CUBO A Mathemati al Journal Vol.15, N o 03, (45�50). O tober 2013 On entralizers of standard operator algebras with involution Maja Fo²ner, Benjamin Mar en Fa ulty of Logisti s, University of Maribor, Mariborska esta 7 3000 Celje Slovenia, maja.fosner�fl.uni-mb.si, benjamin.mar en�fl.uni-mb.si Nej �irovnik Fa ulty of Natural S ien es and Mathemati s, University of Maribor, Koro²ka esta 160 2000 Maribor Slovenia. nej .sirovnik�uni-mb.si ABSTRACT The purpose of this paper is to prove the following result. Let X be a omplex Hilbert spa e, let L(X) be the algebra of all bounded linear operators on X and let A(X) ⊂ L(X) be a standard operator algebra, whi h is losed under the adjoint operation. Let T : A(X) → L(X) be a linear mapping satisfying the relation 2T(AA∗A) = T(A)A∗A + AA∗T(A) for all A ∈ A(X). In this ase T is of the form T(A) = λA for all A ∈ A(X), where λ is some �xed omplex number. RESUMEN El propósito de este artí ulo es probar el siguiente resultado. Sea X un espa io de Hilbert omplejo, sea L(X) el álgebra de todos los operadores lineales a otados sobre X y sea A(X) ⊂ L(X) la álgebra de operadores lási a, la ual es errada bajo la opera ión adjunto. Sea T : A(X) → L(X) una apli a ión lineal satisfa iendo la rela ión 2T(AA∗A) = T(A)A∗A+AA∗T(A) para todo A ∈ A(X). En este aso, T es de la forma T(A) = λA para todo A ∈ A(X), donde λ es un número omplejo �jo. Keywords and Phrases: ring, ring with involution, prime ring, semiprime ring, Bana h spa e, Hilbert spa e, standard operator algebra, H∗-algebra, left (right) entralizer, two-sided entralizer. 2010 AMS Mathemati s Subje t Classi� ation: 16N60, 46B99, 39B42. 46 Maja Fo²ner, Benjamin Mar en & Nej �irovnik CUBO 15, 3 (2013) This resear h has been motivated by the work of Vukman, Kosi-Ulbl [5℄ and Zalar [13℄. Throughout, R will represent an asso iative ring with enter Z(R). Given an integer n ≥ 2, a ring R is said to be n-torsion free if for x ∈ R, nx = 0 implies x = 0. An additive mapping x 7→ x∗ on a ring R is alled involution if (xy)∗ = y∗x∗ and x∗∗ = x hold for all pairs x, y ∈ R. A ring equipped with an involution is alled a ring with involution or ∗ -ring. Re all 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 ase aRa = (0) implies a = 0. We denote by Qr and C the Martindale right ring of quotients and the extended entroid of a semiprime ring R, respe tively. For the explanation of Qr and C we refer the reader to [2℄. An additive mapping T : R → R is alled a left entralizer in ase T(xy) = T(x)y holds for all pairs x, y ∈ R. In ase R has the identity element, T : R → R is a left entralizer i� T is of the form T(x) = ax for all x ∈ R, where a is some �xed element of R. For a semiprime ring R all left entralizers are of the form T(x) = qx for all x ∈ R, where q ∈ Qr is some �xed element (see Chapter 2 in [2℄). An additive mapping T : R → R is alled a left Jordan entralizer in ase T(x2) = T(x)x holds for all x ∈ R. The de�nition of right entralizer and right Jordan entralizer should be self-explanatory. We all T : R → R a two-sided entralizer in ase T is both a left and a right entralizer. In ase T : R → R is a two-sided entralizer, where R is a semiprime ring with extended entroid C, then T is of the form T(x) = λx for all x ∈ R, where λ ∈ C is some �xed element (see Theorem 2.3.2 in [2℄). Zalar [13℄ has proved that any left (right) Jordan entralizer on a semiprime ring is a left (right) entralizer. Let us re all that a semisimple H∗-algebra is a omplex semisimple Bana h∗-algebra whose norm is a Hilbert spa e norm su h that (x, yz∗) = (xz, y) = (z, x∗y) is ful�lled for all x, y, z ∈ A. For basi fa ts on erning H∗-algebras we refer to [1℄. Vukman [10℄ has proved that in ase there exists an additive mapping T : R → R, where R is a 2-torsion free semiprime ring satisfying the relation 2T(x2) = T(x)x + xT(x) for all x ∈ R, then T is a two-sided entralizer. Kosi-Ulbl and Vukman [9℄ have proved the following result. Let A be a semisimple H∗−algebra and let T : A → A be an additive mapping su h that 2T(xn+1) = T(x)xn + xnT(x) holds for all x ∈ R and some �xed integer n ≥ 1. In this ase T is a two-sided entralizer. Re ently, Benkovi£, Eremita and Vukman [3℄ have onsidered the relation we have just mentioned above in prime rings with suitable hara teristi restri tions. Kosi-Ulbl and Vukman [9℄ have proved that in ase there exists an additive mapping T : R → R, where R is a 2-torsion free semiprime ∗-ring, satisfying the relation T(xx∗) = T(x)x∗ (T(xx∗) = xT(x∗)) for all x ∈ R, then T is a left (right) entralizer. For results on erning entralizers on rings and algebras we refer to [4�13℄, where further referen es an be found. Let X be a real or omplex Bana h spa e and let L(X) and F(X) denote the algebra of all bounded linear operators on X and the ideal of all �nite rank operators in L(X), respe tively. An algebra A(X) ⊂ L(X) is said to be standard in ase F(X) ⊂ A(X). Let us point out that any standard operator algebra is prime, whi h is a onsequen e of a Hahn-Bana h theorem. In ase X is a real or omplex Hilbert spa e, we denote by A∗ the adjoint operator of A ∈ L(X). We denote CUBO 15, 3 (2013) On entralizers of standard operator algebras with involution 47 by X∗ the dual spa e of a real or omplex Bana h spa e X. Vukman and Kosi-Ulbl [5℄ have proved the following result. Theorem 0.1. Let R be a 2-torsion free semiprime ring and let T : R → R be an additive mapping. Suppose that 2T(xyx) = T(x)yx + xyT(x) (1) holds for all x, y ∈ R. In this ase T is a two-sided entralizer. In ase we have a ∗ -ring, we obtain, after putting y = x∗ in the relation (1), the relation 2T(xx∗x) = T(x)x∗x + xx∗T(x). It is our aim in this paper to prove the following result, whi h is related to the above relation. Theorem 0.2. Let X be a omplex Hilbert spa e and let A(X) be a standard operator algebra, whi h is losed under the adjoint operation. Suppose T : A(X) → L(X) is a linear mapping satisfying the relation 2T(AA∗A) = T(A)A∗A + AA∗T(A) (2) for all A ∈ A(X). In this ase T is of the form T(A) = λA, where λ is a �xed omplex number. Proof. Let us �rst onsider the restri tion of T on F(X). Let A be from F(X) (in this ase we have A∗ ∈ F(X)). Let P ∈ F(X) be a self-adjoint proje tion with the property AP = PA = A (we also have A∗P = PA∗ = A∗). Putting P for A in (2) we obtain 2T(P) = T(P)P + PT(P). Left multipli ation by P in the above relation gives PT(P) = PT(P)P. Similarly, right multipli ation by P in the above relation leads to T(P)P = PT(P)P. Therefore T(P) = T(P)P = PT(P) = PT(P)P. (3) Putting A + P for A in the relation (2) we obtain 2T(A2) + 2T(AA∗ + A∗A) + 4T(A) + 2T(A∗) = = T(A)(A + A∗) + T(A)P + T(P)A∗A + T(P)(A + A∗)+ + (A + A∗)T(A) + PT(A) + AA∗T(P) + (A + A∗)T(P). Putting −A for A in the above relation and omparing the relation so obtained with the above relation gives 2T(A2) + 2T(AA∗ + A∗A) = = T(A)(A + A∗) + T(P)A∗A + (A + A∗)T(A) + AA∗T(P) (4) 48 Maja Fo²ner, Benjamin Mar en & Nej �irovnik CUBO 15, 3 (2013) and 4T(A) + 2T(A∗) = = T(A)P + PT(A) + T(P)(A + A∗) + (A + A∗)T(P). (5) So far we have not used the assumption of the theorem that X is a omplex Hilbert spa e. Putting iA for A in the relations (4) and (5) and omparing the relations so obtained with the above relations, respe tively, we obtain 2T(A2) = T(A)A + AT(A), (6) 4T(A) = T(A)P + PT(A) + T(P)A + AT(P). (7) Putting A∗ for A in the relation (5) gives 4T(A∗) + 2T(A) = = T(A∗)P + PT(A∗) + T(P)(A + A∗) + (A + A∗)T(P). Putting iA for A in the above relation and omparing the relation so obtained with the above relation leads to 2T(A) = T(P)A + AT(P). Comparing the above relation and (7), we obtain 2T(A) = T(A)P + PT(A). (8) Right (left) multipli ation by P in the above relation gives T(A)P = PT(A)P and PT(A) = PT(A)P, respe tively. Hen e, PT(A) = T(A)P, whi h redu es the relation (8) to T(A) = T(A)P. From the above relation one an on lude that T maps F(X) into itself. We therefore have a linear mapping T : F(X) → F(X) satisfying the relation (6) for all A ∈ F(X). Sin e F(X) is prime, one an on lude, a ording to Theorem 1 in [10℄ that T is a two-sided entralizer on F(X). We intend to prove that there exists an operator C ∈ L(X), su h that T(A) = CA (9) for all A ∈ F(X). For any �xed x ∈ X and f ∈ X∗ we denote by x ⊗ f an operator from F(X) de�ned by (x ⊗ f)y = f(y)x, y ∈ X. For any A ∈ L(X) we have A(x ⊗ f) = (Ax) ⊗ f. Now let us hoose su h f and y that f(y) = 1 and de�ne Cx = T(x ⊗ f)y. Obviously, C is linear and applying the fa t that T is a left entralizer on F(X), we obtain (CA)x = C(Ax) = T((Ax) ⊗ f)y = T(A(x ⊗ f))y = T(A)(x ⊗ f)y = T(A)x for any x ∈ X. We therefore have T(A) = CA for any A ∈ F(X). As T is a right entralizer on F(X), we obtain C(AB) = T(AB) = AT(B) = ACB. We therefore have [A, C]B = 0 for any CUBO 15, 3 (2013) On entralizers of standard operator algebras with involution 49 A, B ∈ F(X), when e it follows that [A, C] = 0 for any A ∈ F(X). Using losed graph theorem one an easily prove that C is ontinuous. Sin e C ommutes with all operators from F(X), we an on lude that Cx = λx holds for any x ∈ X and some �xed omplex number λ, whi h gives together with the relation (9) that T is of the form T(A) = λA (10) for any A ∈ F(X) and some �xed omplex number λ. It remains to prove that the relation (10) holds on A(X) as well. Let us introdu e T1 : A(X) → L(X) by T1(A) = λA and onsider T0 = T −T1. The mapping T0 is, obviously, additive and satis�es the relation (2). Besides, T0 vanishes on F(X). It is our aim to show that T0 vanishes on A(X) as well. Let A ∈ A(X), let P ∈ F(X) be a one- dimensional self-adjoint proje tion and S = A + PAP − (AP + PA). Su h S an also be written in the form S = (I − P)A(I − P), where I denotes the identity operator on X. Sin e S − A ∈ F(X), we have T0(S) = T0(A). It is easy to see that SP = PS = 0. By the relation (2) we have T0(S)S ∗S + SS∗T0(S) = = 2T0(SS ∗S) = = 2T0((S + P)(S + P) ∗(S + P)) = = T0(S + P)(S + P) ∗(S + P) + (S + P)(S + P)∗T0(S + P) = T0(S)S ∗S + T0(S)P + SS ∗T0(S) + PT0(S). We therefore have T0(S)P + PT0(S) = 0. Considering T0(S) = T0(A) in the above relation, we obtain T0(A)P + PT0(A) = 0. (11) Multipli ation from both sides by P in the above relation leads to PT0(A)P = 0. Right multipli ation by P in the relation (11) and onsidering the above relation gives T0(A)P = 0. Sin e P is an arbitrary one-dimensional self-adjoint proje tion, it follows from the above relation that T0(A) = 0 for all A ∈ A(X), whi h ompletes the proof of the theorem. We on lude the paper with the following onje ture. Conje ture 0.3. Let R be a semiprime ∗-ring with suitable torsion restri tions and let T : R → R be an additive mapping satisfying the relation 2T(xx∗x) = T(x)x∗x + xx∗T(x) for all x ∈ R. In this ase T is a two-sided entralizer. Re eived: April 2013. A epted: September 2013. 50 Maja Fo²ner, Benjamin Mar en & Nej �irovnik CUBO 15, 3 (2013) Referen es [1℄ W. Ambrose: Stru ture theorems for a spe ial lass of Bana h algebras, Trans. Amer. Math. So . 57 (1945), 364-386. [2℄ K. I. Beidar, W. S. Martindale 3rd, A. V. Mikhalev: Rings with generalized identities, Mar el Dekker, In ., New York, (1996). [3℄ D. Benkovi£, D. Eremita, J. Vukman: A hara terization of the entroid of a prime ring, Studia S i. Math. Hungar. 45 (3) (2008), 379-394. [4℄ I. Kosi-Ulbl, J. Vukman: An equation related to entralizers in semiprime rings, Glas. Mat. 38 (58) (2003), 253-261. [5℄ I. Kosi-Ulbl, J. Vukman: On entralizers of semiprime rings, Aequationes Math. 66 (2003), 277-283. [6℄ I. Kosi-Ulbl, J. Vukman: On ertain equations satis�ed by entralizers in rings, Internat. Math. J. 5 (2004), 437-456. [7℄ I. Kosi-Ulbl, J. Vukman: Centralizers on rings and algebras, Bull. Austral. Math. So . 71 (2005), 225-234. [8℄ I. Kosi-Ulbl, J. Vukman: A remark on a paper of L. Molnár, Publ. Math. Debre en. 67 (2005), 419-421. [9℄ I. Kosi-Ulbl, J. Vukman: On entralizers of standard operator algebras and semisimple H∗- algebras, A ta Math. Hungar. 110 (3) (2006), 217-223. [10℄ J. Vukman: An identity related to entralizers in semiprime rings, Comment. Math. Univ. Carol. 40 (1999), 447-456. [11℄ J. Vukman: Centralizers of semiprime rings, Comment. Math. Univ. Carol. 42 (2001), 237- 245. [12℄ J. Vukman: Identities related to derivations and entralizers on standard operator algebras, Taiwan. J. Math. Vol. 11 (2007), 255-265. [13℄ B. Zalar: On entralizers of semiprime rings, Comment. Math. Univ. Carol. 32 (1991), 609- 614.