Clanek&_nbsp_20.dvi CUBO A Mathematical Journal Vol.12, No¯ 01, (95–102). March 2010 An Identity Related to Derivations of Standard Operator Algebras and Semisimple H∗-Algebras1 Irena Kosi-Ulbl Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, Maribor, Slovenia email : irena.kosi@uni-mb.si and Joso Vukman Department of Mathematics and Computer Science, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor, Slovenia email : joso.vukman@uni-mb.si ABSTRACT In this paper we prove the following result. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators on X, and let A(X) ⊂ L(X) be a standard operator algebra. Suppose D : A(X) → L(X) is a linear mapping satisfying the relation D(An) = n ∑ j=1 A n−j D(A)Aj−1 for all A ∈ A(X). In this case D is of the form D(A) = AB − BA, for all A ∈ A(X) and some B ∈ L(X), which means that D is a linear derivation. In particular, D is continuous. We apply this result, which generalizes a classical result of Chernoff, to semisimple H∗−algebras. This research has been motivated by the work of Herstein [4], Chernoff [2] and Molnár [5] and is a continuation of our recent work [8] and [9] .Throughout, R will represent an associative ring. Given an integer n ≥ 2, a ring R is said to be n−torsion free, if for x ∈ R, nx = 0 1This research has been supported by the Research Council of Slovenia 96 Irena Kosi-Ulbl and Joso Vukman CUBO 12, 1 (2010) implies x = 0. Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear mapping D : B → A is called a linear derivation in case D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R. In case we have a ring R an additive mapping D : R → R is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R and is called a Jordan derivation in case D(x2) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R, such that D(x) = ax − xa holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [4] asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. Cusack [3] generalized Herstein’s result to 2−torsion free semiprime rings. Let us recall that a semisimple H∗−algebra is a semisimple Banach ∗−algebra whose norm is a Hilbert space norm such that (x, yz∗) = (xz, y) = (z, x∗y) is fulfilled for all x, y, z ∈ A (see [1]). Let X be a real or complex Banach space and let L(X) and F (X) denote the algebra of all bounded linear operators on X and the ideal of all finite rank operators in L(X), respectively. An algebra A(X) ⊂ L(X) is said to be standard in case F (X) ⊂ A(X). Let us point out that any standard algebra is prime, which is a consequence of Hahn-Banach theorem. RESUMEN En este artículo nosotros provamos el seguiente resultado. Sea X un espacio de Banach real o complejo, sea L(X) a algebra de todos los operadores linares acotados sobre X, y sea A(X) ⊂ L(X) una algebra de operadores estandar. Suponga D : A(X) −→ L(X) una aplicación lineal verificando la relación D(An) = n ∑ j=1 A n−j D(A)Aj−1 para todo A ∈ A(X). En este caso D es de la forma D(A) = AB − BA, para todo A ∈ A(X) y algún B ∈ L(X), lo que significa que D es una deriviación lineal. En particual, D es continua. Nosotros aplicamos este resultado el cual generaliza un resultado clásico de Chernoff, para H∗-algebras semisimple. Este trabajo fué motivado por un trabajo de Herstein [4], Chernoff [2] y Molnár [5] y este una continuación de nuestro reciente trabajo [8] y [9]. Key words and phrases: Prime ring, semiprime ring, Banach space, standard operator algebra, H∗–algebra, derivation, Jordan derivation. Math. Subj. Class.: 46K15, 46H99, 13N15. Let us start with the following result proved by Chernoff [2] (see also [6] and [8]). THEOREM A. Let X be a real or complex Banach space and let A(X) be a standard operator algebra on X. Let D : A(X) → L(X) be a linear derivation. In this case D is of the form D(A) = AB − BA, for all A ∈ A(X) and some B ∈ L(X). In particular, D is continuous. It is our aim in this paper to prove the following result which generalizes Theorem A. THEOREM 1. Let X be a real or complex Banach space and let A(X) be a standard operator CUBO 12, 1 (2010) An Identity Related to Derivations of Standard Operator ... 97 algebra on X. Suppose D : A(X) → L(X) is a linear mapping satisfying the relation D(An) = n ∑ j=1 An−j D(A)Aj−1. for all A ∈ A(X). In this case D is of the form D(A) = AB − BA, for all A ∈ A(X) and some B ∈ L(X), which means that D is a linear derivation. In particular, D is continuous. Proof. We have the relation D(An) = n ∑ j=1 An−j D(A)Aj−1. (1) Let A be from F (X) and let P ∈ F (X), be a projection with AP = P A = A. From the above relation one obtains D(P ) = P D(P ) + (n − 2) P D(P )P + D(P )P. (2) Right multiplication of the relation (2) by P gives P D(P )P = 0. (3) Putting A + P for A in the relation (1), we obtain n ∑ i=0 ( n i ) D ( A n−i P i ) = ( n−1 ∑ i=0 ( n − 1 i ) A n−1−i P i ) D(A + P )+ ( n−2 ∑ i=0 ( n − 2 i ) A n−2−i P i ) D(A + P )(A + P )+ ( n−3 ∑ i=0 ( n − 3 i ) A n−3−i P i ) D(A + P )(A + P )2 + · · · + (A + P )2D(A + P ) ( n−3 ∑ i=0 ( n − 3 i ) A n−3−i P i ) + (A + P )D(A + P ) ( n−2 ∑ i=0 ( n − 2 i ) A n−2−i P i ) + D(A + P ) ( n−1 ∑ i=0 ( n − 1 i ) A n−1−i P i ) . (4) Using (1) and rearranging the equation (4) in sense of collecting together terms involving equal number of factors of P we obtain: 98 Irena Kosi-Ulbl and Joso Vukman CUBO 12, 1 (2010) n−1 ∑ i=1 fi (A, P ) = 0, where fi (A, P ) stands for the expression of terms involving i factors of P. Replacing A by A + 2P, A + 3P, ..., A + (n − 1) P in turn in the equation (1), and expressing the resulting system of n − 1 homogeneous equations of variables fi (A, P ), i = 1, 2, ..., n − 1, we see that the coefficient matrix of the system is a van der Monde matrix        1 1 · · · 1 2 22 · · · 2n−1 ... ... ... ... n − 1 (n − 1) 2 · · · (n − 1) n−1        . Since the determinant of the matrix is different from zero, it follows that the system has only a trivial solution. In particular, fn−2 (A, P ) = n (n − 1) D ( A 2 ) − (n − 1) (n − 2) ( A 2 D(P ) + D(P )A2 ) − ((n − 2) (n − 3) + (n − 3) (n − 4) + · · · + 3 · 2 + 2 · 1) ( A2D(P )P + P D(P )A2 ) − 2 ((n − 2) + (n − 3) + (n − 4) + · · · + 3 + 2 + 1) (AD (A) P + P D (A) A) − 4 (1 · (n − 2) + 2 · (n − 3) + 3 · (n − 4) + · · · + (n − 3) · 2 + (n − 2) · 1) AD(P )A− 2 (n − 1) (AD (A) + D (A) A) = 0, and fn−1 (A, P ) = nD (A) − (P D (A) + D (A) P ) − (n − 1) (AD(P ) + D(P )A) − ((n − 2) + (n − 3) + (n − 4) + · · · + 2 + 1) (AD(P )P + P D(P )A) − (n − 2) P D (A) P = 0. The above equations reduce to n (n − 1) D ( A 2 ) = (n − 1) (n − 2) ( A 2 D(P ) + D(P )A2 ) + 1 3 (n − 3) (n − 2) (n − 1) ( A2D(P )P + P D(P )A2 ) + (n − 2) (n − 1) (AD (A) P + P D (A) A) + 4 (1 · (n − 2) + 2 · (n − 3) + 3 · (n − 4) + · · · + (n − 3) · 2 + (n − 2) · 1) AD(P )A+ 2 (n − 1) (AD (A) + D (A) A) , (5) CUBO 12, 1 (2010) An Identity Related to Derivations of Standard Operator ... 99 and 2nD (A) = 2 (P D (A) + D (A) P ) + 2 (n − 1) (AD(P ) + D(P )A) + (n − 2) (n − 1) (AD(P )P + P D(P )A) + 2 (n − 2) P D (A) P, (6) respectively. Multiplying the relation (3) from both sides by A we obtain AD(P )A = 0, (7) which reduces the relation (5) to n (n − 1) D ( A2 ) = (n − 1) (n − 2) ( A2D(P ) + D(P )A2 ) + 1 3 (n − 3) (n − 2) (n − 1) ( A 2 D(P )P + P D(P )A2 ) + (n − 2) (n − 1) (AD (A) P + P D (A) A) + 2 (n − 1) (AD (A) + D (A) A) . (8) Applying the relation (3) and the fact that AP = P A = A, we have P D(P )A = (P D(P )P )A = 0. Similarly one obtains that AD(P )P = 0. The relations (8) and (6) can now be written as nD ( A2 ) = (n − 2) ( A2D(P ) + D(P )A2 ) + (n − 2) (AD (A) P + P D (A) A) + 2 (AD (A) + D (A) A) , (9) and nD (A) = P D (A) + D (A) P + (n − 1) (AD(P ) + D(P )A) + (n − 2) P D (A) P = 0, (10) respectively. Right multiplication of the relation (10) by P gives D(A)P = D(P )A + P D(A)P. (11) Similarly one obtains P D(A) = AD(P ) + P D(A)P. (12) Multiplying the relation (11) from the right side and the relation (12) from the left side by A, we obtain D(A)A = D(P )A2 + P D(A)A, (13) and AD(A) = A2D(P ) + AD(A)P. (14) Combining relations (9), (13) and (14) we obtain nD ( A2 ) = (n − 2) ( D(P )A2 + P D (A) A ) + (n − 2) ( A2D(P ) + AD (A) P ) + 2 (AD (A) + D (A) A) = (n − 2) (AD (A) + D (A) A) + 2 (AD (A) + D (A) A) . 100 Irena Kosi-Ulbl and Joso Vukman CUBO 12, 1 (2010) We have therefore D(A2) = D(A)A + AD(A) (15) for any A ∈ F (X). From the relation (10) one can conclude that D(A) ∈ F (X) for any A ∈ F (X). We have therefore a Jordan derivation on F (X). Since F (X) is prime it follows that D is a derivation by Herstein’s theorem. Applying Theorem A one can conclude that D is of the form D(A) = AB − BA, (16) for all A ∈ A(X) and some B ∈ L(X). It remains to prove that the relation (16) holds on A(X) as well. Let us introduce D1 : A(X) → L(X) by D1(A) = AB − BA and consider D0 = D − D1. The mapping D0 is, obviously, linear and satisfies the relation (1). Besides, D0 vanishes on F (X). It is our aim to prove that D0 vanishes on A(X) as well. Let A ∈ A(X), let P be an one-dimensional projection and S = A + P AP − (AP + P A). We have D0(S) = D0(A). and SP = P S = 0. We have D0(A n) = n ∑ j=1 An−j D0(A)A j−1 (17) for all A ∈ A(X). Applying the above relation we obtain n ∑ j=1 Sn−j D0(S)S j−1 = D0(S n) = D0(S n + P ) = D0((S + P ) n) = n ∑ j=1 (S + P )n−j D0(S + P )(S + P ) j−1 = n ∑ j=1 (S + P )n−j D0(A)(S + P ) j−1 = n ∑ j=1 (Sn−j + P )D0(S)(S j−1 + P ) = n ∑ j=1 Sn−j D0(A)S j−1+ n ∑ j=1 P D0(A)S j−1 + n ∑ j=1 S n−j D0(A)P + P D0(A)P. We have therefore n ∑ j=1 P D0(A)S j−1 + n ∑ j=1 Sn−jD0(A)P + P D0(A)P = 0. (18) Multiplying the above relation from both sides by P we obtain P D0(A)P = 0, (19) which reduces the relation (18) to n ∑ j=1 P D0(A)S j−1 + n ∑ j=1 S n−j D0(A)P = 0. (20) CUBO 12, 1 (2010) An Identity Related to Derivations of Standard Operator ... 101 Right multiplication of the above relation by P gives n ∑ j=1 Sn−jD0(A)P = 0. (21) Let us prove that n−1 ∑ j=1 kj S n−1−jD0(A)P = 0 (22) holds where kj = 2 n−1−j − 2n−1, j = 1, 2, ..., n − 1. Putting in the relation (21) 2A for A we obtain n ∑ j=1 2n−jSn−j D0(A)P = 0. Multiplying the relation (21) by 2n−1 and subtracting the relation so obtained from the above relation we obtain the relation (22). Since the relation (21) implies the relation (22) one can conclude by induction that D0(A)P = 0. Since P is an arbitrary one-dimensional projection, it follows that D0(A) = 0, for any A ∈ A(X), which completes the proof of the theorem. Let us point out that in case n = 3 Theorem 1 reduces to Theorem in [9]. THEOREM 2. Let A be a semisimple H∗−algebra and let D : R → R be a linear mapping satisfying the relation D(xn) = n ∑ j=1 xn−j D(x)xj−1 for all x ∈ R. In this case D is a linear derivation. Proof. The proof goes through using the same arguments as in the proof of Theorem in [5] with the exception that one has to use Theorem 1 instead of Lemma in [5]. Since in the formulation of the results presented in this paper we have used only algebraic concepts, it would be interesting to study the problem in a purely ring theoretical context. We conclude with the following conjecture. CONJECTURE. Let R be a semiprime ring with suitable torsion restrictions and let D : R → R be an additive mapping satisfying the relation D(xn) = n ∑ j=1 xn−j D(x)xj−1 for all x ∈ R. In this case D is a derivation. In case R has the identity element the conjecture above was proved in [8]. Since semisimple H∗−algebras are semiprime, Theorem 2 proves the conjecture above in a special case. Received: June, 2008. Revised: October, 2009. 102 Irena Kosi-Ulbl and Joso Vukman CUBO 12, 1 (2010) References [1] Ambrose, W., Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc., 57 (1945), 364–386. [2] Chernoff, P.R., Representations, automorphisms, and derivations of some operator alge- bras, J. Funct. Anal., 2 (1973), 275–289. [3] Cusack, J., Jordan derivations on rings, Proc. Amer. Math. Soc., 53 (1975), 321–324. [4] Herstein, I.N., Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), 1104– 1119. 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