Derivations on Banach *-ideals in von Neumann algebras

It is well known [1, Lemma 4.1.3] that every derivation on a C *-algebra A is norm continuous. In fact, this also easily follows from another well known fact [1, Corollary 4.1.7] that every derivation on A realized as a *-subalgebra iii the algebra B(H ) of all bounded linear operators on a Hilbert space H is given by a reduction of an inner derivation on a von Neumann algebra M = A (the closure of A in the weak operator topology on B(H )). In the special setting when A = K (H ) (the ideal of all coinpact operators on H) a nd M = B(H ). the latter result states that for every derivation 5 on A there exists an operator a G B(H ) such that 5(x) = [a, x] for every x G K(H ). The ideal K(H) is a classical example of a Banach operator ideal in B(H ) (see [2, 3, 4, 5]). Any such ideal E = K(H ) is a Вanach *-algebra (albeit not a C *-algebra) and a natural question immediately suggested by this discussion is as follows.

Question 1. Let (E, || • ||e ) C K(H ) be a Banach ideal of compact operators on H and lei 5 : E ^ E he a derivation on E. Is 5 continuous with i■aspect to a norm || • ||e о n E? If this fact is true, then does there exist an operator a G B(H ) such that 5 ( x ) = [ a, x ] for every x G E

The positive answer to Question 1 was obtained in the paper [6] (see also [7]).

Let now M L>e an arbitrary win Neumann algebra. An *-ideal E of M is called a, Banach *-idecA if E is equipped wildi a Banach norm || • ||e- such that kaxbkE 6 kakM · kxkE · kbkM for all x E E aiid a, b E M.

It is natural to pose the following variant of question 1.

Question 2. Let M be an arbitrary von Neumann algebra and let ( E, || • ||e ) be a Banach *-ide: E ^ E be a derivalion on E. Is 6 continuous with i-aspect t o a norm || • ||e on E? If this fact is true, then does there exist an operator a E M such that 6(x) = [a, x] for every x E E?

The following theorem, the main result of this paper, gives a positive answer to Question 2.

Theorem 1. Let (E, || • ||e ) be a Вanach *-ideal of the von .Neumann algebra M and let 6 : E ^ E be a derivation on E. Then there exists an element a E E such that 6 ( x ) = [ a, x ] for all x E E. Moreover. we can choose such an element a as fol lows: ^a^M 6 | 6 |e ^e.

• 2.    Preliminaries

For details on the von Neumann algebra theory, the reader is referred to e.g. [1, 8, 9].

Lei H l >e a Hilbert spa ce oxer the field C of complex ilumbers. lei B(H ) lie the *-algebra of all bounded linear operators on H, 1 et M be a von Neumann subalgebra in B(H ) and let P(M ) = {p E M : p² = p = p * } lie the lattice of all projections in M. The center of a win Neumann algebra M will lie denoted by Z(M ).

Let A lie an arbitrary - subalgebra in M. A linear mapping 6 : A ^ M is called derivation on A with wdues in M if the equality 6(xy) = 6(x)y + x6 ( y ) holds for all x,y E A. It is not difficult to verify that for every a E A the mapping 6_a(x) = [ a, x ] = ax — xa, x E A, defines a derivation on A, in addition 6_a (A) C A. Such derivations 6_a are called inner derivations on A.

If A is a *-subalgebra in M then a derivation 6 : A ^ M is said I,о lie a *-derivation if 6(x* ) = 6 ( x )* for all x E A. For every derivation 6 : A ^ M of’ a *-alge lira A in to M we define mappings

W)~         A   6 im( x ): =    2 , x E A.

• Il,    is easy l,o see that 6 r₆ aiid 6 j_m ai-e *-derivations on A. тогеол-ег 6 = 6 r₆ + i6 j_m.

Lei, E lie a two-sided ideal in M. Then E is an *-idea 1 in M and the c-onditions x E M. y E E. | x | 6 | y | imph- that x E E.

We need the following property of two-sided ideals in von Neumann algebras.

Proposition 1 [10, Proposition 2.4.22]. If E is wo-closed two-sided ideal in a von Neumann algebra M then there exists a central projection z E Z(M )) such that E = z • M.

A non-zero two-sided ideal E of M, equipped with a Banach norm | • ||e, is called a Banach *-tdcal. if kaxbkE 6 kakM · kbkM · kxkE whenever x E E aird a,b E M.

It should be observed that any a Banach *-id eal (E, | • ||e ) is *-closed a nd that x E M, y E E aird |x| 6 |y| imph" that x E E aird | x |e 6 | у |е-

Let A be a C *-subalgebra in the C *-algebra B(H ). By [1, Lemma 4.1.3] every derivation 5 : A ^ A is a || • ||b(h )-continuous. The following Theorem gives an extension of the derivation 5 l.o Hie von Neumann algebra A wo. wl iere A wo is a wo-closure of C *-subal gebra A ii i B ( H).

Theorem 2 [10, Proposition 3.2.24], [1, Theorem 4.1.6, Corollary 4.1.7], [11, Theorem 2]. Lei A lie a C*-subalgebra in the C*-algelira B(H ) and let 5 : A ^ A be a derivation on A. Then there exists an element a in A = N such that 5(x) = 5a (x) = [a, x] for all x G A and ||5 | a _w a = ||5a |n ^n- Moreover we can choose such an element a G N as follows: kakN ⁶ 2 * ^|5aUN^- '

• 3.    Main Results

Throughout this section M is an arbitrary von Neumann algebra. We recall that a projection p G P (M ) is called ап atom if 0 = q G P (M ), q 6 p imply that q = p. If q is an atom then q • M • q = q • C

Proposition 2. Let (E, || • I e ) be a Вanach *-ideal in the von Neumann algebra M and let 5 : E ^ E lae a derivation on E. T1 ion 5 is a eontinuous mapping on (E, || • I e )•

<1 ’Without loss of general!ty. we may assume that 5 is a *-derivatic)ii. Since (E, || • I e ) is a Banach space, it is sufficient to prove that the graph of 5 is closed. Suppose a contrary. Then there exist a sequence { а_п } П=₁ C E and an element 0 = a G E such that a = a*, ||an k E ^ 0 and || 5(a_n) — a |E ^ 0 as n ^ ra.

Let a = a ₊ — a- be an orthogonal decomposition of a. that is a+,a— G E. a₊,a— >  0. and a ₊ a - = 0. "Without loss of generality. we may assume that a ₊ = 0. otherwise we consider the sequence {— a _n}n=₁. Siiice a G E. there exists a projection p G M such that pap > Xp for some X >  0. Replaciiig an with an we may assume pap > p. Hence, for some operator c G M. we have p = c*papc G E.

There are two possible cases:

• (i)    There exists an atom 0 = q G P(M ) such that q 6 p:

• (ii)    The 1 attice P(M ) does riot coutairi atoms q = 0 such that q 6 p

In the case ( i). we have q G E aiid q 6 qaq. Since q is an atom, it follows qanq = Xnq. Xn G C. arid we immediately deduce that lim n ,^ X n = 0 from the assuriiptioii | a _n|E ^ 0. Since

5(qan q) = 5(q)a n q + q5(a n q)) = 5(q)a n q + q5(a n )q + qa n 5(q)

it follows that

|5(qanq) — q5(an)q|E 6 2|5(q)|M|an|E ^ 0, as n ^ ra, and q 6 qaq = || • Ie - lim 5(qanq) = || • |e — lim 5(Xnq) = 5(q) lim Xn = 0. n→∞             n→∞         n→∞

This contradicts with the assumption that q = 0.

In the case (ii). there exists a pairwise orthogonal sequence {еп}П=1 C P(M) such that 0 = en 6 p for all n > 1. Clearly, we have {en}^ C E aiid enaen > en for any natural number n G N. Let {mn}n=1 be any sequence of positive integers such that mn > (2n + 1)/|enIe, n > 1.

Passing to a. subsequence if necessary, we may assume without loss of generality that

||a n|E < m- ¹ 2^{- n} ,    | 5(a n ) — akE < m- ¹

and that

IKIk < 2-1nm-1|5(en)kM1

whenever n >  1 is sueh that 5(e_n ) = 0. Let us define an element

∞ c := ^mnenanen E E n=1

where the series converges in the norm | • |e, since vre have |m_ne_na_ne_n|E < 2^-n. We intend to obtain a contradiction by showing that the norm |5(c) |e is larger than an у positive integer n.

Indeed, fixing such n >  1. we 1 iave | 5(c) |e > | e_n5(c)e_n I e and

| en5(c)en I e = | 5(en c)en - 5(en)cen I e = mn | 5(en anen)en - 5(en)enan€nkE

= m_n | e_n5(e_na_ne_n ME = m_n | e_n5(a_n )en + en 5(en )anen + e_na_n8(en )en | E

• > mnken(5(an ) - a)en + enaen I e - m_n\\e_n5(en)anenkE - m_n\\e_na_n8(e_n)e_n I e

• > mn(kenaenkE - | en(5(an) - a)en I e ) - 2mnkankE | 5(en ) |m

• > mn(kenaenkE - P(an) - akE ) - n > mn | enaen |e - 1 - n >  mn | en |e - 1 - n > n.

This shows that 5 is a continuous mapping on (E , | • |e ). B

Proposition 3. Let ( E, | • |e ) be a Вanach *-ideal in the von Neumann algebra M and lei 5 be a deri^uion on E. T1 ion 5 is a continuous mapping on (E , | • |m ) aiid | 5 |^ := “^'■'.PHOL,) ⁶ *11- *» ^|5 | = J⁵« ₍E >Me ₎^₍E_J|.kE,.

C By Proposition 2. a derivation 5 : E м E is a coiitinuous on (E, | • |e ). in particular. ¹⁵¹ ,^:= H^E U E < ”;, ,   ,

Let x E E a nd 5(x) = 0. Let 0 < e <  | 5(x) |m and de note by px the spectral projection of the operator | 5 ( x )| corresponding I,о the segment [ | 5(x) |m - e, | 5(x) |m ]. Using Gelfand- Naimark theorem, one can obtain that px = 0.

We have that 0 <  ( | 5(x) |m - e)p_x 6 | 5(x) | p x. Tlien px E E and

| 5(x) | px = (px | 5(x) | ² px)¹/² = (px5(x)* 5(x)px)^{1 / 2} = | 5(x)px | .

Since the norm | • |e is monotone, we obtain that

^(|5(x)|M ^{- e)|}px^kE ^{6 |5(x)}px ^kE = ^|5(xpx) ^{- x5(}px^)kE ^{6 |5(}xpx^)kE ^{+ |x5(}px^)kE

6 | 5(xpx) Ie + | х | м | 5(px) |e 6 | 5 || xpx |e + | х | м | 5 || px |e

6 | 5 || x |m I p x I e + | x |m | 5 || px |e = 2 | 5 || x |m We , that is

( | 5(x) |m - e) | px | E 6 2 | 5 || x |m | px | E.

Dirddiiig by | p _x |e and using arIritrariiiess of e. we infer t hat,

^{| 5 ( x )|} m 6 адим •

Thus the operator 5 is Irounded with re sped, I,о the norm | • |m. in ad( litioii. |5|^ 6 2 |5|. B Now we give a proof of Theorem 1.

C Proof of Theo rem 1. Denote l>y E ai id E the closure of tlie ideal E with respec£to the uniform and weak operator topology, respectively. Then E C E C E. It is clear that E is a C *-subalgebrajn M and the derivation 5 extends by continuity (see Proposition 3) up to a derivation 5 : E м E. in additioii |5|^ = |5|^.

Since E is a wo-closcd lwо-sided in M. it follows, by Proposition 1. that E = z•M for some central project.юн z in M. Then E is a W *-subalgebra wit h the identity z. By Theorem 2. the derivation 5 extends up to a derivation 5 : E —> E, in addition, there exists an element a E E such that 5 ( x ) = 5a ( x ) = [ a, x ] for all x E E aiid kakM 6 2|| 5 a||^ = 2 ||5 |^ 6 ||5||. B

Corollary 1 (cf. [6, Theorem 3.2]). Let (E, k • ||e ) be a Banach ideal of compact operators in B(H ) and lei 5 : E ^ E be a derivation on E. Then there exi'sis an operator a E B(H ) such that 5(x) = [ a, x ] for all x E E. Moreover. we can choose such an element a as follows: ^|I " ⁵ " E"*E' . . .

Corollary 2 (cf. [12, Theorem 5.2]). Let M be a commutative von Neumann algebra and let (E, || • I e ) 5e a Вanach *-idead in M. Then any derivation 5 оn E vanishes.

A detailed study of derivations on the ideals in commutative AW*-algebras is given in the paper [12]. In particular, it is shown here that if the Boolean algebra P(M ) of all projections in the commutative AW*-algebra M is not ст-distributive then there exists a nonzero derivation on ideals in M with values in a commutative *-algebra C^(Q~) Ф i • C^(Q) where Q is a Stone cornpacturn corresponding I,о the Boolean algebra P(M ). An analogous result for derivations on an algebra C^(Q, C) was earlier obtained by A. G. Kusraev [13] for a. general Stone compactum.