A note on semiderivations in prime rings and C*-algebras

Throughout the paper unless otherwise stated, R is the prime ring with centre Z ( R ) , Q is the Martindale quotient ring of R and C is the extended centroid R (for further details see [1]). For given x,y G R , the symbol [ x,y ] and x о y stands for the commutator and anticommutator of x and y defined as xy — yx and xy + yx , respectively. We also note that a ring R is said to be a prime ring if aRb = { 0 } implies that either a = 0 or b = 0 . For any subsets A and B of R , [ A, B ] stands for the additive subgroup generated by [ a, b ] with a G A and b G B . Also, an additive subgroup L of R is said to be Lie ideal of R if [ u, r ] G L for all u G L and r G R . A mapping g : R ^ R is said to be commuting (resp. centralizing) on a subset S of R if [ g ( x ) ,x ] = 0 (resp. [ g ( x ) ,x ] G Z (R )) for all x G S . An additive mapping D : R ^ R is called a derivation on R , if D ( xy ) = D ( x ) y + x D ( y ) holds for all x, y G R .

In [2], Bergen introduced the notion of semiderivation. An additive mapping F : R ^ R is called a semiderivation associated with a mapping G : R ^ R , whenever

F (xy) = F (x)G (y) + xF (y) = F (x)y + G (x)F (y)

• ^# For the second author, this research is supported by the National Board of Higher Mathematics (NBHM), India, Grant № 02011/16/2020 NBHM (R. P.) R & D II/ 7786.

and F ( G ( x )) = G ( F ( x )) holds for all x,y G R . For G = 1 r , the identity map on R , F is clearly a derivation. Bre˘ser [3] proved that the only semiderivations of prime rings are ordinary derivations and mappings of the form F ( x ) = y(x - G ( x )) , where Y G C and G is an endomorphism.

Let us briefly recall the motivation behind this study. In [4], Posner studied the centralizing derivations of prime rings and proved that if R is a prime ring and D is a non-zero derivation of R such that [ D ( x ) ,x ] G Z ( R ) , for all x G R , then R is commutative. This result due to Posner was then extended to Lie ideals by Lanski [5]. In [6], Daif and Bell showed that a semiprime ring R must be commutative if it admits a derivation D such that either D ([ x, y ]) — [ x, y ] = 0 for all x,y G R or D ([ x, y ]) + [ x, y ] = 0 for all x,y G R . In 2002, Ashraf and Rehman [7] obtained the same conclusion if the commutator is replaced by an anti-commutator which stated that if a prime ring R admits a derivation D such that D ( x ) о D ( y ) = x о y for all x, y G R , then R is commutative. In [8], Herstein proved that a ring R is commutative if it has no nonzero nilpotent ideal and there is a fixed integer n >  1 such that ( xy ) ⁿ = x n y n for all x,y G R . In [9], Bell proved that a prime ring R with nonzero center, for which char( R ) = 0 or char( R ) > n , where n >  1 , must be commutative if it admits a nonzero derivation D such that D ([ x ⁿ ,y ] — [ x,y ⁿ ]) G Z(R ) for all x,y G R . Further, Ali et al. [10] showed that if R be a 2 -torsion free semiprime ring and it admits a derivation D such that D ( x m о y n ) G Z ( R ) for all x,y G R , then R is commutative (for additional associated results [11-14]).

On the other hand, recently Haung [15] proved that a prime ring R satisfies S4 , the standard identity in four variables if char( R ) > n + 1 or char( R ) = 0 and F ( x ) ⁿ = 0 holds, where x G L , a noncentral Lie ideal of R and F is a semiderivation associated with an automorphism ξ of R .

Given the above discussions, we investigate and describe the structure of a ring R which satisfies certain identities involving automorphisms and semi-derivations. Also, we discuss the nature of C ^∗ -algebras. To be more specific, we obtain the following theorems:

Theorem 1.1. Let R be a prime ring of char( R ) = 2 and m, n G Z ⁺ . If an automorphism Z of R satisfies ( x m о y ⁿ ) ^z G Z ( R ) for all x, y G R , then R satisfies S4, the standard identity in four variables.

Theorem 1.2. Let R be a prime ring of char( R ) = 2 and m, n G Z ⁺ . If a semiderivation F associated with an automorphism £ such that F ( x m о y n ) G Z(R ) . Then R satisfies S4, the standard identity in four variables.

Theorem 1.3. Let A be a primitive C * -algebra and m, n G Z ⁺ . If an automorphism £ : A ^ A satisfies the relation ( x m о y ⁿ ) ^z g Z ( A ) for all x,y G A , then A is C * — W4- algebra.

Theorem 1.4. Let A be a primitive C * -algebra and n G Z ⁺ . If an anti-automorphism Z : A ^ A satisfies the relation ( x n ) ^z + x n * G Z ( A ) for every x,y G A , then A is C ^* — W 4 -algebra.

2.    Preliminaries

Before proving our main results, we fix some notions which are required for the exposition of our main results. An automorphism ξ is called Q -inner if there exists an invertible element q G Q such that £ ( x ) = qxq ^-1 for all x G R . Also, the standard identity S4 in four variables is defined as follows:

where ( — 1) ^ is a sign of permutation ^ of the symmetric group of degree 4. Further we mention the following results which are crucial in developing the proof of our main theorem.

Fact 2.1. Let R be a prime ring and I a two sided ideal of R. Then I , R, Q satisfy the same generalized polynomial identities with coefficients in Q ( see [16]) . Furthermore, I , R and Q satisfy the same generalized polynomial identities with automorphisms ( see [17, Theorem 1]) .

Fact 2.2. Let R be a prime ring with extended centroid C . Then the following conditions are equivalent:

• (i)    dime RC < 4.

• (ii)    R satisfies s4, the standard identity in four variables.

• (iii)    R is commutative or R embeds in M2(F) for F a field.

• (iv)    R is algebraic of bounded degree 2 over C.

• (v)    R satisfies [[x²,y], [x,y]] = 0.

3.    Main Results

Fact 2.3. Let R be a prime ring and L a be non-central Lie ideal of R . If char( R ) = 2 , by [18, Lemma 1] there exists a nonzero ideal I of R such that 0 = [ I, R ] C L . If char( R ) = 2 and dimC- RC >  4 , i.e., char( R ) = 2 and R does not satisfy S 4 , then by [19, Theorem 13] there exists a nonzero ideal I of R such that 0 = [ I , R ] C L . Thus if either char( R ) = 2 or R does not satisfy s4 , then we may conclude that there exists a nonzero ideal I of R such that [ I , I ] C L .

Proposition 3.1. Let R be a dense subring of End (V D ) and Z : R ^ R be an automorphism of R . If R satisfies ([ xi,X2 ] о [ yi,y2 ]) ^z € Z(R ) for all Х1,Х2,У1,У2 € R , then either dim(V D ) ^ 2 or Z is an identity map on End( V D ) .

• <1 First assume that V d be a right vector space over a division ring D . Let End( VD ) the ring of D -linear transformations on V D . Thus in view of classical Jacobson Theorem [20, Isomorphism Theorem, p. 79], we have s ^z = P s P ^-1 for every s € End( V D ) , where Z is an automorphism of End( V D ) and P is an invertible semi-linear transformation. Hence, for all v € V , Z € D , P ( v^ ) = ( Pv ) Z ( ^ ) . Given by the hypotheses, we obtain

0= [ [ xi,X2 ] ^Z [ yi,y2 ] ^Z + [ yi,y2 ] ^Z [ xi,X2 ] ^Z ,z ] = [ P [ xi,X2 ][ yi,y2 ] P - 1 + P [ У1, У2 ][ Х1, X2 ] P - 1,z ] for every xi, X2, yi, У2, z € End( V D ) . Let us assume that v and P ^-1 v are D -dependent for every v € V .In view of [21, Lemma 1], we find that P ^-1 v = vx , where x € D and v € V . Hence, for all s € End( V D ) , P ^-1 ( sv ) = svx and sv = P ( svx ) = P ( s ( vx )) = P s P ^-1 ( v ) = s ^z v for all s € End( V D ) , v € V . Therefore, we find that ( s ^z — s ) V = (0) for every s € End( V D ) . Hence, s ^z = s for every s € End( V D ) . This shows that Z is an identity map on End( V D ) , as required.

Thus, there exists v € V such that v and P-1v are linearly D -independent. Firstly, we assume that dim(VD) ^ 4. Then we may take w, Pv € V such that {w,v, Pv, P-1 v} is D-independent. Let x,y € End(VD) such that x1v = 0, x1P-1v = 0, x1w = v, y1P-1v = 0, zv = 0;

x ₂ v = w, x ₂ P ^-1 v = v, y 1 v = v, y ₂ P ^-1 v = v, z P v = w.

[ x1, x2 ] v = v and hence, our assumption

We notice that [ x1,x ₂ ] P 1 v = 0 , [ y1,y ₂ ] P 1 v = v , yields

0=( P^ [ X1,X2 ][ y1,y2 ] P ¹ + P [ y1,y2 ][ X1,X2 ] P 1>z])v = -W,

a contradiction, implying that dim( V D ) ^ 3 .

Secondly, we assume that dim(VD) = 3. Take Pv E V such that {v, Pv, P-1 v} is D-independent and then {v, Pv, P-1 v} forms a D-basis of V. If P(v + P-1v + Pv) E vD and P(P-1v + Pv) E vD, then Pv, P(P-1 v + Pv) E vD and then v, P-1v + Pv E P-1(vD) = P-1(v)Z-1(D) = P-1vD, contradicting the fact that {v, P-1v + Pv} is D-independent. Therefore, one can pick p E {0,1} such that u = pv + P-1v + Pv and Pu / vD. Write Pu = va + P-1ve + Pvy, where а, в, Y E D and в, Y both are not zero. By density of theorem, there exist xi, X2, yi, У2, z E End(VD) such that x1v = 0, x2v = Pv,  y1v = v,  y2v = v,  zv = 0;

x 1 P ^-1 v = v,   x ₂ P ^-1 v = 0 , y 1 P ^-1 v = 0 , y ₂ P ^-1 v = v,   z P ^-1 v = v ;

x 1 P v = u, x ₂ P v = 0 ,   y 1 P v = v, y ₂ P v = v, z P v = u.

That is x i u = ( p + 1) v + P ^-1 v + P v , X 2 u = P v, yiu = ( p + 1) v and y 2 u = — uy . Therefore, we can see that [ x1,x ₂ ] P ^-1 v = —P ^-1 v , [ y1,y ₂ ] P ^-1 v = v , [ x1,x ₂ ] v = u , [ y1,y ₂ ] P ^-1 v = 0 . Also, z P u = ve + uY . As в , Y are not both zero and v , u are D -dependent, so it is easy to see that z P u = 0 . Thus in all, we see that

0=

([P [ xi,X2 ][ yi,y2 ] P 1 + P [ yi,y2 ][ xi,X2 ] P 1,z])v =

—z P u,

a contradiction, implying that dim( V D ) D 2 . >

Theorem 3.1. Let R be a non-commutative prime ring of characteristic different from two and Z be an automorphism of R . If R satisfies ([ xi,X2 ] о [ yi,y2 ]) ^z E Z(R ) for all x i , Х 2 ,У 1 ,У 2 E R , then R satisfies S4, the standard identity in four variables.

• <1 Firstly, we assume that Z is an inner automorphism of R , i.e., s ^z = psp ^-1 for every s E R .As Z is the non-identity map, so p / C . Then

Ф( г ) = P [ X i ,X 2 ][ y i ,y 2 ]P 1 + P [ y i ,y 2 ][ x i ,X 2 ]P 1,z]

is a non-trivial generalized polynomial identity (GPI) of R and hence of Q as well. By Martindale’s theorem [22], Q is isomorphic to dense subring of the ring of linear transformations of a vector space V over D , where D is a finite dimensional division ring over C . By Proposition 3.1, we have dim( V D ) D 2 . Thus it follows that either Q = D or Q = M 2 (D ) , the ring of 2 x 2 matrices over D . More generally, we assume that Q = M k ( D ) , for k D 2 .

If C is finite, then D is field by Wedderburn’s theorem. On the other hand, if C infinite, let F be the algebraic closure of C , therefore by the Van der monde determinant argument, we see that Q ^c F satisfies the generalized polynomial identity Ф( г ) = 0 . Moreover, Q® C F = M k ( D ) 0c F = M k ( D 0c F ) = M _t (F ) , for some t ^ 1 . Considering Proposition 3.1 and the fact that Q is not commutative, we assert that t = 2 , yields the required conclusion.

Secondly, we assume that ζ is an outer automorphism. By [17, Theorem 1], Q and hence R satisfy [[ x _i ,x ₂ ] ^z [ y _i ,y ₂ ] ^z + [ y _i ,y ₂ ] ^Z [ x _i ,x ₂ ] ^z , z ] = 0 . As x ^z , y ^z -word degree <  char( R ) , then by [23, Theorem 3], R satisfies [[ xi, x ‘₂ ][ yi, y2 ] + [ yi, y2 ][ xi, x ‘ 2 ] , z ] = 0 . That is, R is a polynomial identity (PI) ring. Thus, R and M _t (F ) satisfy the same polynomial identities [24, Lemma 1], i. e., for each xi,x2,yi, y2, z E M _t (F ) , [[ x ‘ , x ₂ ][ yi, y2 ] + [ yi, У2 ][ x ‘ , x2 ] , z ] = 0 . Take k ^ 3 and e ij , the usual unit matrix. Therefore, for x = e23 , У = e32 , z = eii , s = e12 , we get a contradiction 0 = [[ xi,x2 ][ yi,y2 ] + [ y ‘ ,y2 ][ xi,x2 ] ,z ] = [[ ei i,e12 ][ e23,e32 ] + [ e23,e32 ][ ei i,e12 ] , [ e23 032 ]] = e12 = 0 . Hence t = 2 , i.e., R satisfies S4 , the standard identity in four variables. This completes the proof. >

<1 Proof Of Theorem 1.1. We are given that ( x m о y ⁿ ) ^z g Z ( R ) for every x,y g R . Let Si = {r m : r g R } and S2 = {r ⁿ : r g R} be the additive subgroups. It implies that ( a о b ) ^z g Z ( R ) for all a g S _i , b g S ₂ . In view of [25, Main theorem], and since char( R ) = 2 , either Si have a non-central Lie ideal Li of R or r m g Z ( R ) for all r g R . The latter case concludes R to be commutative. Similarly, assume that there exists a Lie ideal L2 C Z ( R ) such that L2 C S ₂ - Moreover, in view of Fact 2.3, there exist Ii and I2 nonzero two-sided ideals of R such that 0 = [ Ii, R ] C L i and 0 = [ I2, R ] C L ₂ . Also, R is non-commutative as Li, L2 are non-central Lie ideal of R . Therefore ( x о y ) ^z g Z ( R ) for all x g [ Ii, Ii ] , y g [ I2, I? ] - Since Ii , I2 and R satisfy the same differential identities (see [24, Theorem 3]), so we have ( xоy ) ^z g Z ( R ) for all x,y g [ R,R ] . By Theorem 3.1, we get the required result. >

Using the same technique as used in Theorem 1.1 and Theorem 3.1, we can write in view of above result

Theorem 3.2. Let R be a non-commutative prime ring of characteristic different from two and F be a non-zero semiderivation associated with an automorphism ξ of R. If R satisfies F ([ xi,X2 ] о [ yi,У2 ]) g Z ( R ) for all x i ,X 2 ,y i ,y 2 g R, then R satisfies S4, the standard identity in four variables.

⊳ First we note that if ξ is an identity map on R , then F is not more than a derivation. In view of previous discussion, we have nothing to prove. Hence, we proceed by assuming that ξ is not an identity map on R . Hence in view of Bresar [3], F ( x ) = 7 ( x—x ^z ) for all x g R , where 0 = y g C . Thus by our hypothesis we can write y([x ₁ ,x ₂ ] ◦ [ yi,У2 ] — ([ x _i ,x ₂ ] ◦ [ y _i ,y ₂ ]) ^z ) g Z ( R ) which can be rewritten as 7 (([ x _i , x ₂ ] ◦ [ y _i , y ₂ ]) IR — ([ x _i , x ₂ ] о [ y _i , y ₂ ]) ^z ) g Z(R ) , where Ir is the identity map on R . It is well known that if £ is an automorphism of R , then £ + kl R ( k is an any integer) is also an automorphism on R . Thus, we set £ — Ir = Z- Therefore, the last relation can be written as 7 ([ xi,x2 ] ◦ [ yi,y2 ]) ^Z g Z(R ) for all x i ,x 2 ,y i ,y 2 g R . Since 0 = y g C , the above identity reduces to ([ x _i ,x ₂ ] о [ y _i ,y ₂ ]) ^z g Z ( R ) for all x _i ,x ₂ ,y _i ,y ₂ g R and hence in view of Theorem 3.1, we get the desired conclusion. ⊲

Proof Of Theorem 1.2. We are given that F ( x m о y ⁿ ) g Z ( R ) for every x,y g R . Let Si = {r ^m : r g R} and S2 = {r ⁿ : r g R} be the additive subgroups. It is easy to see that F ( x о y ) g Z ( R ) for each x g Si , y g S2 . Since char( R ) = 2 and by main theorem of [25], we have either r m g Z(R ) for every r g R or Si contains a non-central Lie ideal Li of R . The first case concludes that R to be commutative. Similarly, assume that there exists a Lie ideal L2 C Z ( R ) such that L2 C S ₂ . According to Fact 2.3, there exist nonzero two-sided ideals Ii and I2 of R such that 0 = [ Ii,R ] C Li and 0 = [ I2,R ] C L ₂ . Since Li, L2 are non-central Lie ideal of R , so R is non-commutative. Hence, F ( x о y ) g Z ( R ) for all x g [ Ii, Ii ] , y g [ I2, I2 ] . Since Ii , I2 and R satisfy the same differential identities (see [24, Theorem 3]), so we have F ( xоy ) g Z ( R ) for all x,y g [ R,R ] . Applying Theorem 3.2, we are done.

Corollary 3.1. Let R be a prime ring of characteristic different from two, m be fixed positive integer and F be a nonzero semiderivation associated with an automorphism ξ of R . If F ( x m ) g Z ( R ) for all x, y g R , then R satisfies S4, the standard identity in four variables.

Corollary 3.2. Let R be a prime ring of characteristic not two. If R admits an automorphism Z of R such that ( x ⁿ ) ^z g Z ( R ) for all x g R, then R satisfies S4, the standard identity in four variables.

Theorem 3.3. Let R be a prime ring of characteristic not two. If R admits an automorphism Z of R such that ( x ⁿ ) ^z + x ⁿ g Z ( R ) for all x g R , then R satisfies S4, the standard identity in four variables.

• <1 It is well known that if Z is an automorphism of R, then Z + klR (k is an any integer) is also an automorphism on R. We have given that (xn)^z+ xn E Z(R) for all x E R which can be rewritten as (xn)^z + (xn)IR E Z(R), where Ir is the identity map on R. Thus, we set Z - Ir = £. Therefore, the last relation can be written as (xⁿ)^zE Z(R) for all x E R and hence by Corollary 3.2 we have done. >

4.    Result Based on C ∗-Algebras

A Banach algebra is a linear associate algebra which, as a vector space, is a Banach space with norm || • || satisfying the multiplicative inequality; ||xy || С ||x||||y || for all x and y in A . A Banach algebra A is a PI-algebra if and only if there exists n E N and a polynomial q E W_n^ q = 0 , such that q ( xi, x2,..., x n ) = 0 for all xi, x2,..., x n E A , where W n is the set of all complex polynomials in n non-commuting variables. An involution on an algebra A is a map x ।—> x * of A onto such that the following conditions are hold: (i) ( xy)* = y * x * , (ii) ( x * ) * = x , and (iii) ( x + Ay ) * = x * + Ay * for all x,y E A and A E C the field of complex number, where A is the conjugate of λ . Of course the prototypical example of an involution on a Banach algebra is the adjoint operation on B(H ) , the set of bounded linear operators on Hilbert space H . Another important example is complex conjugation on C(X) , the set of all continuous complex valued functions on X , a compact Hausdroff space defined as f * ( x ) := f ( x ) .

An algebra equipped with an involution is called a * -algebra or algebra with involution. A Banach * -algebra is a Banach algebra A together with an isometric involution |x * | = |x| for all x E A . A Banach * -algebra is called a C * -algebra A if |x * x| = |x| ² for all x E A . A C ^∗ -algebra A is primitive if its zero ideal is primitive, that is, if A has a faithful nonzero irreducible representation. Let W _n denote the standard polynomial of degree n in n non-commuting variables, W n = S _CTe s _n sign ( a ) a_CT(i)a ^ (2) ••• a_CT(_n), where S_n is the set of all permutations of { 1 , 2 , 3 , ••• , n} and sign ( a ) = ± 1 for a even (odd) (see [26, 27] and references therein). An algebra A is said to be an C * -W_n -algebra if W n ( ai,a2, • • • , a n ) = 0 for each choice of elements ai, a2, • • • , a n E A . In particular, an algebra is C * — W4 -algebra if it satisfies the standard identity W4 ( ai, a2, аз, 04 ) = 0 for all 01,02,03,04 E A . Moreover, an algebra is C ^* — W 2 -algebra if and only if it is commutative, i. e., a C ^* — W 2 -algebra is commutative if it satisfies the standard identity W 2 ( ai , a2 ) = 0 for all ai , a2 E A . Many researcher discussed Gelfand’s theory for Banach algebra and C ^∗ -algebra namely, Banach- W 2_n -algebra and C ^* — W 2_n -algebra. Throughout the present section, C ^* -algebras are assumed to be nonunital unless indicated otherwise.

• <    Proof Of Theorem 1.3. We have given that Z : A ^ A is an automorphism of A and A is a primitive C * -algebra such that ( x ^m о y ⁿ ) ^z E Z ( A ) for all x,y E A . Therefore, A is prime by [28, Theorem 5.4.5] because A is primitive C ^∗ -algebra. Hence, A is a prime ring since A is a prime C ^∗ -algebra. By application of Theorem 1.1 get the required conclusion, thereby proving the theorem. >

• <    Proof Of Theorem 1.4. We have ( x ⁿ ) ^z + x n * E Z ( A ) for all x E A . Replace x ^* for x , to get ( x ⁿ* ) ^z + x n E Z ( A ) for all x E A . Now, a map n : A ^ A by x ⁿ = x *^Z for every x E A . It is easy to see that ( xy ) ⁿ = x ⁿ y ⁿ for all x, y E A , that is, n is an automorphism of A and hence we find that ( x ⁿ ) ⁿ + x n E Z ( A ) for every x E A . Therefore, A is prime by [28, Theorem 5.4.5] because A primitive C ^∗ -algebra. Hence, A is a prime ring since A is a prime C * -algebra. Application of Theorem 3.3 yields the required conclusion. >