Derivations on Banach ∗-ideals in von Neumann Algebras Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2018, Volume 20, Issue 2, P. 23-28

УДК 517.98

DOI 10.23671 /VNC.2018.2.14715

DERIVATIONS ON BANACH *-IDEALS IN VON NEUMANN ALGEBRAS A. F. Ber1, V. I. Chilin2, F. A. Sukochev3

1 Institute of Mathematics of Republica of Uzbekistan; 2 National University of Uzbekistan;

3 School of Mathematics and Statistics, University of New South Wales

Abstract. It is known that any derivation S : M ^ M on the von Neumann algebra M is an inner, i.e. S(x) := Sa(x) = [a, x] = ax — xa, x £ M, for some a £ M. If H is a separable infinite-dimensional complex Hilbert space and K(H) is a C*-subalgebra of compact operators in C*-algebra B(H) of all bounded linear operators acting in H, then any derivation S : K(H) ^ K(H) is a spatial derivation, i.e. there exists an operator a £ B(H) such that S(x) = [x, a] for all x £ K(H). In addition, it has recently been established by Ber A. F., Chilin V. I., Levitina G. B. and Sukochev F. A. (JMAA, 2013) that any derivation S : E ^ E on Banach symmetric Meal of compact operators E С K(H) is a spatial derivation. We show that the same result is also true for an arbitrary Banach *-ideal in every von Neumann algebra M. More precisely: If M is an arbitrary von Neumann algebra, E be a Banach *-ideal in M and S: E ^ E is a derivation on E, then there exists an e lement a £ M such th at S(x) = [x, a] for all x £ E, S

Key words: von Neumann algebra, Banach *-ideal, derivation, spatial derivation. Mathematical Subject Classification (2010): 46L57, 46L51, 46L52.

1. Introduction

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 in 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 Ж = Aw° (the closure of A in the weak operator topology on Ш(Н)). In the special setting when A = K(H) (the ideal of all compact operators on H) and M = B(H), the latter result states that for every derivation ¿on A there exists an operator a G B(H) such that S(x) = [a, ж] 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 Banach *-algebra (albeit not a C*-algebra) and a natural question immediately suggested by this discussion is as follows.

Question 1. Let (E, || ■ ||E) С K (H) be a Banach ideal of compact op era tors on H and let 5: E ^ E be a derivation on E. Is 5 continuous with respect to a norm || ■ ||e on E? If this fact is true, then does there exist an operator a G B(H) such th at 5(x) = [a, ж] for every x G E?

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

© 2018 Ber A. F, Chilin V. I, Sukochev F. A.

Let now M be an arbitrary von Neumann algebra. An *-ideal E of M is called a Banach *-ideal, if E is equipped with a Banach norm || ■ ||g, such that

\\axb\\s ^ ||a|M ■ ||x||g ■ ||b|M

for all x G E and a, b G 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, || ■ ||g) be a Banach *-ideaI of M. Le t 5 : E ^ E be a derivation on E. Is 5 continuous with respect to a norm || ■ ||g on E? If this fact is true, then does there exist an operator a G M such that 5(x) = [a, x] for every x G E?

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

Theorem 1. Let (E, || • \\g ) be a Banach * -ideal of the von Neumann algebra Jl and let ô: E —» E be a derivation on E. Then there exists an element a G E such that S(x) = [a, x] for all x G E. Moreover, we can choose such an element a as follows: ||a||M ^ ||5||g^g-

2. Preliminaries

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

Let # be a Hilbert space over the field С of complex numbers, let B(H) be the *-algebra of all bounded linear operators on H, let M be a von Neumann subalgebra in B(H) and let P(M) = {p G M : p2 = p = p*} ^e ^te of all projections in M. The center of a von

Neumann algebra M will be denoted by Z(M).

Let A be an arbitrary subalgebra in M. A linear map ping §: A ^ M is called derivation on A with values in M if the equality §(xy) = §(x)y + x§(y) holds for all x,y G A. It is not difficult to verify that for every a G A the mapping §a(x) = [a,x] — ax xai x G A a derivation on A, in addition §a (A) С A Such derivations §a are called inner derivations A

If A is a *-subalgebra in M then a derivation §: A ^ M ^s said to be a *-derivation if §(x*) = §(x)* for all x G A. For every derivation §: A ^ M of a *-algebra A into M we define mappings

It is easy to see that §Re and §im аде on A, moreover § = §Re + i§im-

Let E be a two-sided ideal in M. Then E is an *-ideal in M and the conditions x G M, y G E, |x| ^ |y| imply that x G E.

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

Proposition 1 [10, Proposition 2.4.22]. IfEis wo-closed two-sided ideal in a von Neumann algebra M then there exists a central p rejection z G Z (M)) such th at E = z • M.

A non-zero two-sided ideal E of M, equipped with a Banach norm || • ||e, is called a Banach *-ideal, if

||e ^ ||a||M • ЦЬЦм • ||x|e

whenever x G E and a,b G M.

It should be observed that any a Banach *-ideal (E, || ■ \\g^s ^^^^^^^^ and that x G M, y G E and |x| ^ |y| imply that x G E and ||x||e ^ ||yne-

Let A be a C*-subalgebra in the C*-algebra ). 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 to the von Neumann algebra Aw°, where A w° is a wo-closure of C*-subalgebra A in £i§(H).

Theorem 2 [10, Proposition 3.2.24], [1, Theorem 4.1.6, Corollary 4.1.7], [11, Theorem 2]. Let A be a C*-sub^gebra in the C*-algebra B(H) and let 5: A ^ A be a derivation on A. Then there exists an element a in Aw° = such that 5(x) = 5a(x) = [a,x] for all x £ A and ||5||a^a = ||5aMoreover we can choose such an element a e N as follows:

IMU ^ 2 ' II./K—k/K-

3. Main Results

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

Proposition 2. Let (E, || ■ ||E) be a Banach *-ideal in the von Neumann algebra M and let 5: E ^ E be a derivation on E. Then 5 is a continuous mapping on (E, || ■ ||e)•

< Without loss of generality, we may assume that 5 is a *-derivation. Since (E, || ■ ||E) is

5

there exist a sequence |an}^=1 C E and an element 0 = a e E such that a = a*, 11an |e ^ 0 and ||5(an) — a||E ^ 0 as n ^ to.

Let a = a+ — a_ be an orthogonal decomposition of a, that is a+,a_ e E, a+,a_ ^ 0, and a+a_ = 0. Without loss of generality, we may assume that a+ = 0, otherwise we consider the sequence {—Sinee a e E, there exists a proj ection p e M such that pap ^ Ap for some A > 0. Replacing an with ^ we may assume pap ^ p. Hence, for some operator c e Jl, we have p = c*papc e E.

There are two possible cases:

(i) There exists an atom 0 = q e P (M) such th at q ^ p;

(ii) The latt i ce P (M) does not contain atoms q = 0 such th at q ^ p.

In the case (i), we have q e E and q ^ qaq. Since q is an atom, it follows qanq = Anq, An e C, and to immediately deduce that limn^^ An = 0 from the assumption ||an||E ^ 0. Since

5(qa„ q) = 5(q)a„ q + q5(a„q)) = 5(q)a„ q + q5(a„)q + qa„5(q)

it follows that

||5(qanq) — q5(an)q|U ^ 2||5(q)||M||an|U ^ 0, as n ^ to,

and

q ^ qaq = || ■ ||e — lim 5(qa„q) = || ■ ||e — lim 5(A„q) = 5(q) lim A„ = 0.

q=0

In the case (ii), there exists a pairwise orthogonal sequence {en}^=1 C P(M) such that 0 = en ^ p for all n ^ 1. Clearly, we have {en}^=1 C E and enaen ^ en for any natural number n e N. Let {mn}^=1 be any sequence of positive integers such that

mn > (2n +1)/|en||e, n ^ 1.

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

||an||E <m_12_n, ||5(an) — a||E < m-1

and that

||a„||E < 2-1nm-1||£(e„)||M whenever n ^ 1 is such that 5(en) = 0. Let us define an element

c := ^ mnen,anen G E

n=1

where the series converges in the norm || ■ ||E, since we h ave ||mnenanen||E < 2-n. We intend to obtain a contradiction by showing that the norm ||ô(c)||e is larger than any positive integer n. Indeed, fixing such n ^ 1, we have ||ô(c)||E ^ ||enô(c)en||E and

||en^(c)en||e = ||ô(enc)en - ô(en)ce,n||e = mn||ô(enanen)en - b(e,n)enanen||E

= mnWenSfenane-n )en||E = mn||enô(an )e,n + e,n ô(e n )anen + enanô(en )en\\E

^ mn|en(^(an) - a)en + e,nae,n||e - mn||en5(e,n)anen||E - m-aHe-na,n5(e,n)en||e ^ mn(|enaen|E - ||en(ô(an) - a)en||e) - 2mn||an||E||ô(en)||m ^ mn(|enaen|E - ||ô(an) - a||E) - n> mn\\enaen||e - 1 - n ^ mn||en||e - 1 - n>n.

This shows that ô is a continuous mapping on (E, || ■ ||E). >

Proposition 3. Let (E, || ■ ||E) be a Banach *-ide^ in the von Neumann algebra M and let ô be a derivation on E. Then ô is a continuous mapping on (E, || ■ ||m) and ||ô|x := ||ô||(E,IhImH(e,|hIM) ^ 2||ô|\, w^ere ||ô|\ = ||ô||(e;|M|EH(e,|h|E)■

< By Proposition 2, a derivation ô : E ^ E is a continuous on (E, || ■ ||E), in particular, ||ô|| := ||ô||e^e < œ.

Let x G E and ô(x) = 0. Let 0 < e < ||ô(x)||m and denote by px the spectral projection of the operator |ô(x)| corresponding to the segment [||ô(x)||m - e, ||ô(x)||m]• Using Gelfand-Naimark theorem, one can obtain that px = 0.

We have that 0 < (||ô(x)||M - e)px ^ |ô(x)|px. The n px G E and

|Ô(x)|px = (Px|ô(x)|2px)1/2 = (Pxô(x)* ô(x)px)1/2 = |ô(x)px|.

Since the norm || ■ is monotone, we obtain that

(||ô(x)||m - e)|px|E ^ ||ô(x)px||e = ||ô(xpx) - xô(px)||e ^ ||ô(xpx)||e + ||xô(px)||e < ||Ô(xpx )||e + ||x|m ||Ô(px )||e < ||ôWWxPx ||e + \|x\|м ||ô||||px ||e

< ||ô||||x\UHpxHe + ||x\m\|ô||||px||e = 2||ô||||x||m\\px\U,

that is

(||ô(x)\U - e)|px\E < 2||ô||||x\UHpxHe.

Dividing by ||px||E and using arbitrariness of e, we infer that

||ô(x)\u < 2||ô||||x\u.

Thus the operator ô is bounded with respect to the norm || ■ ||m ™ addition, 11ô|^ ^ 2||ô||. > Now we give a proof of Theorem 1.

< Proof of Theorem 1. Denote by E and E the closure of the ideal E with respect to the uniform and weak operator topology, respectively. Then E C E C E. It is clear that E is a C*-subalgebraJn M the derivation ô extends by continuity (see Proposition 3) up to a derivation ô : E ^ E, in addition ||ô||x = 11ô|^-

Since E is a wo-closed two-sided in M, it follows, by Proposition 1, that E = z-M for some central projection z in M. Then E is a W*-subalgebra with 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 such that S(x) = 5a(x) = [a,x] for all x G E and ||a||_# ^ = ^Halloo ^ IHI- >

Corollary 1 (cf. [6, Theorem 3.2]). Let (E, || ■ ||E) be a Banach ideal of compact operators in B(H) and 1 et 5: E ^ E be a derivation on E. Then there exists 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:

||a||M ^ ||5|| e^e-

M

and let (E, || ■ ||E) be a Ban a ch *-ideal in M. Then any derivation 5 on 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 ^-distributive then there exists a nonzero derivation on ideals in M with values in a commutative *-algebra C^(Q) © i ■ C^(Q), where Q is a Stone compactum corresponding to 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.

References

1. Sakai S. C*-Algebras and W*-Algebras, Berlin, Springer-Verlag, 1971.

2. Gohberg I., Krein M. Introduction to the Theory of Linear Nonself'adjoint Operators, Providence (R.I.), Amer. Math. Soc., 1969, Translat. of Math. Monogr., vol. 18.

3. Kalton N., Sukochev F. Symmetric Norms and Spaces of Operators. J. Reine Angew. Math., 2008, vol. 621, pp. 81-121.

4. Schatten R. Norm Ideals of Completely Continuous Operators. Second printing, Ergebnisse der Mathematik und ihrer Grenzgebiete, band 27, Berlin, Springer-Verlag, 1970, 98 p.

5. Simon B. Trace Ideals and Their Applications. Second edition, Math. Surveys and Monogr., vol. 120, Providence (R.I.): Amer. Math. Soc., 2005.

6. Ber A. F., Chilin V. I., Levitina G. B. and Sukochev F. A. Derivations with Values in Quasi-Normed Bimodules of Locally Measurable Operators. J. Math. Anal. Appl., 2013, vol. 397, no. 2, pp. 628-643. DOI: 10.101610/j.jmaa.2012.07.068.

7. Ber A. F., Chilin V. I. and Levitin a G. B. Derivations with Values in Quasi-Normed Bimodules of Locally Measurable Operators. Sib. Adv. Math., 2015, vol. 25, no. 3, pp. 169-178. DOI: 10.3103/S1055134415030025.

8. Strätilä S., Zsido L. Lectures on von Neumann Algebras, Bucharest, Editura Academiei, 1979.

9. Takesaki M. Theory of Operator Algebras I, Berlin etc., Springer-Verlag, 1979.

10. Bratelli O., Robinson D. W. Operator Algebras and Quantum Statistical Mechanics 1, N. Y., SpringerVerlag, 1979.

W*

pp. 147-150.

12. Chilin V. I., Levitin a G. B. Derivations on Ideals in Commutative AW *-Algebras. Sib. Adv. Math., 2014, vol. 24, no. 1, pp. 26-42. D01.10.3103/S1055134414010040.

13. Kusraev A. G. Automorphisms and Derivations on a Universally Complete Complex /-Algebra. Sib. Math. J., 2006, vol. 47, no. 1, pp. 77-85. DOI: 10.1007/sll202-006-0010-0.

Received March 21, 2018 Aleksey F. Ber

Institute of Mathematics of República of Uzbekistan, Mirzo Ulughbek Street, 81, Tashkent 100170, Uzbekistan E-mail: aberi960@mail .ru, Aleksey .BerOmicros .uz

Vladimir I. Chilin National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan E-mail: vladimirchiliagmail.com, chilin@ucd.uz

Fedor A. Sukochev

School of Mathematics and Statistics,

University of New South Wales,

Ms Marina Rambaldini, RC-3070, Sidney 2052, NSW, Australia E-mail: f.sukochev@unsw.edu.au

Владикавказский математический журнал 2018, Том 20, Выпуск 2, С. 23-28

ДИФФЕРЕНЦИРОВАНИЯ В БАНАХОВЫХ ♦-ИДЕАЛАХ АЛГЕБР ФОН НЕЙМАНА

Бер А. Ф., Чилин В. И., Сукочев Ф. А.

Аннотация. Известно, что любое дифференцирование 3 : М ^ М на алгебре фон Неймана М является внутренним, т. е. 3(х) := 3а(х) = [а, х] = ах — ха, х £ М, для некоторого а £ М. Если Н сепарабельное бесконечномерное гильбертово пространство и К(Н) теть С*-подалгебра компактных операторов в С*-алгебре В(Н) всех ограниченных линейных операторов, действующих в Н, то каждое дифференцирование 3 : К(Н) ^ К(Н) есть специальное дифференцирование, т. е. существует такой оператор а £ В(Н), что 3(х) = [х, а] для всех х е К(Н). В недавней работе А. Ф. Вера, В. И. Чилина, Г. В. Левитиной, Ф. А. Сукочева (.ШЛА. 2013) установлено, что каждое дифференцирование 3: Е ^ Е на любом банаховом симметричном идеале компактных операторов Е С К(Н) также является пространственным. Мы показываем, что аналогичный результат верен и для произвольных банаховых ^-идеалов в любой алгебре фон Неймана М. Более точно: Если М любая алгебра фон Неймана, Е банаховый *-идеал в М и 3: Е ^ Е есть дифференцирование на Е, то существует такой элемент а £ М, что 3(х) = [х, а] для всех х £ Е, т. е. 3 есть пространственное дифференцирование.

Ключевые слова: алгебра фон Неймана, банахов *-идеал, дифференцирование, пространственное дифференцирование.