TERNARY HOMOMORPHISMS BETWEEN UNITAL TERNARY C * -ALGEBRAS

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1 THE PULISHING HOUSE PROCEEDINGS OF THE ROMNIN CDEMY, Series, OF THE ROMNIN CDEMY Volume, Number/0, pp TERNRY HOMOMORPHISMS ETWEEN UNITL TERNRY C -LGERS M. ESHGHI GORDJI, Th. M. RSSIS Departmet of Mathematics, Sema Uiversity, P. O. ox 595 6, Sema, Ira Departmet of Mathematics, Natioal Techical Uiversity of thes, Zografou, Campus 5780 thes, Greece madjid.eshaghi@gmail.com Let, be two uital terary C -algebras. We prove that every almost uital almost liear mappig h: which satisfies [ u vy] ) [ ( ) ( h u h for all uv, U( ), all y, ad all 0,,,..., is a terary homomorphism. lso, for a uital terary C -algebra of real rak zero, every almost uital almost liear cotiuous mappig h: is a terary homomorphism whe [ u vy] ) [ ( ) ( h u h holds for all uv, I( sa), all y, ad all 0,,,.... Furthermore, we ivestigate the Hyers-Ulam-Rassias stability of terary homomorphisms betwee uital terary C -algebras. Key words: Terary homomorphism, Terary C -algebra.. INTRODUCTION Terary algebraic operatios were cosidered i the 9th cetury by several mathematicias ad physicists such as Cayley [5] who itroduced the otios of cubic matrix, which i tur ([,7,,4,,5]) was geeralized by Kapraov at al. []. Followig the termiology of Ref. [8], a oempty set G with a terary operatio[.,.,.] : G G is called a terary groupoid ad is deoted by (G,[.,.,.]). The terary groupoid (G,[.,.,.]). is called commutative if [ x, x, x] [ xσ() xσ() xσ() ] for all x, x, x G ad all permutatios σ of {,,}. If a biary operatio is defied o G such that [ x, y, z] ( x y) z for all x, y, z G, the we say that [.,.,.] is derived from. We say that (G,[.,.,.]) is a terary semigroup if the operatio [.,.,.] is associative, i.e., if [[ x, y, z], u, v] [ x,[ y, u], v] [ x, y,[ u, v]] holds for all x, y, u, v G (see Ref. [,, ]). C -terary algebra is a complex aach space, equipped with a terary produc ( X, Y, Z) ( X, Y, Z) of ito, which is C -liear i the outer variables, cojugate C-liear i the middle variable, ad associative i the sese that [ x, y,[ w, v]] [ x,[ w, y], v] [[ x, y, z], w, v] ad satisfies [ x y, z] x. y. z, ad [ x, xx, ] x. If a C -terary algebra (,[.,.,.]) has a idetity, i.e., a elemet e such that x [ x, e, e] [ e, e, x] for all x, the it is routie to verify that, edowed with xoy : [ x, e, y] ad x : [ e, x, e ], is a uital C -algebra. Coversely, if (, o) is a uital C -algebra, the [ x y, z] : xo y oz, makes ito a C -terary algebra. C-liear mappig H : is called a C -terary algebra homomorphism if H ([ x, y, z]) [ Hx), H( y), H( z)],

2 90 M. Eshaghi Gordji, Th. M. Rassias for all x, y, z. Terary structures ad their geeralizatio the so-called -ary structures, raise certai hops i view of their applicatios i physics [, 0,,, 6]. The study of stability problems origiated from a famous talk give by S. M. Ulam [4] i 940: uder what coditio does there exist a homomorphism ear a approximate homomorphism? I the ext year 94, D. H. Hyers [5] aswered affirmatively the questio of Ulam. This stability pheomeo is called the Hyers-Ulam stability of the additive fuctioal equatio g ( x + y) g( x) + g( y). geeralized versio of the theorem of Hyers for approximately liear mappigs was give by Th.M. Rassias []. The stability pheomeo that was itroduced ad proved by Th. M. Rassias is called Hyers-Ulam- Rassias stability. The stability problems of several fuctioal equatios have bee extesively ivestigated by a umber of authors ad there are may iterestig results cocerig this problem [6,9,,,4 8,0,7 ]. Throughout this paper, let be a uital terary C -algebra with uit e, ad a uital terary aach algebra with uit elemet e. Let U () be the set of uitary elemets i, sa : { x x x }, ad I( sa) { v sa v, v Iv( )}. I this paper, we prove that every almost uital almost liear mappig h : is a homomorphism whe h ( [uvy] ) [ y )] for all ) u, v U (, all y, ad all 0,,,.... lso, for a uital terary C -algebra of real rak zero, every almost uital almost liear cotiuous mappig h : is a terary homomorphism whe h ( [uvy] ) [ y )] holds for all u ), v I ( sa, all y, ad all 0,,,.... Furthermore, we ivestigate the Hyers-Ulam-Rassias stability of terary -homomorphisms betwee uital terary C -algebras. Note that a uital terary C -algebra is of real rak zero, if the set of ivertible self-adjoit elemets is dese i the set of self-adjoit elemets [4]. We deote the algebraic ceter of by Z ().. TERNRY HOMOMORPHISMS ON UNITL TERNRY C -LGERS Followig the same approach as i [6], we obtai the ext theorem. Theorem.. Let f : be a mappig such that f ( 0) 0 ad that f ( [uvy] ) [ f ( f ( ) vf ( y )], (.) for all u, v U ( ), all y, ad all 0,,,... ssume as well that there exists a fuctio ( ) [ ) φ: {0} 0, such that φ ( xy, ) φ ( x, y) < for all x, y {0 } ad that 0 µ x+µ y f ( ) µ f( x) µ f( y) φ ( x, y) (.) f ( e) for all µ T ad all x, y. If lim I( sa) Z( ), the the mappig f : is a terary homomorphism. Proof. Set µ i (.), it follows from Theorem of [9] that there exists a uique additive mappig h : such that f ( x) x) ( φ ( x, x) +φ ( x, x) ) (.) f ( x) for all x {0}. This mappig is give by h ( x) lim for all x. y the same reasoig as i the proof of Theorem of [6], h is C-liear. It follows from (.) that

3 Terary homomorphisms betwee uital terary C -algebras 9 f ([uvy] ) [ f ( f ( f ( [ uvy] ) lim lim [ f (, (.4) 9 9 for all u, v U ( ), all y. Sice h is additive, the by (.4), we have [ uvy] ) [ uv( [ f ( for all u, v U ( ) ad all y. Hece, f ( y) [ uvy] ) lim[ ] [ (.5) f ( e) for all u, v U ( ) ad all y. y the assumptio, we have e) lim U ( ) hece, it follows by (.4) ad (.5) that [ e) e) [ eey] ) [ e) e) f ( for all y. We deote the uit elemet of by e. Sice h (e) belogs to I( sa ), the hy ( ) [ e e hy ( )] [[ he () e he ( )] ehy ( )] [ he () [ e he ( ) e ] hy ( )] [ he () [ e e he ( )] hy ( )] [() he e [ e hehy ( ) ( )] ] [ he () [ eehe ( )] hy ( )] [() he e[ ehehy ( ) ( )] ] [ he () e[ [ he () ehe ( )] he ( ) hy ( )] ] [ he () e [ ( ) [ ( ) ( ) ( )] ] ] he e hehehy [ he () e[ he ( ) e[ hehe ( ) ( ) f( ] ] [ () [[ () ( )] he () f( he e he ehe ] [ he () [ e e he ( )] hy ( )] [ he () e [ e he ( ) f( ] [ he () [ ehe ( ) e] f( [[ he () ehe ( )] f( e [ eef( f( y), for all y. We have to show that f is a terary homomorphism. For every a, b, we defie a b : [ aeb]. The : is a biary product for which (, ) may be cosidered as a (biary) C -algebra. lso, we have a U(, [] ) if ad oly if a U ((, )) for all a. Now, let a, b. y Theorem 4..7 of [], m a, b are fiite liear combiatios of uitary elemets, i.e., a ciui, b d jv j( ci, d j C, ui, v j U( )), it i j follows from (.5) that m f([ aby] ) [ aby] ) [( cu i i)( d jvj) y]) i j m m h [ cid juivjy] h cid j [ uv i jy] i j i j m m cid j [ uv i j] ) cid j [ ui) vj) i j i j m [ cidj ui) vj) y) ] [ cu i i) d jvj) i j i j [ hahbhy ( ) ( ) ( )], for all y. m

4 9 M. Eshaghi Gordji, Th. M. Rassias 4 This completes the proof of theorem. Corollary.. Let p (0,), θ [0, ) be real umbers. Let f : be a mappig such that f ( 0) 0 ad that f ( [uvy] ) [ f ( f ( ) vf ( )] for all u, v U ( ), all y, ad all 0,,,.... Suppose that µ x+µ y p p f( ) µ f( x) µ f( y) θ ( x + y ) for all µ T f ( e) ad all x, y. If lim I( sa), the the mappig f : is a terary homomorphism. p p Proof. Set ( x, y): ( x y ) φ + all x, y. The by Theorem. we get the desired result. Theorem.. Let be a terary C -algebra of real rak zero. Let f : be a cotiuous mappig such that f ( 0) 0 ad that f ( [uvy] ) [ f ( f ( ) vf ( y )] (.6) for all u, v I( sa) all y, ad all 0,,,.... Suppose that there exists a fuctio φ: ( {0} ) [ 0, ) satisfyig (.) ad φ ( xy, ) < for all x, y {0 } f ( e). If lim I( sa), the the mappig f : is a terary homomorphism. Proof. y the same reasoig as i the proof of Theorem., there exists a uique C-liear mappig h : satisfyig (.). It follows from (.6) that f ([uvy] ) [ f ( f ( f ( [ uvy] ) lim lim [ f ( (.7) 9 9 for all u, v I( sa), ad all y. y additivity of h ad (.7), we obtai that [ uvy] ) [ uv( [ f (, for all u, v ( ) I sa ad all y. Hece, f ( y) [ uvy] ) lim[ ] [, for all u, v I( sa) ) ad all y. (.8) y the assumptio, we have f ( e) e) lim U ( ). Similar to the proof of Theorem., it follows from (.7) ad (.8) that h f o. So h is cotiuous. O the other had is real rak zero. Oe ca easily show that I( sa ) is dese i { x sa : x }. Let u, v { x sa : x } There are { t },{ z} i I( sa ) such that limt u, limz v. Sice h is cotiuous, it follows from (.8) that [ uvy] ) lim( t z y)) lim[( t z ) lim[ t ) z ) [, (.9)

5 5 Terary homomorphisms betwee uital terary C -algebras 9 a+ a b+ b for all y. Now, let a, b. The we have a a+ i a, b b+ ib, where a:, b: a a b b ad a :, b : are self-adjoit. First cosider a b 0, a, b 0. Sice h is C-liear, it i i follows from (.9) that ( ) ([ ] a b f( [ aby] ) h [ aby] h ab y ) h a b a b a b a b a b h a b h h y) a b a b a b h a h b hy ( ) a ) b ) hy ( ) f ( a) f( b) f( y), for all y. for all a b Now, cosider a b 0, a, b 0. Sice h is C-liear, it follows from (.9) that a b f ([ aby] ) [ aby] ) [ iaiby] ) h a b a b a b a b a b h a b h h y) a b a b a b h i a h i b y) ia ) ib ) y) a b ( ) ( ) ( y) f a f b f, for all y. Suppose a b 0, a, b 0. The by (.9), we have a b f ([ aby] ) [ aby] ) ( ) a ib y ) h i a b a b a b a b i a b h i a b h h y) a b a b a b h a h i b y) [ a ) ib ) y) ] a b ( ) ( ) ( y) f a f b f, for all y. Similarly we ca show that ([ ] ) ( ) ( ) ( ) f aby f a f b f y, y if a b 0, a, b 0. I the case that b 0, a, a, b 0, we have ([ ] ) [ ] ( ) ([( ) ] ) ([ ] ) ([ ] ) f aby h aby h a + ia b y h ab y + ih a b y a b a b h y + ih y a b a b a b a b

6 94 M. Eshaghi Gordji, Th. M. Rassias 6 for all a b a b a b h y + i a b h y a b a b a b h h y + i a b h h y a b a b h a h b y + i h a h b y a b a b ( ) [ h h h y ] i a ) b ) y) h a ia b ) y) a b + + ( ) ( ) ( ) ( ) ( ) ( y) f a f b f, for all y. y a same reasoig above, we ca show that ([ ] ) ( ) ( ) ( ) f aby f a f b f y y if a 0, a, b, b 0. Now cosider b 0, a, a, b 0. The by (.9), we have ([ ] ) [ ] ( ) [ ] ( ) ( )( ) ( ) ([ ] ) f aby h aby h a + ia ib y h ia b y h a b y a b a b ih y h y a b a b a b a b a b a b i a b h y a b h y a b a b i a b h h y) a b h h y) a b a b ( + ( h a ih h y ih ih h y b a b a b a b ( ) ( ) ( ) ( ) ( ) ( ) ( a a) ( b) ( ) h a ih b h y + ih a ih b h y h + i h i h y (.0) ( ) ( ) ( y) f a f b f, for all y. lso, by a same reasoig, we ca see that ([ ] ) [ ( ) ( ) ( )] f aby f a f b f y, for all y if a 0, a, b, b 0. Fially cosider that a, a, b, b 0. The by (.9), we have ( ) f ([ aby] ) [ aby] ) h ( a ia)( b ib) y + + ([ ] ) ([ ] ) ([ ] ) ([ ] ) h ab y + h iab y + h ia b y h ia b y a b a b h a b y ih a b y a b + + ih a b a b a b a b a b a b a b + a b + h i h a b a b a b h a b a b

7 7 Terary homomorphisms betwee uital terary C -algebras 95 a b a b + i a b h a b h a b a b a b h h y + i a b h h y a b a b + i a b h h y a b h h y a b a b h a h b y + h a ih b y a b a b + ih a h b y + ih a ih b y a b a b [ a) b) y) ] + [ ) i ) y) ] a b [ i ] [ ] a) b) y) i ) i ) y) a b [ h a+ ia h b+ ib h y ] [ f a f b f y ], for all y. ( ) ( ) ( ) ( ) ( ) ( ) Hece, f ([ aby] ) [ f ( a) f ( b) f ( for all a, b, y ad f is terary homomorphism. Corollary.4. Let be a terary C -algebra of real rak zero. Let p ( 0,), θ [0, ) be real umbers. Let f : be a mappig such that f ( 0) 0 ad that f ( [uvy] ) [ f ( f ( ) vf ( y )] (.) for all u, v I( sa), all y, ad all 0,,,.... Suppose that µ x+µ y p p f f( x) f( y) µ µ θ x + y f ( e) for all µ T ad all x, y. If lim U ( ), the the mappig f : is a terary homomorphism. p p Proof. Set φ ( xy, ): x + y for all x, y. The by Theorem. we get the desired result. + REFERENCES. V. bramov, R. Kerer ad. Le Roy, Hypersymmetry a Z graded geeralizatio of supersymmetry, J. Math. Phys., 8, 650, F. agarello, G. Morchio, Dyamics of mea-field spi models from basic results i abstract differetial equatios, J. Stat. Phys., 66, , 99.. N. azuova,. orowiec ad R. Kerer, Uiversal differetial calculus o terary algebras, Lett. Math. Phys., 67, 95, L. row ad G. Pederse, C -algebras of real rak zero, J. Fuct. al., 99,.49, Cayley, O the 4 cocomitats of the terary cubic, m. J. Math., 4,, S. Czerwik, Fuctioal Equatios ad Iequalities i Several Variables, World Scietific Publishig Compay, Sigapore, New Jersey, Lodo, Y. L. Daletskii ad L.. Takhtaja, Leibiz ad Lie algebra structures for Nambu algebra, Lett. Math. Phys., 9, 7, S. Duplij, Terary Hopf lgebras, Symmetry i Noliear Mathematical Physics, Part, (Kyiv, 00), 49448; Pr. Ist. Mat. Nats. kad. Nauk Ukr. Mat. Zastos. 4, Part, ; Nat. kad. Nauk Ukrai, Ist. Mat. Kiev, M. Eshaghi Gordji ad H. Khodaei, Solutio ad stability of geeralized mixed type cubic, quadratic ad additive fuctioal equatio i quasi-aach spaces, Noliear alysis.-tm., 7, (009).

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