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The Decomposition of Lie Derivations Irish Math. Soc. Bulletin 51 (2003), 11–20 11 The Decomposition of Lie Derivations CHARLES J. READ Abstract. A linear map D : A! E, where A is an alge- bra and E an A-bimodule, is a Lie derivation if D([a; b]) = [a; D(b)] + [D(a); b] for every a; b 2 A. Two examples are a derivation and a centre-valued trace; there are several theo- rems asserting that, in certain cases, a Lie derivation from A to E can be written (in “standard form”) as a sum D = d+¿, where d is a derivation and ¿ is a centre-valued trace. These go back to a ring-theoretic theorem of M. Bresar [1]. In the context of Banach algebras, one has a positive result by Miers [5] in the case when D : M!M where M is a von Neumann algebra. M. Mathieu and A. R. Villena [6], [7] have generalised this to a full, positive result in the case of general C¤-algebras. Questions of automatic continuity also arise in the Banach algebra setting, and positive results are achieved by Berenguer and Villena ([2], [3]) when D : A!A where A is a semisimple Banach algebra. In this paper, we start at the other end of this line of enquiry and give some examples of Lie derivations on Banach algebras that cannot be written in standard form. In a related result, we give some examples of discontinuous derivations from semisimple Banach algebras A, such that the separating subspace of the derivation does not lie in the centre of the bimodule. 1. Indecomposable, Continuous Lie Derivations Definition 1. A Lie derivation D will be said to be standard if it can be written as a sum D = d + ¿ as in the Abstract; we do not require d or ¿ to be continuous. Otherwise, D is nonstandard. 2000 Mathematics Subject Classification. Primary 46H25; Secondary 46H40, 46H70, 16W10. Key words and phrases. Derivation, Lie derivation, Banach algebra, bimod- ule, operator algebra, automatic continuity. 12 Charles J. Read In this section we shall exhibit some continuous, nonstandard Lie derivations on Banach algebras. In the radical case the underlying algebra can be finite dimensional, but not in the semisimple case. In both cases, the infinite dimensional version of the example can be varied so that it becomes discontinuous. Before considering Banach algebras, let us first consider a graded L1 (n) algebra A = n=0 A (i.e., every x 2 A is uniquely written as a PN (n) (n) (n) (n) (m) (n+m) finite sum n=0 x , x 2 A , and A ¢ A ½ A ). It is PN (n) PN (n) a fact that the map d : A!A with d( n=0 x ) = n=0 nx is a derivation. In the normed algebra context, it is never continuous unless the subspaces A(n) are eventually zero — which would mean that, assuming A(0) = C, A would necessarily be a nilpotent algebra with unit adjoined. But though the sequence of subspaces must still terminate, the conclusion that A must be nilpotent is no longer true if we make a similar definition with Lie derivations in mind. L1 (p) Definition 2. A Lie graded algebra is an algebra A = n=0 A n, PN (p) such that every x 2 A is uniquely written as a finite sum n=0 x n, x(p)n 2 A(p)n, and the commutators [A(p)n; A(p)m] ½ A[n+m] for every n and m.A Lie graded Banach algebra is the completion of a Lie graded algebra A equipped with an algebra norm such that the subspaces A[n] are closed. Let’s give an example of a Lie graded algebra that is not (with the same grading) a graded algebra. Let F be the free unital complex algebra on three generators a; b and c. Let I1 ½ F be the 2-sided ideal generated by the commutators [F; F], and let I2 ½ I1 be the 2-sided ideal generated by the commutators [F;I1]. Let A = F=I2, so there are natural quotient maps q2 : F!A and q1 : A!F=I1. L1 (n) (0) Now F is a graded algebra; F = n=1 F , where F = C, and for n > 0, F n is the finite dimensional linear subspace of F spanned by the “words” of length n in the three generators a; b and c. In r s r+s view of the fact that [F ; F ] ½ F it is plain that I1 is a graded L1 n (n) (n) (r) (s) ideal, I1 = n=1 I1 where I1 = F \ I1. Then [F ;I1 ] ½ (r+s) (r+s) L1 (n) F \ I1 = I1 , so I2 also is graded, I2 = n=1 I2 where (n) (n) I2 = F \ I2. The quotient maps q2 and q1 are homomorphisms of graded algebras. The Decomposition of Lie Derivations 13 There is a natural inner product on F such that, if w; w0 are “words” in the three generators, we have ( 1 ; if w = w0 2 F (n); < w; w0 >= n! 0; otherwise: 1 The presence of the factors n! ensures that the Euclidean norm k ¢ k2 associated with this inner product is an algebra norm, and the de- L1 n composition F = n=1 F is of course orthogonal. This is the norm we shall use on F when we want to get a radical Banach algebra with unit adjoined, indeed, a radical operator algebra. Now every graded subspace M ½ F such as I1 or I2 is closed in (F; k ¢ k2) (after all, the grading is an orthogonal direct sum and the components of the grading are finite dimensional), and there is an orthogonal projection P : F!M because it is none other P1 (i) (i) than i=1 Pi, where Pi is the orthogonal projection F !M . So when we need a continuous projection P : A! I1=I2 = ker q1, plainly we can use the orthogonal projection P : F! I1. But when we want a semisimple example we shall go for the l1 Pn norm on F, that is if x = i=1 ¸iwi 2 F where the wi are distinct words in the three generators, we define X n k x k1 = ¡i = 1 j¸ij: In this case it is useful to us that in fact P is k ¢ k1-continuous as well. The reason is as follows. If w = ®1®2 : : : ®n is a word of length n (each ®i 2 fa; b; cg), and ¼ ¼ 2 Sn, the symmetric group on n elements, then we write w = ® ® : : : ® ; and plainly each difference w ¡ w¼ lies in the ¼(1) ¼(2) ¼(n) P commutator ideal I , whereas the average 1 w¼ is orthogonal 1 n! ¼2Sn to u1[u2; u3]u4 for any 4 words u1; u2; u3 and u4 (including the trivial word 1), and so is orthogonal to I1. Thus the orthogonalP projection P : F! I sends each k ¢ k unit vector w to w ¡ 1 w¼; 1 1 n! ¼2Sn hence k I ¡ P k1 = 1; so the projection P : A = F=I2 ! I1=I2 also satisfies k I ¡ P k1 = 1. With respect to either norm, we have a decomposition A = Ker P © Im P with a continuous projection P . Writing A[1] = ker P and A[2] = Im P , the decomposition A = A[1] © A[2] is a Lie grading, because if x; y 2 A[1] then [x; y], being a commutator, lies in A[2], while if [2] x 2 A = F=I2 and y 2 A = I1=I2 then [x; y] 2 [F;I1]=I2 = 14 Charles J. Read [1] [2] I2=I2 = (0). So there is a Lie derivation D : A!A, D(a + a ) = a[1] + 2a[2] for any a[i] 2 A[i]. D extends continuously to a Lie derivation on the completion of either normed algebra (A; k ¢ k1) or (A; k ¢ k2). We claim that in neither case can D be decomposed (continuously or otherwise) into a sum d + ¿, with d a derivation and ¿ a centre-valued trace. Let’s write Ai for the completion of (A; k ¢ ki)(i = 1; 2). Now the subspaces A(n) are finite-dimensional and mutually orthogonal, and with respect to the k ¢ k1 norm the natural projection Pn onto Ln (n) i=0 A has norm 1; so for any finite-dimensional subspace U of the original, incomplete normed algebra F, the orthogonal projection from A onto q2(U) will be continuous with respect to the k ¢ k1 norm as well as the Euclidean norm k ¢ k2 (for one obtains it by composing PN , for suitably large N, with a necessarily continuous operator on the finite-dimensional normed space Im PN ). Let us abuse notation slightly by suppressing the quotient map q2 and regarding a; b etc. as elements of A. With respect to either norm, if D = d + ¿ : Ai !Ai we may write d(a) = ¸a + ®; d(b) = ¹b + ¯; (1) where ® ? a and ¯ ? b. Now ¿(a) = (1 ¡ ¸)a ¡ ®, hence ¿(a)b = (1 ¡ ¸)ab ¡ ®b (with ®b ? ab; ba); and b¿(a) = (1 ¡ ¸)ba ¡ b® with b® ? ab; ba. Therefore [¿(a); b] = (1 ¡ ¸)[a; b] + (b® ¡ ®b); (2) with b® ¡ ®b orthogonal to lin(ab; ba). Now any “double commuta- tor” x 2 [F;I1] is a linear combination of words of length at least 3, (2) so in our original grading we have I2 = (0). It followsp that even in A = F=I2, we have k [¿(a); b] ki ¸ 2j1 ¡ ¸j (if i = 1), or 2j1 ¡ ¸j (if i = 2).
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