Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 206165, 9 pages

http://dx.doi.org/10.1155/2011/206165

## Ternary Weighted Function and Beurling Ternary Banach Algebra

Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran

Received 26 July 2011; Accepted 29 August 2011

Academic Editor: GabrielΒ Turinici

Copyright Β© 2011 Mehdi Dehghanian and Mohammad Sadegh Modarres Mosadegh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let be a ternary semigroup. In this paper, we introduce our notation and prove some elementary properties of a ternary weight function on . Also, we make ternary weighted algebra and show that is a ternary Banach algebra.

#### 1. Introduction

The notion of an n-ary group was introduced by DΓΆrnte [1] (inspired by E. NΓΆther) and is a natural generalization of the notion of a group and a ternary group considered by Certaine [2] and Kasner [3].

In the first section, which have preliminary character, we review some basic definitions and properties related to ternary groups and semigroups (cf. also Belousov [4] and Rusakov [5]).

Dudek [6], FeΔzullaev [7], Kim and Fred [8], and Lyapin [9] have also studied the properties of the ternary semigroups.

The present paper may be described as an introduction to harmonic analysis on ternary semigroups. In Section 2, we introduce our notation and prove some elementary properties of a ternary weight function. In Section 3, we make ternary weighted algebra and show that is a ternary Banach algebra.

*Definition 1.1. *A nonempty set with one ternary operation is called a ternary groupoid and denoted by .

We say that is a ternary semigroup if the operation is associative, that is, if
hold for all .

*Definition 1.2. *A ternary semigroup is a ternary group if for all , there are such that

One can prove (post [10]) that elements , and are uniquely determined. Moreover, according to the suggestion of post [10], one can prove (cf. Dudek et al. [11]) that, in the above definition, under the assumption of the associativity, it suffices only to postulate the existence of a solution of , or equivalently, of .

In a ternary group, the equation has a unique solution which is denoted by and called the skew element to (cf. DΓΆrnte, [1]). As a consequence of results obtained in DΓΆrnte [1], we have the following theorem.

Theorem 1.3. *In any ternary group for all , the following identities take place:
*

Other properties of skew elements are described in Dudek [12] and I. Dudek and W. A. Dudek [13].

*Definition 1.4 (see [14]). *Let be a ternary group, itβs inverse operation, and be equipped with a topology . Then, we say that is a topological ternary group if and only if(i)ternary operation is continuous in , and(ii)the 2-operation is continuous in .

Let be a ternary group and any subset of . We denote by denote set of all (skew element) such that , that is,

*Definition 1.5. *A ternary Banach algebra is a complex Banach space , equipped with a ternary product of into , which is associative in the sense that , and satisfy .

Let be a ternary Banach algebra and , and subsets of . We define

Let and be a ternary Banach algebra. A linear mapping is called to be a ternary homomorphism if .

*Definition 1.6. *Let be a ternary semigroup, and let denote the set of mappings of into such that
with pointwise addition and scalar multiplication and with the norm

Let be a ternary semigroup; for , we define and it is called ternary convolution product on .

Theorem 1.7 (see [15]). *Let be a ternary semigroup, then ternary convolution product on is associative.*

Theorem 1.8 (see [15]). *Let be a ternary semigroup, then under the usual norm is a ternary Banach algebra.*

#### 2. Ternary Weight Function on Ternary Semigroup and Ternary Group

*Definition 2.1. *A ternary weight on ternary semigroup is a positive real function such that

*Remark 2.2. *If and are two ternary weight function, then is a ternary weight function.

*Example 2.3. *Let with be a ternary group. For , define . Then, is a weight ternary function.

Theorem 2.4. *Let be a compact subset of a topological ternary group , a ternary weight function on , and the interior of is nonempty for some . Then, there exists such that
**
for all .*

*Proof. *First we establish the existence of . To that end, for , let
Clearly . Choose such that (interior ). Fix , and let . Then is an open neighborhood of the , and, hence, by compactness of , there exist such that
Now, define by
If , then for some and , and, hence,
Thus, , for all .

Next let
and suppose that . Then, there exists a sequence in such that . Since
we must have , which contradicts boundedness of on the compact set .

Proposition 2.5. *Let be a ternary weight function on a ternary semigroup such that is finite for some . Then , for all .*

*Proof. *Suppose that for some . Choose such that . Then
Thus, is an infinite subset of contrary to the hypothesis. Thus, the result follows.

Corollary 2.6. *Let be a ternary weight function on a compact topological ternary group , and the interior of is nonempty for some . Then, for all .*

*Proof. *This follows trivially from Theorem 2.4 and Proposition 2.5.

Let be the set of all complex-valued continuous functions on , the space of all bounded functions in under the supremum norm and a continuous ternary weight function on . We define the space of ternary weighted continuous functions by with the norm given by

Define for all .

Lemma 2.7. *Let be a continuous ternary weight function on a ternary semigroup . Then, , for all and if and only if is finite.*

*Proof. *To establish the necessary condition, we note that and so if , we have . Hence,
Conversely, suppose that is finite, for all , and let
and so . Hence, .

Corollary 2.8. *Let be a continuous ternary weight function on a compact topological ternary group , and the interior of is nonempty for some . Then, for all and .*

*Proof. *For all and in , we have
and so . By Lemma 2.7, our result follows.

#### 3. Ternary Beurling Algebra

Let be a ternary semigroup. In [15] introduce ternary Banach algebra . Now, we make ternary Beurling algebra and show some elementary properties.

*Definition 3.1. *Let be a ternary semigroup, let be a ternary weight on , and let denote the set of mappings of into such that
with pointwise addition and scalar multiplication, with ternary convolution
and with the norm

Theorem 3.2. *Let be a ternary semigroup and be a ternary weight function on . Then, is a ternary Banach algebra.*

*Proof. *By Theorem 1.7, we need only to check that
for all . This justifies the last inequality of the following calculation:

The ternary algebra is called the ternary Beurling algebra on associated with the ternary weight .

If , we obtained .

If for all , then is a ternary subalgebra of , and if for all , then is a ternary subalgebra of .

Proposition 3.3. *Let and are ternary weights on and is a continuous nonzero ternary homomorphism. If is norm dense in . Then, the norm closure of contains the norm closure of for all in .*

*Proof. *Since is norm dense in , we can find a sequence in for which , with the limit taken in the norm topology. By the continuity of , this implies that
Hence, , and, therefore, the norm closure of as well, belongs to the norm closure of . This completes the proof of the proposition.

Proposition 3.4. *Let and are ternary weights on , and is a continuous nonzero ternary homomorphism. If is norm dense in . Then, the norm closure of contains the norm closure of for all in .*

*Proof. *Since is norm dense in , we can find sequences and in for which , with the limit taken in the norm topology. By the continuity of , this implies that

Hence, , and, therefore, the norm closure of as well, belongs to the norm closure of . This completes the proof of the proposition.

*Definition 3.5. *Let be a ternary group, let be a ternary weight on , and let denote the set of mappings of into such that
with pointwise addition and scalar multiplication, with ternary convolution
and with the norm

It is clear that is a ternary Banach algebra.

Theorem 3.6. *Let be a topological ternary group, a ternary weight function on such that interior not empty for some . Then,*(i)*every compactly supported function in belongs to and*(ii)*For every and , and .*

*Proof. *(i) It is immediately since is bounded on compact subsets of by Theorem 2.4.

(ii) It follows simply from submultiplicativity of

Let be a ternary group and be a ternary weight function, define such that For , we have and .

Theorem 3.7. *Let be a discrete uncountable ternary group. Then, there exists set such that and .*

*Proof. *Since is uncountable, for some , the set is also uncountable. Note that . Now, if , then
so that .

#### References

- W. Dörnte, βUntersuchungen über einen verallgemeinerten Gruppenbegriff,β
*Mathematische Zeitschrift*, vol. 29, no. 1, pp. 1β19, 1929. View at Publisher Β· View at Google Scholar - J. Certaine, βThe ternary operation $(abc)=a{b}^{-1}c$ of a group,β
*Bulletin of the American Mathematical Society*, vol. 49, pp. 869β877, 1943. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - E. Kasner, βAn extension of the group concept,β
*Bulletin of the American Mathematical Society*, vol. 10, pp. 290β291, 1904. View at Google Scholar - V. D. Belousov,
*n-ary Quasigroups*, Shtintsa, Kishinev, Moldova, 1972. View at Zentralblatt MATH - S. A. Rusakov,
*Some Applications of n-ary Group Theory*, Belaruskaya Navuka, Minsk, Belarus, 1998. - I. M. Dudek, βOn ternary semigroups connected with para-associative rings,β
*Ricerche di Matematica*, vol. 35, no. 2, pp. 191β203, 1986. View at Google Scholar Β· View at Zentralblatt MATH - R. B. Feĭzullaev, βTernary semigroups of locally homeomorphic mappings,β
*Doklady Akademii Nauk SSSR*, vol. 252, no. 5, pp. 1063β1065, 1980. View at Google Scholar - K. H. Kim and W. R. Fred, βTernary semigroups associative on each pair of factors,β
*Simon Stevin*, vol. 54, no. 2, pp. 65β74, 1980. View at Google Scholar Β· View at Zentralblatt MATH - E. S. Lyapin, βRealization of ternary semigroups in binary semigroups,β in
*Modern Algebra*, pp. 43β48, leningrad University, Leningrad, Russia, 1981. View at Google Scholar Β· View at Zentralblatt MATH - E. L. Post, βPolyadic groups,β
*Transactions of the American Mathematical Society*, vol. 48, pp. 208β350, 1940. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - W. A. Dudek, K. Gazek, and B. Gleichgewicht, βA note on the axioms of n-Groups,β in
*Universal Algebra*, vol. 29, pp. 195β202, Colloquia Mathematica Societatis Janos Bolyai, Esztergom, Hungary, 1977. View at Google Scholar - W. A. Dudek, βAutodistributive n-Groups,β
*Annales Societatis Mathematicae Polonae*, vol. 23, no. 1, pp. 1β11, 1993. View at Google Scholar - I. Dudek and W. A. Dudek, βOn skew elements in n-Groups,β
*Demonstratio Mathematica*, vol. 14, no. 4, pp. 827β833, 1981. View at Google Scholar - J. Ušan, βOn topological n-groups,β
*Mathematica Moravica*, vol. 2, pp. 149β159, 1998. View at Google Scholar - M. Dehghanian and S. M. S. Modarres Mosadegh, βTernary Banach algebra ${l}^{1}(S)$ and approximate identity,β In press.