In this paper functional equations related to derivations on semiprime rings and standard operator algebras are investigated. We prove, for example, the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X into itself and let A(X) ⊂ L(X) be a standard operator algebra. Suppose there exist linear mappings D,G:A(X) → L(X) satisfying the relations D(A3)=D(A2)A+A2G(A),G(A3)=G(A2)A+A2D(A) for all A A(X). In this case there exists B L(X) such that D(A)=G(A)=[A,B] holds for all A A(X)

Nejc Širovnik

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Nejc Sirovnik

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Glasnik Matematicki

On functional equations related to derivations in semiprime rings and standard operator algebras

Širovnik, Nejc

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GLASNIK MATEMATIČKIVol. 47(67)(2012), 95 – 104ON FUNCTIONAL EQUATIONS RELATED TODERIVATIONS IN SEMIPRIME RINGS AND STANDARDOPERATOR ALGEBRASNejc ŠirovnikUniversity of Maribor, SloveniaAbstract. In this paper functional equations related to derivationson semiprime rings and standard operator algebras are investigated. Weprove, for example, the following result, which is related to a classicalresult of Chernoff. Let X be a real or complex Banach space, let L(X)be the algebra of all bounded linear operators of X into itself and letA(X) ⊂ L(X) be a standard operator algebra. Suppose there existlinear mappings D,G : A(X) → L(X) satisfying the relations D(A3) =D(A2)A+A2G(A), G(A3) = G(A2)A+A2D(A) for all A ∈ A(X). In thiscase there exists B ∈ L(X) such that D(A) = G(A) = [A,B] holds for allA ∈ A(X).Throughout, R will represent an associative ring with center Z(R). Asusual we write [x, y] for xy − yx. Given an integer n ≥ 2, a ring R is saidto be n−torsion free, if for x ∈ R, nx = 0 implies x = 0. Recall that aring R is prime, if for a, b ∈ R, aRb = (0) implies a = 0 or b = 0, and issemiprime in case aRa = (0) implies a = 0. We denote by Qs the symmetricMartindale ring of quotients. For the explanation of Qs we refer the readerto [2]. Let A be an algebra over the real or complex field and let B be asubalgebra of A. A linear mapping D : B → A is called a linear derivationin case D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R. In case wehave a ring R, an additive mapping D : R → R is called a derivation, ifD(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R and is called a Jordanderivation in case D(x2) = D(x)x + xD(x) is fulfilled for all x ∈ R. Aderivation D is inner in case there exists a ∈ R such that D(x) = [x, a] holds2010 Mathematics Subject Classification. 16N60, 46B99, 39B42.Key words and phrases. Prime ring, semiprime ring, Banach space, standard operatoralgebra, derivation, Jordan derivation, Jordan triple derivation.This research has been supported by the Research Council of Slovenia.9596 N. ŠIROVNIKfor all x ∈ R. Every derivation is a Jordan derivation. The converse is ingeneral not true. A classical result of Herstein ([9]) asserts that any Jordanderivation on a 2−torsion free prime ring is a derivation. A brief proof ofHerstein’s result can be found in [5]. Cusack ([8]) generalized Herstein’s resultto 2−torsion free semiprime rings (see also [3] for an alternative proof). Forresults in the spirit of Herstein’s theorem we refer to [1,6,10,11]. An additivemapping D : R → R, where R is an arbitrary ring, is called a Jordan triplederivation in case D(xyx) = D(x)yx + xD(y)x + xyD(x) holds for all pairsx, y ∈ R. One can easily prove that any Jordan derivation D on an arbitrary2−torsion free ring R is a Jordan triple derivation (see, for example, [5]).Let X be a real or complex Banach space and let L(X) and F (X) denotethe algebra of all bounded linear operators on X and the ideal of all finiterank operators in L(X), respectively. An algebra A(X) ⊂ L(X) is said to bestandard in case F (X) ⊂ A(X). Let us point out that any standard operatoralgebra is prime, which is a consequence of Hahn-Banach theorem.Brešar ([4]) has proved the following result.Theorem 1. Let R be a 2−torsion free semiprime ring and let D : R → Rbe a Jordan triple derivation. In this case D is a derivation.Motivated by Theorem 1, Vukman, Kosi-Ulbl and Eremita ([16]) haveproved the following result.Theorem 2. Let R be a 2-torsion free semiprime ring. Suppose thereexists an additive mapping T : R → R such thatT (xyx) = T (x)yx− xT (y)x+ xyT (x)for all x, y ∈ R. Then there exists q ∈ Qs such that 2T (x) = qx + xq for allx ∈ R.In the same article, Vukman, Kosi-Ulbl and Eremita proved the resultbelow, which is an immediate consequence of Theorem 1 and Theorem 2.Corollary 3. Let R be a 2−torsion free semiprime ring. If D,G : R →R are additive mappings such thatD(xyx) = D(x)yx− xG(y)x + xyD(x),G(xyx) = G(x)yx − xD(y)x+ xyG(x)for all x ∈ R, then there exists a derivation S : R → R and q ∈ Qs such that4D(x) = qx+ xq + S(x),4G(x) = qx+ xq − S(x)for all x ∈ R.Theorem 1 also motivated Vukman ([20]), who has recently proved thefollowing result.FUNCTIONAL EQUATIONS IN SEMIPRIME RINGS 97Theorem 4. Let R be a 2−torsion free semiprime ring and let D : R → Rbe an additive mapping. Suppose that eitherD(xyx) = D(xy)x + xyD(x)orD(xyx) = D(x)yx + xD(yx)holds for all pairs x, y ∈ R. In both cases D is a derivation.It is our aim in this paper to generalize Theorem 4, but first we will provethe following theorem, which is conceptually linked with Theorem 2.Theorem 5. Let R be a 2−torsion free semiprime ring and let T : R → Rbe an additive mapping. Suppose that either(1) T (xyx) = T (xy)x− xyT (x)or(2) T (xyx) = T (x)yx− xT (yx)holds for all pairs x, y ∈ R. In both cases T = 0.Proof. We will restrict our attention on the relation (1), the proof incase we have the relation (2) is similar and will therefore be omitted.Linearization of the relation (1) givesT (xyz + zyx) = T (xy)z + T (zy)x− xyT (z)− zyT (x), x, y, z ∈ Rand in particular for z = x2(3) T (xyx2 + x2yx) = T (xy)x2 + T (x2y)x− xyT (x2)− x2yT (x), x, y ∈ R.Putting xy + yx for y in (1) and applying the relation (1) we obtainT (xyx2 + x2yx) = T (x2y + xyx)x − x2yT (x)− xyxT (x)= T (x2y)x+ T (xy)x2 − xyT (x)x− x2yT (x)− xyxT (x), x, y ∈ R.We have therefore(4)T (xyx2+x2yx) = T (x2y)x+T (xy)x2−xyT (x)x−x2yT (x)−xyxT (x), x, y ∈ R.By comparing (3) and (4) we obtain(5) xyA(x) = 0, x, y ∈ R,where A(x) = T (x2) − T (x)x − xT (x). Right multiplication of the aboverelation by x and left multiplication by A(x) gives A(x)xyA(x)x = 0 for allpairs x, y ∈ R, whence it follows(6) A(x)x = 0, x ∈ Rby semiprimeness of R. The substitution A(x)yx for y in the relation (5) givesxA(x)yxA(x) = 0 for all pairs x, y ∈ R, which gives(7) xA(x) = 0, x ∈ R.98 N. ŠIROVNIKThe linearization of the relation (6) givesB(x, y)x +A(x)y +B(y, x)y +A(y)x = 0for all pairs x, y ∈ R, where B(x, y) denotes T (xy + yx) − T (x)y − xT (y) −T (y)x − yT (x). Putting in the above relation −x for x and comparing therelation so obtained with the above relation, one obtainsB(x, y)x+A(x)y = 0.Right multiplication of the above relation by A(x) gives, because of therelation (7), A(x)yA(x) = 0 for all pairs x, y ∈ R, whence it follows A(x) = 0for all x ∈ R. In other words, T is a Jordan derivation. By Cusack’sgeneralization of Herstein’s theorem T is a derivation. From the relation(1) we obtain T (x)yx+xT (y)x+xyT (x) = T (x)yx+xT (y)x−xyT (x), whichreduces to(8) xyT (x) = 0, x, y ∈ R.From the relation (8) one obtains(9) T (x)x = 0, x ∈ Rand(10) xT (x) = 0, x ∈ R,(see how the relations (6) and (7) were obtained from the relation (5)). Thelinearization of the relation (9) givesT (y)x+ T (x)y = 0, x, y ∈ R.Right multiplication of the above relation by T (x) gives because of the relation(10) T (x)yT (x) = 0 for all pairs x, y ∈ R, which gives T = 0.We proceed with the following corollary, which is a generalization ofTheorem 4.Corollary 6. Let R be a 2−torsion free semiprime ring and let D;G :R → R be additive mappings satisfying either the relations(11)D(xyx) = D(xy)x + xyG(x),G(xyx) = G(xy)x + xyD(x)or the relations(12)D(xyx) = D(x)yx + xG(yx),G(xyx) = G(x)yx + xD(yx)for all pairs x, y ∈ R. In both cases D and G are derivations and D = G.FUNCTIONAL EQUATIONS IN SEMIPRIME RINGS 99Proof. We will restrict our attention on the relations (11), the proof incase we have the relations (12) is similar and will therefore be omitted. Wehave therefore the relationsD(xyx) = D(xy)x+ xyG(x), x, y ∈ R,(13)G(xyx) = G(xy)x+ xyD(x), x, y ∈ R.(14)Subtracting the relation (14) from the relation (13) one obtains(15) T (xyx) = T (xy)x− xyT (x), x, y ∈ R,where T (x) denotes D(x) −G(x). Applying Theorem 5 gives D = G, whichreduces the relation (13) toD(xyx) = D(xy)x+ xyD(x), x, y ∈ R.Now, it follows from Theorem 4 that D is a derivation, which completes theproof.We proceed with the following conjecture.Conjecture 7. Let R be a semiprime ring with suitable torsion restricti-ons and let D,G : R → R be additive mappings satisfying either the relationsD(x3) = D(x2)x+ x2G(x),G(x3) = G(x2)x + x2D(x)or the relationsD(x3) = D(x)x2 + xG(x2),G(x3) = G(x)x2 + xD(x2)for all x ∈ R. In both cases D an G are derivations and D = G.Our next result is related to the conjecture above.Theorem 8. Let X be a real or complex Banach space and let A(X)be a standard operator algebra on X. Suppose there exist linear mappingsD,G : A(X) → L(X) satisfying either the relations(16)D(A3) = D(A2)A+A2G(A),G(A3) = G(A2)A+A2D(A)or the relations(17)D(A3) = D(A)A2 +AG(A2),G(A3) = G(A)A2 +AD(A2)for all A ∈ A(X). In both cases there exists some fixed B ∈ L(X), such thatD(A) = G(A) = [A,B] holds for all A ∈ A(X), which means that D and Gare linear derivations.100 N. ŠIROVNIKTheorem 8 is related to the result below first proved by Chernoff ([7]) (seealso [12–14,17–19]).Theorem 9. Let X be a real or complex Banach space, let A(X) bea standard operator algebra on X and let D : A(X) → L(X) be a linearderivation. In this case D is of the form D(A) = [A,B] for all A ∈ A(X) andsome fixed B ∈ L(X).In the proof of Theorem 8 we use Herstein’s theorem, Theorem 9, Lemma10 and methods which are similar to those used in [17 - 20].Lemma 10. Let R be a semiprime ring and let f : R → R be an additivemapping. If eitherf(x)x = 0orxf(x) = 0holds for all x ∈ R, then f = 0.For the proof of Lemma 10 we refer to [15].Proof of Theorem 8. We will focus our attention to the relations (16).The proof in case we have the relations (17) is similar and will therefore beomitted. We have therefore the relationsD(A3) = D(A2)A+A2G(A),(18)G(A3) = G(A2)A+A2D(A)(19)for all A ∈ A(X). Subtracting the relation (19) from the relation (18), weobtain(20) T (A3) = T (A2)A−A2T (A)for all A ∈ A(X), where T (A) denotes D(A) − G(A). It is our aim to showthat T (A) = 0 for all A ∈ A(X). Let us first consider the restriction of T onF (X). Let A be from F (X) and P ∈ F (X) a projection with AP = PA = A.Putting P for A in relation (20), we obtain(21) T (P ) = T (P )P − PT (P ).Right multiplication of the relation (21) by P gives(22) PT (P )P = 0.Left multiplication of the relation (21) by P gives because of (22)(23) PT (P ) = 0.Putting A+ P for A in the relation (20), we obtain after some calculations3T (A2) + 3T (A) = T (A2)P + 2T (A)A+ 2T (A)P + T (P )A−A2T (P )− 2AT (A)− 2AT (P )− PT (A).FUNCTIONAL EQUATIONS IN SEMIPRIME RINGS 101Putting −A for A in the above relation and comparing the relation so obtainedwith the above relation, we obtain3T (A) = 2T (A)P + T (P )A− 2AT (P )− PT (A).Left multiplication of the relation (23) by A and considering A = AP givesAT (P ) = APT (P ) = 0, which reduces the relation above to(24) 3T (A) = 2T (A)P + T (P )A− PT (A).Multiplying the relation (24) from both sides by P and considering (23), weobtain(25) PT (A)P = 0.Left multiplication of the relation (24) by P and considering both (23) and(25) leads us to(26) PT (A) = 0.Left multiplication of the relation (26) by A gives(27) AT (A) = 0.Applying (26) in the relation (24) leads us to(28) 3T (A) = 2T (A)P + T (P )A.Right multiplication of the relation (28) by P gives T (A)P = T (P )A, whichreduces the relation (28) to(29) T (A) = T (A)P.From the relation (29) one can conclude that T maps F (X) into itself. Wehave therefore a linear mapping T , which maps F (X) into itself, satisfyingthe relation (27). Applying Lemma 10 we can conclude that T (A) = 0 forall A ∈ F (X). It remains to prove that T (A) = 0 holds for all A ∈ A(X) aswell. The mapping T on A(X) is linear, satisfies the relation (20) and, as wealready showed, vanishes on F (X). Let A ∈ A(X), let P be a one-dimensionalprojection and let us introduce S ∈ A(X) by S = A+PAP − (AP +PA). Wehave SP = PS = 0. It is easy to see that T (S) = T (A) and T (S2) = T (A2).By the equation (20) we now haveT (S2)S − S2T (S) = T (S3) = T (S3 + P )= T ((S + P )3) = T ((S2)(S + P )− (S + P )2T (S)= T (S2)S + T (S2)P − S2T (S)− PT (S).We have therefore T (S2)P −PT (S) = 0 and since T (S) = T (A) and T (S2) =T (A2), we obtain(30) T (A2)P − PT (A) = 0.102 N. ŠIROVNIKPutting −A for A in the relation (30) and comparing the relation so obtainedwith (30), we get(31) PT (A) = 0.As P is an arbitrary one-dimensional projection, we have T (A) = 0 for allA ∈ A(X), which was our intention to prove. We have therefore proved thatD = G. This ascertainment enables us to combine (18) and (19) into onerelation(32) D(A3) = D(A2)A+A2D(A).Let us first consider the restriction of D on F (X). Let A be from F (X) andlet P ∈ F (X) be a projection with AP = PA = A. Putting P for A in therelation (32), we obtain D(P ) = D(P )P +PD(P ). Any multiplication of thelast relation by P leads us to(33) PD(P )P = 0.The substitution A+ P for A in the relation (32) gives(34)3D(A2) + 3D(A) +D(P )= D(A2)P + 2D(A)P + 2D(A)A+D(P )A+D(P )P+A2D(P ) + 2AD(A) + 2AD(P ) + PD(A) + PD(P ).Previously mentioned relation D(P ) = D(P )P +PD(P ) reduces the equation(34) to3D(A2) + 3D(A) = D(A2)P + 2D(A)P + 2D(A)A+D(P )A+A2D(P ) + 2AD(A) + 2AD(P ) + PD(A).Putting −A for A in the above relation and comparing the relation so obtainedwith the above relation, we obtain(35) 3D(A2) = D(A2)P + 2D(A)A+A2D(P ) + 2AD(A)and(36) 3D(A) = 2D(A)P +D(P )A+ 2AD(P ) + PD(A).Right multiplication of the relation (35) by P gives(37) 3D(A2)P = D(A2)P + 2D(A)A+A2D(P )P + 2AD(A)P,which is, after considering (33) and dividing by 2, reduced to(38) D(A2)P = D(A)A+AD(A)P.The relation (35) gives because of (38)(39) 3D(A2) = 3D(A)A+AD(A)P +A2D(P ) + 2AD(A).FUNCTIONAL EQUATIONS IN SEMIPRIME RINGS 103Left multiplication of the relation (36) by A, considering (33) and dividing by2 leads us to(40) AD(A) = AD(A)P +A2D(P ).The relation (39) gives because of (40)3D(A2) = 3D(A)A + 3AD(A),which reduces to(41) D(A2) = D(A)A +AD(A).From the relation (36) one can conclude that D is a linear mapping, whichmaps F (X) into itself. By the relation (41) D is a Jordan derivation on F (X).Since F (X) is prime, it follows that D is a derivation by Herstein’s theorem.Applying Theorem 9 one can conclude that D is of the form(42) D(A) = [A,B]for all A ∈ F (X) and some fixed B ∈ L(X). It remains to prove that therelation (42) holds for all A ∈ A(X) as well. For this purpose we introduceD1 : A(X) −→ L(X) by D1(A) = [A,B] and consider D0 = D − D1. Themapping D0 is linear, satisfies the relation (32) and it vanishes on F (X).Our aim is to prove that D0 vanishes on A(X) as well. Let A ∈ A(X),let P be a one-dimensional projection and let us introduce S ∈ A(X) byS = A + PAP − (AP + PA). We have SP = PS = 0. It is easy to see thatD0(S) = D0(A) and D0(S2) = D0(A2). The relation (32) now leads us toD0(S2)S + S2D0(S) = D0(S3) = D0(S3 + P ) = D0((S + P )3)= D0(S2)S +D0(S2)P + S2D0(S) + PD0(S).We have therefore D0(S2)P + PD0(S) = 0 and since D0(S) = D0(A) andD0(S2) = D0(A2), we arrive at(43) D0(A2)P + PD0(A) = 0.Putting −A for A in the relation (43) and subtracting the relation so obtainedfrom the relation (43), we get(44) PD0(A) = 0.Since P is an arbitrary one-dimensional projection, it follows that D0(A) = 0for all A ∈ A(X), which concludes the proof of the theorem.Acknowledgements.The author wishes to express his thanks to Professor Joso Vukman forproposing the problem and for helpful discussion.104 N. ŠIROVNIKReferences[1] K. I. Beidar, M. Brešar, M. A. Chebotar and W. S. Martindale III, On Herstein’s Liemap Conjectures. II, J. Algebra 238 (2001), 239–264.[2] K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with generalizedidentities, Marcel Dekker, Inc., New York, 1996.[3] M. Brešar, Jordan derivations on semiprime rings, Proc. 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Vukman, An identity related to derivations of standard operatoralgebras and semisimple H∗-algebras, Cubo 12 (2010), 95–102.[13] P. Šemrl, Ring derivations on standard operator algebras, J. Functional Analysis 112(1993), 318–324.[14] J. Vukman, On automorphisms and derivations of operator algebras, Glas. Mat. Ser.III 19(39) (1984), 135–138.[15] J. Vukman, Identities with derivations and automorphisms on semiprime rings, Int.J. Math. Math. Sci. (2005), 1031–1038.[16] J. Vukman, I. Kosi-Ulbl and D. Eremita, On certain equations in rings, Bull. Austral.Math. Soc. Vol. 71 (2005), 53–60.[17] J. Vukman, On derivations of algebras with involution, Acta Math. Hungar. 112(2006), 181–186.[18] J. Vukman, On derivations of standard operator algebras and semisimple H∗-algebras, Studia Sci. Math. Hungar. 44 (2007), 57–63.[19] J. Vukman, Identities related to derivations and centralizers on standard operatoralgebras, Taiwanese J. Math. 11 (2007), 255–265.[20] J. Vukman, Some remarks on derivations in semiprime rings and standard operatoralgebras, Glas. Mat. Ser. III 46(66) (2011), 43–48.N. ŠirovnikDepartment of Mathematics and Computer ScienceFaculty of Natural Sciences and MathematicsUniversity of MariborKoroška 160, 2000 MariborSloveniaE-mail : nejc.sirovnik@uni-mb.siReceived : 14.10.2010.Revised : 22.2.2011.

On functional equations related to derivations in semiprime rings and standard operator algebras

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