CUBO A Mathematical Journal

Vol.10, N
o
¯ 03, (211–222). October 2008

On Two-Sided Centralizers of

Rings and Algebras∗

Joso Vukman

Department of Mathematics and Computer Sciences,

Faculty of Natural Sciences and Mathematics, University of Maribor,

Koroška 160, 2000 Maribor, Slovenia

email: joso.vukman@uni-mb.si

and

Irena Kosi-Ulbl

Faculty of Education, University of Maribor,

Koroška 160, 2000 Maribor, Slovenia

email: irena.kosi@uni-mb.si

ABSTRACT

In this paper we prove the following result. Let A be a semisimple H∗−algebra and

let T : A → A be an additive mapping satisfying the relation (n + 1)T (xnm+1) =

T (x)xnm + xmT (x)x(n−1)m + · · · + xnmT (x), for all x ∈ A and some fixed integers

m ≥ 1, n ≥ 1. In this case T is a two-sided centralizer.

RESUMEN

En este art́ıculo probamos el siguiente resultado. Sea A una H∗−algebra semi-simple

y T : A → A una aplicación aditiva satisfaziendo la relación (n + 1)T (xnm+1) =

∗This research has been supported by the Research Council of Slovenia.



212 Joso Vukman and Irena Kosi-Ulbl CUBO
10, 3 (2008)

T (x)xnm + xmT (x)x(n−1)m + · · · + xnmT (x), para todo x ∈ A y ciertos m ≥ 1, y n ≥ 1

enteros fixados. En este caso T es un centralizador “two-sided”.

Key words and phrases: Prime ring, semiprime ring, Banach space, standard operator algebra,

H∗-algebra, left (right) centralizer, left (right) Jordan centralizer, two-sided centralizer.

Math. Subj. Class.: 16W10, 46K15, 39B05.

Introduction

Throughout, R will represent an associative ring with center 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. As usual the commutator

xy − yx will be denoted by [x, y] .Let us 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. An additive

mapping x 7−→ x∗ on a ring R is called involution in case (xy)∗ = y∗x∗ and x∗∗ = x hold for

all pairs x, y ∈ R. A ring equipped with an involution is called a ring with involution or ∗−ring.

We denote by Qr and C Martindale right ring of quotients and extended centroid of a semiprime

ring R. For the explanation of Qr and C we refer to [3] . An additive mapping T : R → R,

where R is an arbitrary ring, is called a left centralizer in case T (xy) = T (x)y holds for all pairs

x, y ∈ R. The concept appears naturally in C∗-algebras. In ring theory it is more common to work

with module homomorphisms. Ring theorists would write T : RR → RR of a right ring module

R into itself. For a semiprime ring R all such homomorphisms are of the form T (x) = qx, for

all x ∈ R, where q is some fixed element of Qr (see Chapter 2 in [3]). In case R has the identity

element T : R → R is a left centralizer iff T is of the form T (x) = ax, for all x ∈ R, where

a is some fixed element of R.An additive mapping T : R → R is called a left Jordan centralizer

in case T (x2) = T (x)x holds for all x ∈ R.The definitions of right centralizer and right Jordan

centralizer are self-explanatory. We call T : R → R a two-sided centralizer in case T is both a

left and a right centralizer. In case T : R → R is a two-sided centralizer, where R is a semiprime

ring with extended centroid C, then there exists an element λ ∈ C such that T (x) = λx, for all

x ∈ R (see Theorem 2.3.2 in [3]). One of the initial papers using the concept of centralizers (also

called multipliers) is due to Wendel [33] for group algebras. Helgason [9] introduced centralizers

for Banach algebras. Wang [32] studied centralizers of commutative Banach algebras. Johnson

[11] introduced the concept of centralizers for rings. We refer to Busby [7] for a study of so-

called double centralizers in the extension of C∗−algebras. Akemann, Pedersen and Tomiyama

[1] have studied centralizers of C∗-algebras. Several authors have also studied spectral properties

of centralizers on Banach algebras (see [15, 16]). Johnson [12] has studied centralizers on some

topological algebras. Johnson [13] has studied the continuity of centralizers on Banach algebras

(see also [11]). Husain [10] has also investigated centralizers on topological algebras with particular

reference to complete metrizable locally convex algebras and topological algebras with orthogonal

bases. Khan, Mohammad and Thaheem [14] have studied centralizers and double centralizers

on certain topological algebras. Centralizers have also appeared in a variety, among which we



CUBO
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On Two-Sided Centralizers of Rings and Algebras 213

mention representation theory of Banach algebras, the study of Banach modules, Hopf algebras

(see [18, 19]), the theory of singular integrals, interpolation theory, stohastic processes, the theory

of semigroups of operators, partial differential equations and the study of approximation problems

(see Larsen [16] for more details). Zalar [34] has proved that any left (right) Jordan centralizer on

a 2-torsion free semiprime ring is a left (right) centralizer. Molnár [17] has proved that in case we

have an additive mapping T : A → A, where A is a semisimple H∗−algebra, satisfying the relation

T (x3) = T (x)x2 (T (x3) = x2T (x)) for all x ∈ A, then T is a left (right) centralizer. Let us recall

that a semisimple H∗-algebra is a semisimple Banach ∗-algebra A 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 [2]). Benkovič and

Eremita [4] have proved that in case there exists an additive mapping T : R → R,where R is a

prime ring with suitable characteristic restrictions, satisfying the relation T (xn) = T (x)xn−1, for

all x ∈ R and some fixed integer n > 1, then T is a left centralizer. Vukman and Kosi-Ulbl

[26] have proved that any additive mapping T , which maps a semisimple H∗−algebra A into

itself and satisfies the relation 2T (xn+1) = T (x)xn + xnT (x), for all x ∈ A and some fixed integer

n ≥ 1, is a two-sided centralizer (see also [5]). A result of Vukman and Kosi-Ulbl [27] states that

in case 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 (x∗x) = x∗T (x)), for all x ∈ R,then T is a

left (right) centralizer. For results concerning centralizers on prime and semiprime rings, operator

algebras and H∗-algebras we refer to [8, 20 − 31] . 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. We denote by X∗ the dual space of a Banach space X and by I the

identity operator on X.

Vukman [20] has proved the following result.

THEOREM A. Let R be a 2-torsion free semiprime ring and let T : R → R be an additive

mapping. Suppose that

2T (x2) = T (x)x + xT (x)

holds for all x ∈ R. In this case T is a two-sided centralizer.

Vukman and Kosi-Ulbl [23] have proved the result below.

THEOREM B. Let R be a 2-torsion free semiprime ring and let T : R → R be an additive

mapping. Suppose that

3T (xyx) = T (x)yx + xT (y)x + xyT (x)

holds for all pairs x, y ∈ R. In this case T is of the form T (x) = λx, for all x ∈ R and some

fixed element λ from the extended centroid C of R.



214 Joso Vukman and Irena Kosi-Ulbl CUBO
10, 3 (2008)

Motivated by Theorem A and Theorem B Fošner and Vukman [8] have proved the following

theorem.

THEOREM C. Let R be a prime ring and let T : R → R be an additive mapping satisfying

the relation

nT (xn+1) = T (x)xn−1 + xT (x)xn−2 + ... + xn−1T (x),

for all x ∈ R, where n ≥ 2 is some fixed integer. If char(R) = 0, then T is of the form

T (x) = λx, for all x ∈ R and some fixed element λ from the extended centroid C of R.

In the proof of Theorem C Fošner and Vukman used as the main tool the theory of func-

tional identities (Beidar-Brešar-Chebotar theory). The theory of functional identities considers

set-theoretic maps on rings that satisfy some identical relations. When threatening such relations

one usually concludes that the form of the maps involved can be described, unless the ring is very

special (see[6]).

It this paper we consider the following more general relation

(n + 1)T (xnm+1) = T (x)xnm + xmT (x)x(n−1)m + ... + xnmT (x), (1)

where m ≥ 1, n ≥ 1 are some fixed integers. One can notice that the expression (1) for

n = m = 1 is the same as hypothesis of Theorem A. Obviously, any two-sided centralizer on

arbitrary ring satisfies the above relation. We proceed with the following conjecture.

CONJECTURE. Let R be a semiprime ring with suitable torsion restrictions and let T :

R → R be an additive mapping satisfying the relation (1) for all x ∈ R and some fixed integers

m ≥ 1, n ≥ 1. In this case T is a two-sided centralizer.

It is our aim in this paper to prove the above conjecture in semisimple H∗-algebras and in

semiprime rings with the identity element. Our methods differ from those used in [8].

THEOREM 1. Let A be a semisimple H∗-algebra. Suppose T : A → A is an additive

mapping satisfying the relation (1) for all x ∈ A and some fixed integers m ≥ 1, n ≥ 1. In this case

T is a two-sided centralizer.

For the proof of Theorem 1 we need the theorem below which is of independent interest.

THEOREM 2. Let X be a Banach space over the real or complex field F, let A(X) ⊂ L(X)

be a standard operator algebra. Suppose T : A(X) → L(X) is an additive mapping satisfying the

relation

(n + 1)T (Anm+1) = T (A)Anm + AmT (A)A(n−1)m + ... + AnmT (A),



CUBO
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On Two-Sided Centralizers of Rings and Algebras 215

for all A ∈ A(X) and some fixed integers m ≥ 1, n ≥ 1. In this case T is of the form

T (A) = λA, for all A ∈ A(X) and some fixed λ ∈ F.In particular, T is continuous.

Proof. We have the relation

(n + 1)T (Anm+1) = T (A)Anm + AmT (A)A(n−1)m + ... + AnmT (A). (2)

Let us first consider the restriction of T on F (X). 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 T (P ) = P T (P )P, which

gives

T (P )P = P T (P ). (3)

Putting A + P for A in the relation (2), we obtain

(n + 1)

nm+1
∑

i=0

(

nm + 1

i

)

T
(

Anm+1−iP i
)

= (T (A) + B)

(

nm
∑

i=0

(

nm

i

)

Anm−iP i

)

+

(

m
∑

i=0

(

m

i

)

Am−iP i

)

(T (A) + B)





(n−1)m
∑

i=0

(

(n − 1) m

i

)

A(n−1)m−iP i



 + ... + (4)

(

nm
∑

i=0

(

nm

i

)

Anm−iP i

)

(T (A) + B) ,

where B stands for T (P ) . Using (2) and rearranging the equation (4) in sense of collecting

together terms involving equal number of factors of P we obtain

nm
∑

i=1

fi (A, P ) = 0, (5)

where fi (A, P ) stands for the expression of terms involving i factors of P. Replacing A by

A + 2P, A + 3P, . . . , A + nmP in turn in the equation (1), and expressing the resulting system

of nm homogeneous equations of variables fi (A, P ), i = 1, 2, ..., nm, we see that the coefficient

matrix of the system is a van der Monde matrix















1 1 · · · 1

2 22 · · · 2nm

...
...

...
...

nm (nm)
2

· · · (nm)
nm















.



216 Joso Vukman and Irena Kosi-Ulbl CUBO
10, 3 (2008)

Since the determinant of the matrix is different from zero, it follows that the system has only

the trivial solution.

In particular,

fnm−1 (A, P ) = (n + 1)
(

nm+1

nm−1

)

T
(

A2
)

−
(

nm

nm−1

)

T (A) A −
(

nm

nm−2

)

BA2−
(

m

m−2

)(

(n−1)m

(n−1)m

)

A2B −
(

m

m−1

)(

(n−1)m

(n−1)m

)

AT (A) P −
(

m

m−1

)(

(n−1)m

(n−1)m−1

)

ABA−

(

m

m

)(

(n−1)m

(n−1)m−1

)

P T (A)A −
(

m

m

)(

(n−1)m

(n−1)m−2

)

BA2 − · · · −

(

m

m

)(

(n−1)m

(n−1)m−2

)

A2B −
(

m

m

)(

(n−1)m

(n−1)m−1

)

AT (A) P −
(

m

m−1

)(

(n−1)m

(n−1)m−1

)

ABA−

(

m

m−1

)(

(n−1)m

(n−1)m

)

P T (A) A −
(

m

m−2

)(

(n−1)m

(n−1)m

)

BA2−
(

nm

nm−2

)

A2B −
(

nm

nm−1

)

AT (A) = 0

and

fnm (A, P ) = (n + 1)
(

nm+1

nm

)

T (A) −
(

nm

nm

)

T (A) P −
(

nm

nm−1

)

BA−
(

m

m−1

)(

(n−1)m

(n−1)m

)

AB −
(

m

m

)(

(n−1)m

(n−1)m

)

P T (A)P −
(

m

m

)(

(n−1)m

(n−1)m−1

)

BA − · · · −

(

m

m

)(

(n−1)m

(n−1)m−1

)

AB −
(

m

m

)(

(n−1)m

(n−1)m

)

P T (A)P −
(

m

m−1

)(

(n−1)m

(n−1)m

)

BA−

(

nm

nm−1

)

AB −
(

nm

nm

)

P T (A) = 0.

The above equations reduce to

6 (n + 1) (nm + 1) T
(

A2
)

= 12 (T (A)A + AT (A)) + 6 (n − 1) (AT (A) P + P T (A) A) +

(n + 1) ((2n + 1) m − 3)
(

A2B + BA2
)

+ 2m (n − 1) (n + 1) ABA, (6)

and

2 (n + 1) (nm + 1) T (A) = 2 (T (A) P + P T (A)) +

n (n + 1) m (AB + BA) + 2 (n − 1) P T (A) P. (7)

Right multiplication of the relation (7) by P gives

2 (n + 1) (nm + 1) T (A) P = 2 (T (A) P + P T (A)) +

n (n + 1) m (AB + BA) + 2 (n − 1) P T (A) P. (8)

Similarly one obtains

2 (n + 1) (nm + 1) P T (A) = 2 (T (A) P + P T (A)) +



CUBO
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On Two-Sided Centralizers of Rings and Algebras 217

n (n + 1) m (AB + BA) + 2 (n − 1) P T (A) P. (9)

Combining (8) with (9) we arrive at

T (A)P = P T (A),

which reduces the relation (6) to

6 (mn + 1) T
(

A2
)

= 6 (T (A)A + AT (A)) +

((2n + 1) m − 3)
(

A
2
B + BA2

)

+ 2m (n − 1) ABA, (10)

and the relation (7) to

2 (mn + 1) T (A) = 2T (A) P + mn (AB + BA) . (11)

Right multiplication of the above relation by P and combining the relation so obtained with (11)

gives

T (A) = T (A)P.

According to the above relation the relation (11) reduces to

2T (A) = AB + BA. (12)

From the above relation one obtains

2T
(

A2
)

= A2B + BA2. (13)

Right and then left multiplication of the relation (12) by A gives

2T (A) A = ABA + BA2 (14)

and

2AT (A) = A2B + ABA, (15)

respectively. Using the relations (13), (14) and (15) in the relation (10) gives after some calculation

A (m, n) BA2 + A (m, n) A2B − 2A (m, n) ABA = 0,

where A (m, n) stands for mn − m + 3. The above relation reduces to

A
2
B + BA2 − 2ABA = 0. (16)

Applying the relations (13) and (16) in the relation (10) one obtains

2T
(

A2
)

= T (A)A + AT (A). (17)

From the relation (12) one can conclude that T maps F (X) into itself. We have therefore an

additive mapping T : F (X) → F (X) satisfying the relation (17) for all A ∈ F (X). Since F (X)



218 Joso Vukman and Irena Kosi-Ulbl CUBO
10, 3 (2008)

is prime one can apply Theorem A and conclude that T is a two-sided centralizer of F (X). We

intend to prove that there exists an operator C ∈ L(X), such that

T (A) = CA, for all A ∈ F (X). (18)

For any fixed x ∈ X and f ∈ X∗ we denote by x ⊗ f an operator from F (X) defined by (x ⊗ f )y =

f (y)x, for all y ∈ X. For any A ∈ L(X) we have A(x ⊗ f ) = ((Ax) ⊗ f ).Let us choose f and y

such that f (y) = 1 and define Cx = T (x ⊗ f )y. Obviously, C is linear. Using the fact that T is

left centralizer 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, x ∈ X.

We have therefore T (A) = CA for any A ∈ F (X). Since T right centralizer on F (X) we ob-

tain C(AP ) = T (AP ) = AT (P ) = ACP, where A ∈ F (X) and P is arbitrary one-dimensional

projection. We have therefore [A, C] P = 0. Since P is arbitrary one-dimensional projection it

follows that [A, C] = 0 for any A ∈ F (X). Using closed graph theorem one can easily prove that C

is continuous. Since C commutes with all operators from F (X) one can conclude that Cx = λx

holds for any x ∈ X and some λ ∈ F, which gives together with the relation (17) that T is of the

form

T (A) = λA (19)

for any A ∈ F (X) and some λ ∈ F. It remains to prove that the above relation holds on A(X) as

well. Let us introduce T1 : A(X) → L(X) by T1(A) = λA and consider T0 = T − T1. The mapping

T0 is, obviously, additive and satisfies the relation (2). Besides, T0 vanishes on F (X). It is our aim

to prove that T0 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). Note that S can be written in the form S = (I −P )A(I −P ), where

I denotes the identity operator on X, Since, obviously, S − A ∈ F (X), we have T0(S) = T0(A).

Besides, SP = P S = 0. We have therefore the relation

(n + 1)T0(A
nm+1) = T0(A)A

nm + AmT0(A)A
(n−1)m + ... + AnmT0(A), (20)

for all A ∈ A(X). Applying the above relation we obtain

T0(S)S
nm + SmT0(S)S

(n−1)m + ... + SnmT0(S) = (n + 1)T0(S
nm+1) =

(n + 1)T0(S
nm + P ) = (n + 1)T0((S + P )

nm+1) =

T0(S + P )(S + P )
nm + (S + P )mT0(S + P )(S + P )

(n−1)m + ...+

(S + P )(n−1)mT0(S)(S + P )
m + (S + P )nmT0(S + P ) = T0(S)S

nm+

SmT0(S)S
(n−1)m + ... + SnmT0(S) + T0(S)P + S

mT0(S)P +

P T0(S)S
(n−1)m + ... + S(n−1)mT0(S)P +

P T0(S)S
m + P T0(S) + (n − 1)P T0(S)P.



CUBO
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On Two-Sided Centralizers of Rings and Algebras 219

We have therefore

T0(A)P + S
mT0(A)P + P T0(A)S

(n−1)m + ... + S(n−1)mT0(A)P +

P T0(A)S
m + P T0(A) + (n − 1)P T0(A)P = 0. (21)

Multiplying the above relation from both sides by P we obtain

P T0(A)P = 0. (22)

Now right multiplication of the relation (21) by P gives because of (22)

T0(A)P + S
mT0(A)P + ... + S

(n−1)mT0(A)P = 0. (23)

Replacing A by 2A, 3A, . . . , nA in turn in the equation (23), and expressing the resulting

system of n homogeneous equations of variables T0(A)P, S
imT0(A)P , i = 1, 2, ..., n− 1, we see that

the coefficient matrix of the system is a matrix of the form















1 1 · · · 1

1 2m · · · 2(n−1)m

...
...

...
...

1 nm · · · n(n−1)m















.

Since the determinant of the matrix is different from zero, it follows that the system has

only the trivial solution. We have therefore T0(A)P = 0. Since P is an arbitrary one-dimensional

projection, one can conclude that T0(A) = 0, for any A ∈ A(X), which completes the proof of the

theorem.

It should be mentioned that in the proof of Theorem 2 we used some ideas similar to those

used by Molnár in [17]. Let us point out that in Theorem 2 we obtain as a result the continuity

of T under purely algebraic assumptions concerning T, which means that Theorem 2 might be of

some interest from the automatic continuity point of view.

Proof of Theorem 1. The proof goes through using the same arguments as in the proof of

Theorem in [17] with the exception that one has to use Theorem 2 instead of Lemma in [17].

We are ready for our last result.

THEOREM 3. Let n ≥ 1, m ≥ 1 be integers and let R be a 2, m, n, n + 1 and

((n − 1) m + 3)−torsion free semiprime ring with the identity element. Suppose that we have

an additive mapping T : R → R satisfying the relation (1) for all x ∈ R. In this case T is of the

form T (x) = ax, for all x ∈ R and some fixed element a ∈ Z(R).



220 Joso Vukman and Irena Kosi-Ulbl CUBO
10, 3 (2008)

Proof. We have the relation (1). Using similar approach as in the proof of Theorem 2, with

the exception that we use the identity element e instead of a projection, we obtain from the above

relation

6(nm + 1)T
(

x2
)

= 6 (T (x)x + xT (x)) +

((2n + 1) m − 3)
(

x2a + ax2
)

+ 2 (n − 1) mxax, x ∈ R (24)

and

2T (x) = xa + ax, x ∈ R, (25)

where a stands for T (e). In the procedure mentioned above we used the fact that R is m, n and

n + 1-torsion free.

The substitution x2 for x in (25)gives

2T
(

x2
)

= x2a + ax2, x ∈ R. (26)

Multiplying the relation (25) first from the right side then from the left side by x we obtain

2T (x) x = xax + ax2, x ∈ R (27)

and

2xT (x) = x2a + xax, x ∈ R. (28)

Using (26), (27) and (28) in the relation (24)and applying the fact that R is (n − 1)m + 3−torsion

free we obtain after some calculation

x
2
a + ax2 − 2xax = 0, x ∈ R,

which can be written in the form

[[a, x] , x] = 0, x ∈ R. (29)

Putting x + y for x in the above relation we obtain

[[a, x] , y] + [[a, y] , x] = 0, x, y ∈ R. (30)

The substitution xy for y in relation (30) gives because of (29) and (30)

0 = [[a, x] , xy] + [[a, xy] , x] =

= [[a, x] , x] y + x [[a, x] , y] + [[a, x] y + x [a, y] , x] =

= x [[a, x] , y] + [[a, x] , x] y + [a, x] [y, x] + x [[a, y] , x] = [a, x] [y, x] , x, y ∈ R.

Thus we have

[a, x] [y, x] = 0, x, y ∈ R.



CUBO
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On Two-Sided Centralizers of Rings and Algebras 221

The substitution ya for y in the above relation gives [a, x] y [a, x] = 0, for all pairs x, y ∈ R.

Let us point out that so far we have not used the assumption that R is semiprime. Since R is

semiprime, it follows from the last relation that [a, x] = 0, for all x ∈ R. In other words, a ∈ Z (R) ,

which reduces the relation (25) to T (x) = ax, x ∈ R, since R is 2−torsion free. The proof of the

theorem is complete.

Received: May 2008. Revised: August 2008.

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