lie derivations of certain csl algebras

ISRAEL JOURNAL OF MATHEMATICS 155 (2006), 149-156

LIE DERIVATIONS OF CERTAIN CSL ALGEBRAS

BY

FANGYAN LU*

Department of Mathematics, Suzhou University Suzhou 215006, P. R. China e-mail: [email protected]

ABSTRACT

It is shown t h a t each Lie der ivat ion on a reflexive algebra, whose lat t ice

is comple te ly d i s t r ibu t ive and c o m m u t a t i v e , can be un ique ly decomposed

into t he s u m of a der ivat ion and a l inear m a p p i n g wi th image in t he

center of the algebra.

1. In troduct ion

Let ,4 be an associative algebra. A linear mapping 5 : ,4 ~ ,4 is called a Lie

d e r i v a t i o n if 5([A, B]) = [5(A), B] + [A, 5(B)] for all A, B, e ,4, where [A, B] = A B - B A is the usual Lie product. We say a Lie derivation 5 is s t a n d a r d if it

can be decomposed as 5 = d + T, where d is an ordinary derivation and T is a

linear mapping from ,4 into the center of ,4 vanishing on each commutator. The

standard problem, which has been studied for many years, is to find conditions

on ,4 under which each Lie derivation 5 is standard. This problem has been

investigated for prime rings in [3, 4, 17, 20, 21], for C*-algebras and for more

general semisimple Banach algebras in [1, 11, 18, 19], for non-selfadjoint algebras

in [5]. In the present note, we pursue this line of investigation for certain CSL

algebras. More precisely, we shall prove the following result.

THEOREM: Let Alg s be a reflexive a/gebra on a separable Hilbert space 7-I with

completely distributive and commutat ive lattice. Let 5 be a Lie derivation from

* Supported by NNSFC (No. 10571054) and a grant (No. 04KJB110116) from the government of Jiangsu Province of China. Received May 8, 2005

149

150 F. LU Isr. J. Math.

Alg f` into itself. Then there exist a unique derivation d on Alg f` and a unique

linear mapping T from Alg f` into the center of Alg f` such that 5 = d + T.

2. Preliminaries

Let 7-/ be a complex Hilbert space. By B(7-/) and I we mean the set of all

linear bounded operators on 7-/ and the identity operator, respectively. A

subspace la t t i ce f` of 7-/ is a strongly closed collection of projections on

7-I that is closed under the usual lattice operations A and V, and contains 0

and I. A totally ordered subspace lattice is called a nest . A subspace lat-

tice f` is called a c o m m u t a t i v e subspace la t t ice , or a CSL, if each pair of

projections in f` commutes. For standard definitions concerning completely

distributive subspace lattices see [10, 12]. Prom [14] we know that a sub-

space lattice is completely distributive if and only if it is strongly reflexive.

More precisely, a subspace lattice f` is c o m p l e t e l y d i s t r i b u t i v e if and only if

P = v{Q E s : Q_ ~ P} for every P E f` with P ~ 0, which is also equivalent

to the condition P = A{Q_ : L E f` and Q ~ P} for every P c f` with P ~ I,

where Q_ = V{E c f`: E ~ Q}.

For a subspace lattice s on 7-/, the associated subspace lattice algebra AlgE is the set of operators on 7-I that leave invariant every projection in f`.

Obviously, Algs is a unital weakly closed subalgebra of B(7-t). Dually, if A is

a subalgebra of B(~-/) we denote by Lat .A the collection of projections that are

left invariant by all operators in .4. An algebra A is ref lexive if A = Alg Lat A,

and a lattice s is reflexive if s = Lat Alg s Every CSL is reflexive [2]. Clearly,

every reflexive algebra is of the form Alg s for some subspace lattice s and vice

versa. We will call a reflexive algebra Alg s a CSL a lgeb ra if f` is a CSL, and

a C D C a lgeb ra if s is a completely distributive CSL.

Given a subspace lattice, we set

J ( L ) = { P E L : P r 1 6 2

The relevance of J ( f ` ) is due to the following lemma which will be frequently

used.

LEMMA 1 ([15]): Let f` be a subspace lattice on a Hilbert space T/. Then

the rank one operator x | y belongs to Alg f` i f and only i f there is an element

P C i f ( f . ) such that x E P and y E P_~. Here x Q y is defined as (x | = (z, y )x

for z C 7-l.

Let 7~1 (f`) denote the algebra generated by all rank one operators in Alg f`.

It turns out that, for CDC algebras, 7~1 (f`) is 'large enough'.

Vol. 155, 2006 LIE DERIVATIONS OF CERTAIN CSL ALGEBRAS 151

LEMMA 2 ([13]): Let Alg12 be a CDC algebra. Then 741(s is weak*-dense in

Alg s

Recall that a CSL algebra Alg s is irreducible if and only if the commutant

is trivial, i.e. (Algs = CI, which is also equivalent to the condition that

s • = {0, I}, where s177 = {P• : P E s It turns out that any CDC algebra

can be decomposed into the direct sum of irreducible CDC algebras. Let Alg s

be a CDC algebra on a Hilbert space H. We say that two projections P and Q

in ,7(s are c o n n e c t e d if there exist finitely many projections P1, P2 , . . . , Pn

in J(12) such that Pk and Pk+l are comparable for each k = 0, 1 , . . . , n, where

P0 = P and P,~+I = Q. We say that a subset ~7 of J ( s is a c o n n e c t e d

c o m p o n e n t if each pair in ~ is connected and any element in J ( s \ ~ is not

connected with any element in ~ .

LEMMA 3 ([7]): Let Atgs be a CDC algebra on a separable Hitbert

space H. Then there are no more than countably many connected components

{~n : n G A} of ,7(/2) such that i f (E) = U n e h { P : P E ~n}. Moreover, let

Pn = V{P : P C Wn}. Then {Pn : n �9 A} is a subset ofpairwise orthogonal

projections in s n s177 and the algebra Algs can be written as a direct sum:

Alg12 = E ~ ( a l g 1 2 ) P n , n

where each (Alg 12)Pn viewed as a subalgebra of operators acting on the range

of Pn is an irreducible CDC algebra.

The following lemma is a useful characterization of irreducible CDC algebras.

It enables us to prove that each Jordan isomorphism between CDC algebras

is the sum of an isomorphism and an anti-isomorphism in [16]. At present, it

enables us to apply Cheung's results [5].

LEMMA 4 ([16]): Let Alg12 be a non-trivially irreducible CDC algebra on a

Hilbert space H. Then there exists a non-trivial projection P in s satisfying

the following conditions.

(1) P ( A l g s • is faithful, that is, for T �9 Algs T P ( A l g s • = {0}

implies T P = 0 and P(Alg12)PJ-T = {0} implies P • = O.

(2) For every Q �9 12 with Q_ ~ P there exists a non-zero vector YO �9 H such

that x | YO �9 P ( A l g s P• for all x �9 Q. Dually, for every Q �9 12 with

Q ~ P there is a non-zero vector XQ �9 ~ such that XQ | y �9 P(Alg 12)P•

for every Y C QZ_.


3. P r o o f of t h e t h e o r e m

We begin with the irreducible case.

LEMMA 5: Let Algs be an irreducible CDC algebra on a separable Hilbert

space 7-l. Let ~: Alg s -* Alg s be a Lie derivation. Then 6 is standard.

Proof: If s is trivial, then Alg s = B(/-/) is a C*-algebra. It follows from

[18, Theorem 1.1] that 6 is standard.

We may now suppose that s is non-trivial. Let P be a non-trivial projection

in s provided by Lemma 4. Let .4 = P ( A l g s B = P • 1 6 3 • and

AA = P(Alg s Then we can write

[0 B "

Let Z(.4) and Z(B) denote the centers of A and B, respectively. Then by

[5, Proposition 4], 6 takes the form

0 0s (S) + fl.a(A)

where mappings 0A: A --* A, 0s: B ~ B, flA: A --* Z(B), fls: B --* Z(A) and 7: A//--+ A4 are linear and satisfy:

(1) OA is a Lie derivation on A, flA([A1,A2]) = 0, 7(AM) = OA(A)M +

A'~(M) - Mfl.a(A).

(2) ~s is a Lie derivation on B, fls([B1,B2]) = O, ~ ( M B ) = M 0 s ( B ) +

7 ( M ) B - ~/s(B)M.

Since A4 is faithful, by [5, Theorem 11], it suffices to verify that fl.a(A) is a scalar multiple of P • for every A E .4 and ~s (B) is a scalar multiple of P

for every B E B. In the following, we only check the first statement since the

second can be similarly checked. To do this, we define W(.4) to be the set of all

operators A in .4 satisfying that flA(A) is a scalar multiple of P• Then by [5,

Proposition 10], W(.4) contains all idempotents in .4. Hence 7"41(.4) C_ W(.4)

since every rank one operator in a CSL algebra is a sum of finite idempotents

[8], where T41 (A) denotes the linear submanifold of A spanned by the set of all

operators of rank one in .4.

For T, S E .4, we define G(T, S) = O.a(TS) - T~.a(S) - 0x(T)S. Then using

the identity 7 (AM) = OA(A)M + Av(M) - Mi3.4(A), we can verify that

G(T, S ) M = Mt3A(TS) - T M f l A ( S ) - SMf lA(T) , M e .M.


Thus if S �9 7~1(.4), supposing/3a(TS) = AlP • and ~A(S) = A2P • then we

have that

(1) a(T, S )M = SM~3A(T),

where a(T, S) = - (G(T, S) - AlP + A2T).

Note that .4 is also a CDC algebra on the separable Hilbert space PT-L Let

{~k}~=l, n _< co, be the set of connected components of {Q �9 s : Q_ ~ P}.

For each k, let Qk = v{Q : Q �9 ~r and -4k = Qk(AlgZ.)Qk. Then we can

decompose .4 as the direct sum of at most countably many irreducible CDC

algebras {.4k}:

.4 =.41 | | G . . .

Fix A �9 .4. We show below that a(A, S) is a scalar multiple of S for every

S �9 ~1 (.4) A .4k. Let S be in 7~1 (.4) N .4k. For every E �9 ~gk, by Lemma 4

there exists a unit vector YE such that x | YE �9 M for all x �9 E. It follows

from Eq. (1) that

a(A, S)x | y~ = S(x | yE)~3.a(A), x �9 E.

Applying both sides to YE, we obtain a scalar A(E) such that a(A ,S)x =

A(E)Sx. Note that such a scalar A(E) is independent of S. Moreover, since

~k is connected, it follows that A(E) = A(F) for any E, F E ~k. Therefore,

there exists a scalar Ak such that a(A, S)x = AkSx holds for all x E Qk since

U { E : E E ~k} is dense in Qk. Hence this holds on P since a(A, S)x = Sx = 0

for all x c Qj with j r k.

Therefore, there exists a scalar Ak such that a(A, S) = AkS for every S E

7~1 (A) N Ak. Thus AkSM = SM/3A (A) for every S C T~I (A) n Ak. Hence since

S E T~I (.4) rq Ak is weak* dense in Ak by Lemma 2, it follows that

AkQkM = QkMj3.a(A).

Moreover, we have that IAk[ _< [[/~A(A)I I. Now let T = ~-'~AaQk. Then T E A

and T M = M/3.a(A) for every M E A4. Since A4 is faithful, it is not difficult

to verify that T @/~A(A) is in the commutant of AlgE. So ~A(A) is a scalar

multiple of P• because of the irreducibility of Alg s This completes the proof.

LEMMA 6: Let A be a subalgebra of B(TI) containing the identity I and d be

a continuous derivation from A into itself. Let A be in A.

(1) If d(A) is a scalar multiple of I, then d(A) -= O.

154 F. LU Isr. J, Ma th .

(2) For any scalar A, there hold IA[ ___ lid(A)+Alll and IId(A)ll ~ 2lid(A)+A/l[.

Proof: (1) The proof is very similar to that of [9, Problem 232]. Suppose

d(A) = #I for some scalar #. Then d2(A) = 0. Using this fact and the Leibniz

formula, we obtain that dn(A n) = n!(d(A)) n for all n C N. Therefore,

I]-t] n ~ - ~.lidn(An)l I <_ ~lldllnllAll n.

From this, we get # = 0 and then d(A) = O. (2) We may suppose that d(A) # O. Then by (1), there is a vector x in 7-/

such that d(A)x and x are linearly independent. Hence we can suppose that

d(A)x and x are orthogonal for some unit vector x in 7-/. Thus, for all scalars A

IA] = ]IAIII <_ Iid(A)x + Axll <_ lid(A) +AII[

and hence IId(A)II < 21]d(A ) + AI H.

We are now in a position to prove the central result of the paper.

Proof of Theorem: Let Alg s = ~ n (~(Alg s be the irreducible decompo-

sition of Alg s as in Lemma 3. Fix an index n. Let A be an arbitrary operator

in Algs Then for all B C (AIgs we have that

0 = 5([AP•, B]) = ~(AP~)B - B(5(AP~) + AP:5(B) - (5(B)AP~.

Prom this, we obtain 5(AP~)B = Bh(AP~) for all B r (Alg/:)P~, and hence

there exists a scalar An(A) such that (~(AP~)Pn = An(A)Pn. Therefore

a(A)P -- a(APn)Pn + a(AP )Pn = a(AP )Pn + A (A)Pn.

Moreover, since (5(APn)Pn is a Lie derivation from the irreducible CDC algebra

(Alg s into itself, it follows from Lemma 5 that there exist a derivation dn

from (Alg s into itself and a scalar #~(A) such that ~(APn)P~ = d~(APn) + #~(A)Pn. Thus

~(A)Pn = d~(APn) + fn(A)Pn

holds for all A C Alg s where fn is a linear functional on Alg s Furthermore,

since derivations on CSL algebras are continuous [6], by Lemma 6, [Idn(AP~)II <<- 211~(A)I I and If~(A)l _< 1Ih(A)II. Therefore, the mapping d, defined by d(A) = ~n(~dn(APn) for A E Algs is a derivation from Algs into itself, and the

mapping T, defined by T(A) = Y~n (~ fn(A)P~ for A E Alg/2, is a linear mapping

from Alg s into its center. Moreover,

(5(A) = ~ @5(A)P,~ = ~ O(dn(APn) + "rn(A)P,~) = d ( A ) + ~-(A) n n


holds for all A c Alg s From this, one can easily verify t ha t r vanishes on

every commuta to r .

Now it remains to show the uniqueness in the s ta tement . Let 5 = dl + r l =

d2 +3-2 be two decomposi t ions of the Lie derivat ion 5 into the sum of a der ivat ion

and a l inear m a p p i n g with range in the center. Then for all A E Alg s and P~,

we have (dl(A) - d2(A))Pn = (T2(A) - r l (A))Pn is a scalar of Pn. Note t ha t

dl - d : is a cont inuous derivation. Thus, arguing as in the proof of L e m m a 6(1),

we see t ha t (dl (A) - d2(A))Pn = 0 for all A E A l g s and Pn. So dl - d2 = 0

and therefore r l - ~-2 = 0. This comple tes the proof.

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lie derivations of certain csl algebras

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