈K(∆(A)), c b (3.1) PK (eλ) = sup{|φ(eλ)| : φ ∈ K} = keλkK < M. b b Since, each eλ is in C0(∆(A)), relation (3.1) is equivalent to keλk∆(A) < M. Also, A has a weakly bounded c-w approximate units (w.b.c-w.a.u.) of bound b b m, if for each a ∈ A, K ∈K(∆(A)) and ǫ> 0, there exists e ∈ A with kek∆(A) < m and kae − ak < ǫ. K b It isb a routineb calculation that each b.c-w.a.i. is a w.b.c-w.a.i. We will show in the next section that this two concepts are different. For a natural uniform algebra A, one can see immediately that each w.b.c- w.a.i. is a b.c-w.a.i. Remark 1. We can give a more general version of Definition 3.2 as follows: Let τ be a topology on C0(∆(A)) generated by the saturated family (pi) of seminorms on ∆(A); see [3, Definition 1.7] for more details of locally convex topological algebras and saturated family. A net {aα} in A is a w.b.c-w.a.i. for A if {aα} is a b.a.i. for the topological algebra A[τ], that is, for each a ∈ A, a a −→τ a and there exists M > 0 such that for each i and α, p (a ) < M. But in α c b i α the current paper we only focus on the case that τ = τ . d b co c 4 Examples For a locally compact group G and 1 <p< ∞, let Ap(G) denote the Fig`a Talamanca-Herz algebra which is a natural Banach function algebra on G; see [5].