The Extended Abstracts of The 4th Seminar on Functional Analysis and its Applications 2-3rd March 2016, Ferdowsi University of Mashhad, Iran
Oral Presentation PROPER LIE DERIVATIONS ON B(X) AMIRHOSSEIN MOKHTARI1 ∗ AND HAMID REZA EBRAHIMI VISHKI 1
2
Department of Mathematics, Faculty of Mathematics and Statistics, University of Birjand, Birjand, Iran.
[email protected] 2
Department of Pure Mathematics and Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran.
[email protected] Abstract. Cheung [Linear Multilinear Algebra, 51 (2003), 299– 310] characterized Lie derivations on a triangular algebra. In this talk, we study Lie derivations on a generalized matrix algebra and we find some sufficient conditions under which a Lie derivation is proper. Finally we use the results for B(X), the algebra of bounded linear operators on a Banach space X.
1. Introduction Let A be a unital algebra (over a commutative unital ring R) and M be an A−module. A linear mapping D : A → M is said to be a derivation if D(ab) = D(a)b + aD(b)
(a, b ∈ A).
2010 Mathematics Subject Classification. Primary 16W25; Secondary 15A78, 47B47. Key words and phrases. Lie derivation, generalized matrix algebra, bounded linear operator. ∗ Speaker.
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A linear mapping L : A → M is called a Lie derivation if L([a, b]) = [L(a), b] + [a, L(b)]
(a, b ∈ A),
where [a, b] = ab − ba. Every derivation is trivially a Lie derivation. A Lie derivation L is called proper if L is of the form L = D + τ for some derivation D and a linear center valued map τ on A (i.e. τ (A) ⊆ Z(A)). We say that an algebra A has Lie Derivation Property if every Lie derivation on A is proper. Lie derivations on various algebras have been studied by many authors, for example see [1], [2], [3], [4], [4] and references therein. As it has been introduced in [2] (see also [4]), for a Morita context (A, B, M, N , ΦMN , ΨN M ), the set } ( ) {( ) A M a m G= = a ∈ A, m ∈ M, n ∈ N , b ∈ B N B n b forms an algebra under the usual matrix operations, where at least one of two modules M and N is non-zero. The algebra G is called a generalized matrix algebra. In the case where N = 0, G becomes the so-called triangular algebra Tri(A, M, B). A direct verification reveals that the center Z(G) of G is Z(G)={a⊕b|a ∈ Z(A), b ∈ Z(B), am = mb, na = bn for all m ∈ M, n ∈ N }, ) ( a 0 ∈ G. We also define two natural projections where a ⊕ b = 0 b πA : G −→ A and πB : G −→ B by ( ) ( ) a m a m πA : 7→ a and πB : 7→ b. n b n b Obviously, πA (Z(G)) ⊆ Z(A) and πB (Z(G)) ⊆ Z(B). 2. Main results We start with the following result providing the construction of Lie derivations of a generalized matrix algebra. ) ( A M be a generProposition 2.1 ([3, Proposition 4.1]). Let G = N B alized matrix algebra. Then a linear mapping L on G is a Lie derivation if and only if it has the presentation ( ) ( ) a m P (a) − mn0 − m0 n + hB (b) am0 − m0 b + f (m) L = n b n0 a − bn0 + g(n) hA (a) + n0 m + nm0 + Q(b) for some m0 ∈ M, n0 ∈ N and some linear maps P : A −→ A, Q : B −→ B, f : M −→ M, g : N −→ N , hB : B −→ Z(A) and hA : A −→ Z(B) satisfying the following conditions: (a) P, Q are Lie derivations;
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(b) hB ([b, b′ ]) = 0, hA ([a, a′ ]) = 0; (c) f (am) = P (a)m − mhA (a) + af (m), f (mb) = mQ(b) − hB (b)m + f (m)b; (d) g(na) = nP (a) − hA (a)n + g(n)a, g(bn) = Q(b)n − nhB (b) + bg(n); (e) P (mn) − hB (nm) = mg(n) + f (m)n, Q(nm) − hA (mn) = g(n)m + nf (m); for all a, a′ ∈ A, b, b′ ∈ B, m ∈ M and n ∈ N . As a generalization of Cheung’s result [1], we present the following theorem characterizing the proper Lie derivations on a generalized matrix algebra G. Theorem 2.2. Let G be a generalized matrix algebra. A Lie derivation L on G is proper if and only if there exist linear mappings ℓA : A −→ Z(A) and ℓB : B −→ Z(B) satisfying the following conditions: (1) P − ℓA and Q − ℓB are derivations on A and B, respectively; (2) ℓA (a)⊕hA (a) ∈ Z(G) and hB (b)⊕ℓB (b) ∈ Z(G), for all a ∈ A, b ∈ B; (3) ℓA (mn) = hB (nm) and ℓB (nm) = hA (mn), for all m ∈ M, n ∈ N . The next corollary is deduced from Theorem 2.2. Corollary 2.3. Let A be a 2−torsion free unital algebra with a nontrivial idempotent p and q = 1 − p. Then A has Lie derivation property if the following three conditions hold: (1) Z(qAq) = Z(A)q and pAq is a faithful left pAp−module; or pAp = WpAp and pAq is a faithful left pAp−module; or pAp has Lie derivation property and pAp = WpAp . (2) Z(pAp) = Z(A)p and qAp is a faithful right qAq−module; or qAq = WqAq and qAp is a faithful right qAq−module; or qAq has Lie derivation property and qAq = WqAq . (3) One of the following assertions holds: (a) Either pAp or qAq does not contain nonzero central ideals. (b) pAp and qAq are domain. (c) Either pAq or qAp is strongly faithful. As an applilication of Corollary 2.3 we present the next result. Corollary 2.4. Let X be a Banach space of dimension greater than 2. Then B(X) has Lie derivation property. References 1. W.-S. Cheung, Lie derivations of triangular algebras, Linear Multilinear Algebra, 51 (2003), 299–310. 2. Y. Du and Y. Wang, Lie derivations of generalized matrix algebras, Linear Algebra Appl. 437 (2012), 2719–2726. 3. Y. Li and F. Wei, Semi-centralizing maps of generalized matrix algebras, Linear Algebra Appl. 436 (2012), 1122-1153.
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4. A.H. Mokhtari and H.R. Ebrahimi Vishki, More on Lie derivations of generalized matrix algebras, ArXiv:1505.02344v1. 5. Y. Wang, Lie n−derivations of unital algebras with idempotents, Linear Algebra Appl. 458 (2014), 512–525.
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