Abstract
Let A be a ternary Banach algebra. We prove that if A has a left-bounded approximating set, then A has a left-bounded approximate identity. Moreover, we show that if A has bounded left and right approximate identities, then A has a bounded approximate identity. Hence, we prove Altman’s Theorem and Dixon’s Theorem for ternary Banach algebras.
1. Introduction
Ternary algebraic operations were considered in the 19th century by several mathematicians such as Cayley [1] who introduced the notion of cubic matrix which in turn was generalized by Kapranov et al. in [2]. The comments on physical applications of ternary structures can be found in [3–7].
A nonempty set with a ternary operation is called a ternary groupoid and denoted by . The ternary groupoid is called a ternary semigroup if the operation is associative, that is, if holds for all . A ternary semigroup ) is a ternary group if for all , there are such that where the elements are uniquely determined (see [8]).
A ternary Banach algebra is a complex Banach space , equipped with a ternary product of into , which is associative in the sense that and satisfies . An element is an identity of if for all .
Assume that is a ternary Banach algebra a bounded net is a left-bounded approximate identity for if for all . Similarly, a bounded net is a right-bounded approximate identity for if for all . Also, is a middle-bounded approximate identity for if for all . A net is a bounded approximate identity for if is a left-, right-,and middle-bounded approximate identity for .
For ternary Banach algebra , a set is said to be an approximating set for ( and are bounded subsets of ) if for every , and every , there exist , such that .
Existence of bounded approximating set for binary Banach algebras guarantees existing of bounded approximate identity (Altman’s Theorem [9, Proposition 2, page 58] or [10]) and also this notion generalized for commutative Fréchet algebras [11]. For normed algebra with left-bounded approximate identity and right-bounded approximate identity, Dixon [12] proved that has a bounded approximate identity [13, Proposition 2.9.3].
In this paper, we prove Altman’s Theorem and Dixon’s Theorem for ternary Banach algebras. By , we mean the quasiproduct between elements of binary algebra which are defined by .
2. Main Results
We start our work with the following theorem which can be regarded as a version of Altman’s Theorem for ternary Banach algebras.
Theorem 2.1. Let be a ternary Banach algebra and be bounded subsets of such that for given and there are and , . Then possess a left-bounded approximate identity.
Proof. Let , and set
For proof of theorem, it is sufficient to show that for every finite subset , there exists such that for every .
Step 1. Let be singleton. Then, there are and such that , and
Letting , then
Step 2. Let . There is a such that , and for there is a such that
Put and . Then
for .Step 3. Now, suppose that obtained results in Steps 1 and 2 are true for . Let , and let . There exist such that , for , where and are defined as in Step 2. Also, we can choose and such that ,
and . Then for every we have
Let be the collection of all finite subsets of and . Then is a direct set with the following partial order:
Now, we can choose a bounded approximate identity for .
Now, we prove Dixon’s Theorem for ternary Banach algebras. Hence, we prove that if a ternary Banach algebra has both left- and right-bounded approximate identities, then it has a bounded approximate identity.
Theorem 2.2. Let be a ternary Banach algebra and and be bounded left and right approximate identities of , respectively. Then has a bounded approximate identity.
Proof. Consider . We claim that is a bounded approximate identity for . Boundedness of mentioned net is clear. Therefore, we have to show that is a right, left, and middle approximate identity for .
Step 1. is a left approximate identity. Because
where , and .Step 2. is a right approximate identity because
Step 3. By the similar method, we show that the net is a middle approximate identity:
This completes the proof of theorem.