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The kernel and range of derivations

lå)U: NSC 96-2115-M-110-003

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968~1nB977~31n

Abstract

Let δ and δ0 _{be derivations}

of an algebra A. In this project, we connect some relations between the following three conditions: (i) the range of δ0 _{is contained in the}

range of δ; (ii) the kernel of δ is contained in the kernel of δ0_{;(iii) δ}

is an inner derivation generated by

a ∈ A and δ0 _{is an inner derivation}

generated by a polynomial in a.

2d¿b

û˝A‹

Let A be an algebra over a field

F . For every a ∈ A we denote

by δa the inner derivation x 7→

δa(x) = [a, x]. Fix a ∈ A and

con-sider the set R(a) of all b ∈ A satis-fying the range inclusion R(δb) ⊆

R(δa). From δbb0(x) = δ_b(b0x) +

δb0(xb) we see that R(a) is a

subal-gebra of A. Similarly, the set K(a) of all b ∈ A satisfying the ker-nel inclusion Ker(δa) ⊆ Ker(δb)

is a subalgebra of A, also called the double commutant (or the dou-ble centralizer) of a. Since both

R(a) and K(a) contain a, they

also contain hai = {p(a) | p(X) ∈

F [X]}, the subalgebra generated

by a. When are R(a) and K(a) actually equal to hai? More gen-1

erally, we consider the following question: If δ, δ0 _{are derivations of}

A, does (i) R(δ0_{) ⊆ R(δ) or (ii)}

Ker(δ) ⊆ Ker(δ0_{) imply (iii) δ =}

δa and δ0 = δp(a) for some element

a and polynomial p(X) ∈ F [X]?

In the introduction of [7] John-son and Williams observed that

R(a) = K(a) = hai for every a in

the matrix algebra Mn(C). Since

every derivation on Mn(C) is

in-ner, this means that the conditions (i), (ii), and (iii) are equivalent in Mn(C). This was their

start-ing point for considerstart-ing the con-dition (i) in the algebra B(H) of all bounded linear operators on an infinite dimensional Hilbert space

H. Later the results from [7] were

extended to more general operator algebras [6, 8]. In this project, we generalize the observation of John-son and Williams in a different, purely algebraic direction.

Also, in the recent work [9] Kissin and Shulman applied re-sults on the range inclusion of derivations [6, 7, 8] to the study of the so-called range-inclusive maps on C∗_{-algebras. Thereby they}

obtained generalizations of some results on commuting maps, i.e. maps T : L → A satisfying [T x, x] = 0 for every x ∈ L

where L is a subset of a ring (al-gebra) A. There are numerous publications on commuting maps in rings, especially over the last ten years when various applica-tions have been found; in particu-lar the results on commuting maps initiated the development of the theory of functional identities (see e.g. [4]). There has also been some interest in commuting maps on C∗

-algebras [1, 2, 3, 5].

We obtain the following re-sult(and parallel result in semi-prime rings):

Theorem 1 Let A be a prime

ring with char(A) = 0. Let δ and δ0 _{be derivations of A and assume}

that δ is algebraic over C. If either (i) R(δ0_{) ⊆ R(δ), or (ii) Ker(δ) ⊆}

Ker(δ0_{), then there exist a ∈ Q} s

and a polynomial p(X) ∈ C[X] such that δ = δa and δ0 = δp(a).

and we establish some new char-acterizations of commuting maps: Theorem 2 Let A be an algebraic

prime ring. For an additive map T : A → A the following condi-tions are equivalent:

(i) R(δT a) ⊆ R(δa) for all a ∈

A,

(ii) Ker(δa) ⊆ Ker(δT a) for all

a ∈ A,

(iii) T is commuting,

(iv) there exist λ ∈ C and an additive map µ : A → C such that T a = λa + µ(a) for all a ∈ A.

References

[1] P. Ara, M. Mathieu, An ap-plication of local multipliers to centralizing mappings of

C∗_{−algebras, Quart. J. Math.}

Oxford 44 (1993), 129-138.

[2] P. Ara and M. Mathieu, Local

multipliers of C∗_-algebras,

Springer-Verlag, London 2002.

[3] M. Breˇsar, Centralizing map-pings on von Neumann alge-bras, Proc. Amer. Math. Soc. 111 (1991), 501-510.

[4] M. Breˇsar, Functional identi-ties: A survey, Contemporary

Math. 259 (2000), 93-109.

[5] M. Breˇsar, C. R. Miers, Com-mutativity preserving map-pings of von Neumann al-gebras, Canad. J. Math. 45 (1993), 695-708.

[6] C.K. Fong, Range inclusion for normal derivations,

Glas-gow Math. J. 25 (1984),

255-262.

[7] B.E. Johnson, J.P. Williams, The range of a normal deriva-tion, Pacific J. Math. 58 (1975), 105-122.

[8] E. Kissin, V. Shulman, On the range inclusion of nor-mal derivations: Variations on a theme by Johnson, Williams and Fong, Proc.

London Math. Soc. 83 (2001),

176-198.

[9] E. Kissin, V. Shulman, Range-inclusive maps on

C∗_{-algebras, Quart. J. Math.}

Oxford 53 (2002), 455-465.