innerm-ideals in banach algebras

Math. Ann. 291,205-223 (1991)

Mmbn �9 Springer-Verlag 1991

Inner M-ideals in Banach algebras

Wend Werner

Universitfit-GH-Paderborn, Fachbereich 17, Postfach 1621, W-4790 Paderborn, Federal Republic of Germany

Received March 15, 1991

1 Introduction

A subspace J of a Banach space X is called an M-ideal, iff for some subspace Jo of X*,

X* = J ' ~ 1 Jo,

a notion introduced and thoroughly investigated by Alfsen and Effros in [1]. The perhaps most important result of the present paper is the formal solution of a problem (5.2) that had been open for some time ([54, p. 347]): Characterize those Banach spaces for which the space of compact operators behaves with respect to one of its geometric aspects (the one of being an M-ideal) geometrically in the same way as this space does for Hilbert space. (The fact that the space of compact operators on Hilbert space form an M-ideal in the space of all operators, was already known by Dixmier in the 50's.) 5.2 has first been shown in the author's doctoral dissertation 159]. Meanwhile, there have been found some applications [46, 491 which seem to indicate that one is approaching a complete classification of the members in the class in question.

The spaces X for which K(X) is an M-ideal in L(X) have found certain interest in the literature. The reason for this lies partly in the approximation theoretical implications which this property has, 1"2, 24, 25, 34-37, 41, 45, 51, 52], partly in the possible applications in the theory of perturbations [17, 18], and, not in the least part, in the interplay this property has with the geometry of X (see e.g. [4, 6, 10, 15, 16, 23, 29, 30]). One of the most striking results in this direction is the surprising discovery of Cho and Johnson that for subspaces of CP, 1 <p < oo, the compact approximation property of X already implies that K(X) is an M-ideal in L(X) 1"15]. (In the meantime, their result was extended to the case of subspaces and quotients of c o in 1-57].) Prior to this is a result of Harmand and Lima stating that X must possess the compact metric approximation property, whenever K(X) is an M-ideal in L(X). The most restrictive consequences this property has for an arbitrary Banach space X have been found by Fabian, Godefroy, Saab and Li in [23, 29] and

206 w. Werner

[39], where this property is connected to certain resolutions of the identity operator and, for separable spaces, to complementability in superspaces ad- mitting an unconditional finite dimensional Schauder decomposition.

There are two ingredients in the proof of 5.2: First, at a certain point, we need a rather elaborate version of the principle of local reflexivity [7], which in its classical form has already played a central role in similar investigations [28]. The other idea involved is based on the following observation: When X=C(K), the Banach algebra of continuous functions on a compact K, then for each M-ideal J in C(K) there is an idempotent element p~ C(K)** such that J• Note that C(K)** is again a Banach algebra and so, multiplication with p - as well as the property of being idempotent - makes sense. An M-ideal of an arbitrary Banach algebra 9/which shows this behaviour- for a more precise definition one has to use one of the Arens products on 9/** - will here be called inner. The idea of trying to show that a great number of M-ideals in Banach algebras show this behaviour underlies one of the early, and still indispensable investigations of Smith and Ward [53, 54]. They could show that M-ideals in Banach algebras are always subalgebras and that all M-ideals are inner when C(K) is replaced by a commutative Banach algebra or a C*-algebra. Furthermore, for several Banach spaces X it is known that results of this type hold for the algebra L(X). So, work of Cho, Johnson, Flinn, Smith and, somewhat more implicitly, Behrends implies that the M-ideals in L(X) are inner when X is a (real or complex) C(K)-space, a {complex) uniformly smooth or uniformly convex space, or the complex space f l Important here will be the fact that K(X) can only be an inner M-ideal.

A byproduct of the present investigation is a somewhat more systematic and ex- plicit treatment of inner M-ideals. We decided to include some more abstract results (Sect. 3) than actually needed in order to prove 5.2, since with their aid the M-ideals of a number of new examples can be completely calculated. This will be done in Sect. 4, where we shall use a combination of our general results on inner M-ideals and the techniques developed by Smith and Ward .to show that for X being a function algebra, an arbitrary Ll-space, or a predual of one such space all M-ideals in the Banach algebra L(X) are inner. We also show that for an arbitrary function algebra 9/all M-ideals in L(9/) are given by subspaces of the form

~,o,= {TEL(9/)] li kmsup [IT*~kll =0},

where D runs through the family of p-sets for 9/. Similar representations hold in the case of all other Banach spaces treated in Sect. 4. An important cornerstone for our reasoning is a quantitative version of the observation that M-ideals are invarian! under the action of hermitian operators, the latter result being previously known [11, 20, 48, 50]. In particular, this shows that on any function algebra 9 /an operator with an "almost" real numerical range is always close to multiplication with a fixed element a e 92.

2 Notation and useful results

The unit ofa Banach space X is denoted by Bx. For all other unexplained notation of Banach space theory we refer the reader to the standard sources such as [43, 44] or [22]. We shall several times make use of a refinement of Goldstine's theorem, the principle of local reflexivity. This theorem was shown originally in [42]. We

Inner M-ideals in Banach algebras 207

shall apply a version due to Behrends [7]. Compared to the more classical variants, it offers the additional degree of freedom (iv).

2.1 Theorem. Let X be a Banach space, F_~X**, Gc_x*, H ~_L(X) finite dimensional subspaces and put F n : = lin {h**x** I h e H, x** ~ F} + F. Then for each ~ > 0 there is an operator T: FH-~X with

(i) Tlvox = I d (ii) For all g ~ G and each f e F we have g(Tf )=f(g) .

(iii) For all x e F n, Irxll < Txl[ <(1 +e)Ilxli. (iv) For each h e n , II(hT- Th**)lrll <el[hll.

In [7] condition (iii) is replaced by the seemingly weaker 11 T [I II T - l[r<r ) 11 < I + e which, however, implies the above condition after joining an at least one dimensional subspace of X to F. Banach algebras are generally denoted by letters such as 92, ~3 . . . . . The operators of left resp. right multiplication with a given element a are denoted by L, resp. R a. When this distinction is not necessary, i.e. if a commutes with all elements in 92 then we will also write Ma. Important in this note is the fact that the bidual 9.I** of a Banach algebra can be provided with at least two in general different multiplications. For the convenience of the reader, we include the definitions.

2.2 Definition. Let 92 be a Banach algebra. Let a, be92, f g e 9 2 * and F, G~92"*. We then put

f a(b): = f(ab) ,

F f (a ) := F( fa) ,

F G ( f ) : = F(Gf) ,

a f (b):= f(ba) ,

fF (a ) := F(af ) ,

F .G( f ) := G(fF).

The product FG is called the first, F.G. the second Arens multiplication.

In general, these two multiplications do not agree (see e.g. [1 9]). When they do coincide, 9.I is called Arens regular. To emphasize which multiplications is under consideration we shall write ~r resp. R~ to denote right or left multiplication with respect to the i-th Arens multiplication. In the following theorem we collect some results in connection with the Banach algebra 9.1"*. In the proofs of all of them nothing but Definition 2.2 is involved. Nevertheless, some of them are somewhat cumbersome to show.

2.3 Theorem. Let 92 be a Banach algebra. ~1) Whenever 3 is a left (right) ideal in 92, the same is true for 3 •177 in 9.I**

independent of the Arens multiplications under consideration. (ii) I f H : 9 2 ~ is a homomorphism, then H** : 92"*~-~B** also is one if 9.I**

and ~8"* are furnished with the same Arens product. Similarly, if H : 9 2 ~ is an antihomomorphism (i.e. H(ab)=H(b)H(a) for all a, be 92), then H** has the same property whenever 9.I** and ~** are provided with different Arens products.

(iii) For every subalgebra f8, both Arens products on ~** = ~_L• agree with the ones inherited from 9.1"*. In particular, ~•177 is a subalgebra of 92**.

(iv) For all a e 9.I and every F ~ 92"*, we have i,n(a)F=i~(a).F=L**(F) as well as Fi~(a) = F.i~(a) = R**(F).

(v) For f ixed F e 92"*, the mappings R~ and L 2 are w*-continuous.

208 W. Werner

Note that according to (iv), R~'2 as well as Lid 2 are w*-continuous for each a r 92. Let now 92 be a function algebra. Using the fact that 92 embeds isometrically as well as algebraically into a C(K)-space, it follows that 92** again is a function algebra and that both Arens products coincide. (All properties involved pass to subalgebras, see [21].) We will furthermore call a Banach algebra unital, if it contains a two sided unit which we usually denote by e. A left approximate unit for 92 is a net (p~) such that lim p~a = a for all a ~ 92. Right approximate units are defined

gg

in a similar manner. The connection between the concepts unit and approximate unit is given by the next theorem.

2.4 Theorem. Let 92 be a Banach algebra. (i) Suppose that 92 has a bounded right approximate unit (q~). Then each weak*-

cluster point of (q~) is a right unit for 92"* with respect to the first Arens product. The analogous result is oalid for left approximate units, when we substitute the first by the second Arens product.

(ii) I f 92** has a right unit q with respect to any of the Arens multiplications then there exists a right approximate unit (q,) in 92 with lim Itq~lL = Ilqll.

~t

Using 2.3(v) the proof of (i) is straightforward. It also can be found in [13], where a proof for (ii) is given as well. When 2.1 is used, a different argument is possible, which can be extracted from the proof of 3.5. Standard references for the following are [-12] and [14]. We denote by

S~= {tp ~ 92" I t p(1) = 1 = II~P II }

the state space of 92 and by V(a, ~) = {q)(a) Itp e S~) the numerical range ofa ~ 92. We call a hermitian iff V(a, 92)_~R. In the case of the Banach algebra 92 = L(X), the set Sf.(x ) as a whole is highly non accessible in general. However, concerning the set V(T,L(X)) one may restrict one's attention to the subset II(X)g_Sux ) which is defined by

n ( x ) : = {(x, x*) e Sx x sx, I x*(x) = 1}

and define the spatial numerical range of an operator T to be

v ( r ) : = {x*rxl(x*, x) ~ n(X)} .

In this way, one has access to the set V(T, L(X)) by virtue of V(T, L(X)) = ~ V(T). In particular, the notion of a hermitian operator may also be defined by saying that T is hermitian iff V(T)~R.

The concept of a multiplier algebra for Banach spaces has grown out of an attempt to isolate the geometric content of the algebraic definition. Note that both concepts in general do not match When they may be used simultaneously. So here, the Banach algebra Mult(X) is defined for arbitrary Banach spaces X and consists of all those operators T for which each pse x Bx , is an eigenvector of T* with eigenvalue, say, aT(P). It is not difficult to see that Mult(X) is (algebraically isomorphic to) a function algebra on some compact Hausdorff space K (take e.g. the quotient of fl(exBx.) which is obtained by gluing those points together that Mult(X) cannot separate). Those T s M u l t ( X ) that possess a natural adjoint in Mult(X), i.e. for which there exists a T* e Mult(X) such that for all peexBx, a r,(P) = a - ~ (the complex conjugate of ar(p)], are said to belong to the centralizer,


denoted by Z(X). Clearly, when X is a real space, both concepts coincide. Let Z0,1(X) denote the convex set of all T ~ Z(X) with aT(P)e [0, 1] for each p e exBx..

2.5 Theorem. Let X denote a Banach space. O) An operator T belongs to Mult(X)/ff

II Tx--xoll < r

whenever IlOx- xoll <=r for all scalars 0 with lOP < II TIP. (ii) An operator T belongs to Zo, I(X ) iff for all x l , 2 e X

IF Zxl + ( I d - T)x2 II =<max~llxl II, IIx211} �9

A proof of part (i) of the above theorem can be found in [3], a short proof of(ii) is in [61 ], where a slightly more general result is proved. A projection P that belongs to Z(X) is called an M-projection. Amongst the projections on X it may be characterized by the condition [Ixll =max{IrPxPI, [[(Id-P)x[]} for all x eX. The range of P is termed M-summand. Dually, there is the notion of L-projections and L- surnmands, the former being defined by Jl x f[ = I PPxll + II(Id- P)x IF for all x E X. The Cunningham algebra CI(X) of X is the norm closure of the linear spane of the set of L-projections on X. The complementary subspace of an M-summand (L-summand) Y that is uniquely determined by the complementary M-projection (L-projection) I d - P is denoted by Y• M- and L-projections enjoy a certain uniqueness condition in as much as there are no further contractive projections with the same range or kernel, respectively.

2.6 Theorem [31]. Suppose t.hat Q is a contractive projection onto an M-summand J. Then Q is the M-projection with range J. I f Q is a contractive projection the kernel of which is an L-summand J, then Q is the L-projection with kernel J.

The following theorem describes the behaviour of Z(X) and Mult(X) for dual spaces:

2.7 Theorem. For dual spaces X the algebras Z(X) and Mult(X) are dual spaces as n

well and Z(X*) is the closure of the set containing the elements ~, 2iP i where the Pi i = 1

are disjoint M-projections. In addition, an operator T ~ L(X*) belongs to Z(X*) if and only if there is an 7", ~ CI(X) with T* = T.

A complete proof for this theorem for the algebra Z(X) can be found in [9]. Let us take the opportunity to include (the sketch of) a short proof for this slightly more general result:

As is well known, L(X)= (X. ~ X ) * whenever X* = X. What suffices to show (i), is that Mult(X) is a w*-closed subspace in this duality. But this is an easy consequence of 2.5, giving the result for Z(X) (real scalars) and Mult (X) (complex scalars). We leave the case of Z(X) for complex scalars to the reader. Since, in any case, Z(X) ~ C(K) for some extremally disconnected space K, the first part follows from the Stone-WeierstraB Theorem, and for the second statement one uses the fact that M-projections are w*-continuous on dual spaces, see e.g. [9].

2.8 Definition. A subspace J ~ X is called an M-ideal iffJ • is an L-summand of X*.

Note that an M-ideal is not necessarily an M-summand [Look at the case where X = C(K).] A characterization of M-ideals that does not explicitly resort to dual Spaces has been found by Alfsen, Effros, and Lima (see [3,40]):

210 w. Werner

2.9 Theorem. Let J be a subspace of X. The following are equivalent: (i) J is an M-ideal.

(ii) For arbitrary j t, 2, 3 ~ S j, x e Bx and ~ > 0 there is an element j ~ J such that

I l j~+x- j l [< l+e Vi=1,2 ,3 .

The concepts "M-ideal" and "Muir(X)" are loosely connected by

2.10 Proposition. Let 3 be an M-ideal of a subalgebra ~ of Mult(X). Then

X z : = - l ~ { j x l j ~ 3, x ~ X }

is an M-ideal and Xz = {ix IJ E 3, x ~ X } .

The proof of this theorem is known for the case that ~ is a subalgebra of Z(X), !-3]. We will obtain 2.10 in full generality as a corollary to 3.8.

We end up this section with an example: Let 9.1 be a function algebra on some compact space K. Then Mult(9.i) consists of all the multiplication operators Ma where a runs through the elements of 9.L Furthermore, the M-ideals are the spaces of the form 3o = { f ~ 9.I I ~D = 0), where D runs through the p-sets of 9/. Recall that D is called a peak set for ~ iff there is some a e 9~ with I[a II = 1 such that aid = 1 and

lalr\D I < 1, and that D is called a p-set iffD = ~ D~ for peak sets D,. Particularly, all

subspaces of the form 3k = ( f e 93 1 f (k) = 0}, where k e chg.I, the Choquet bound- ary of~I are M-ideals. All this is known [-4, 33]. Whereas the proof for Mult(gJ) is easy, the isolation of the M-ideals needs some skill. We shall present a proof of this in the next section which makes use of the techniques to be developed there and which is different from the one given in i-33].

We shall need the following characterization of the 2-approximation property. Since this is well known, we omit the proof.

2.11 Proposition. Let X be a Banach space and denote by A(X) the algebra of approximable operators on X. Then the following are equivalent.

(i) X has the ~ - A P . (ii) A(X) contains a left 2-bounded approximate unit.

(ii) There is a net (T~) in A(X) with II Z~ll < 2 converging to the identity in the weak operator topology.

3 Inner M-ideals

3.1 Definition. Let 9.1 be a Banach algebra. An element T~ Mult(g2) is called left (right) inner, if there is an element t~9.I such that for all a t 9J the relation T(a)=Lt(a)=ta and, respectively, T(a)=Rt(a)=at holds. The subalgebra of 91 consisting of all those elements of 9.I that give rise to a left (right) inner element in Mult(9.I) is denoted by

Multlnn(9.t) (Mult~,,(9~)).

Analogously we define Zl.n(9.1) and 2q'i,n(9.I).

Note that Mult~n.(9.I) and Mult~.,(9.I) are subalgebras of 9.1, whereas Mult(91) is a subalgebra of L(9.I). The fact that there are two different reasonable multiplications on 9.I** makes the corresponding definition for 9.I** somewhat unpleasant.

Inner M-ideals in Banach algebras 21 1

We denote by

Multlh~(92** ) (Mult~(92**)) --inn,--Z l'~ t91"'/, (Zinn(~ [ r ' i * * ))

the respective subalgebra of 92** furnished with the i-th Arens product.

3.2 Definition. An M-ideal ~g92 is called right (left) inner, if the M-projection P:gA**--+~ •177 is left (right) inner with respect to the second (first) Arens multiplication on 92**.

A subspace `3 is called a two-sided inner M-ideal iff there is p ~ 92** such that L 2 as well as R~ define M-projections 92"*~`3•177 We call an M-ideal inner if it is either left or right inner.

The interest in this type of M-ideals has been engendered by the fact that for instance in C*-algebras and commutative Banach algebras, all M-ideals are known to be of this type [53, 54]. [This is also true for the algebra L(X) for a number of Banach spaces, see 4.1.] We think that some remarks are in order:

1. The necessity for using different multiplications in the definition of an inner M-ideal is due to the fact that M-projections in dual spaces are always w*- continuous (2.7), a property that is shared only by one of the respective Arens multiplications (2.3). In fact, when e.g. also a**~-+a**.p defines an M-projection on 9.I**, then by its w*-continuity, 2.3(iv) and (v), we must have for all a** e 92"*, being approximated in the weak*-topology by (x~) in B~,

a**.p=w*- lim x~.p=w*- lim x~p=a**p. gt gt

we find that Zinn(92 )~ 7"/, 1(~[**~ 2. With the same reasoning as above, z 2 ** _-~n~,~ , as well as _~,~.z~'~tgt**~)T'2fgl**l,=_~,~ ,. It is, for the time being, not clear whether a similar

Multi~(92 ) and Multi~(92 ) (see Problem I in relation holds for the algebras ~.i ** , i ** the last section). Nevertheless, a glimpse at the proof of 3.7 should convince the reader that ~. 2 ** Multi,,(92 ) and Mult [,;~(92"*) are the more pleasant objects to work with.

3. Let ~ be a two-sided inner M-ideal of 92 and p be the element in 92** that gives rise to the M-projections R~ and L 2 onto `3• It follows by 2.6 that p.a = ap for all a e 92** which is "almost central" behaviour.

4. Note further that, at least when 9g** is unital, every M-ideal .3 being a left as well as a right inner M-ideal is a two-sided inner M-ideal. This follows, since according to 2.6, for every pair of M-projections L 2 and R~ with `3..L as a common range we have p = L2(e)= R~(e)= ~.

5. Consider a Banach space X containing a non-trivial M-ideal J. Converting X into a Banach algebra by putting xy = 0 for any x, y ~ X, shows that there is an abundance of M-ideals, which are ideals without being inner.

Suppose that you want to find out, whether a given M-ideal `3 in a unital Banach algebra 92 is inner. A natural way to decide this is to look at the element P(e), where P denotes the M-projection 92'*-}`3•177 You then would have to check, whether multiplication with this element from an appropriate side and with the help of one of the Arens products gives rise to an M-projection. This motivates the following result, which collects the known properties of the mapping

A~:Z(92)--,92, A~(T):= T(e).

When this is unlikely to cause any confusion, we occasionally will omit the index and simply write A. Observe that we may restrict our attention to Z(92) even when

212 w. Werner

dealing with M-ideals. This is due to the fact that M-ideals in 91 correspond to certain M-projections in Z(9~**). In fact, a satisfactory characterization of this correspondence is the subject of part (iii) in the following.

3.3 Theorem. Let 91 be a unital Banach algebra and denote by d :Z(92)~92 the mapping T~--+ T(e).

(i) The range of ZR(92) under A is contained in the set of hermitian elements o f 92.

(ii) A(P) is a projection for each M-projection P ~ Z(91). Furthermore, in the bidual 92** of a Banach algebra 92, the M-projections P arising from M-ideals ~ in 92 via P(92"*)= ~• are characterized by the condition that the function A(P)Is~ is lower semicontinuous.

(iii) A is multiplicative. (iv) A is an isometry. (v) A preserves spectral radii.

Proof. (i) This follows from the fact that all operators in ZR(92) are hermitian (this follows by a Krein-Milman type argument) and that for each (a, a*) e//(92) the pair (e, a| belongs to H(L(92)).

(ii) 1-54, 26]. (iii) Since by bitransposition Z(92) is a subalgebra of Z(92"*) [5] and since the

map A for 92 is just the restriction of the mapping A for 9.f** to Z(92), it is sufficient to prove the claim for dual Banach algebras only. In this case, however, Z(92) is

generated by elements ofthe form ~ ~PiwheretheP~denoteM-projectionswith i - - 1

PiPj=PjP~=O (see 2.7.). Put A(Pi)=z ~. Now, it is clearly enough to show that P1P2=P2Px =0 implies zlz2=z2z ~ =0. But by assumption, P~ +P2 is an M- projection and so, according to (ii), we have (zl + z2) 2= zi + z2. This leads to ztz: +Z2Zx =0 and

( - ZlZ2) 2 = z~z2zlz2 = - zl z~z2z~ = - z~z2.

In the same way it follows that -z2z~ is idempotent. But -ZxZ 2 = z2zl and hence,

- z ~z2 = ( - z ~z2) 2 = ( z ~ z ~ ) 2 = _ z 2 z ~ .

This implies z lz2 = z2zl =0. (iv) This is a special case of 1-62, Satz 7.18], which you may, however, prove

directly proceeding similarly to the above, see [32].

(v) Using the formula 0(a)= lim"~/tia"ll, this follows from (iii) and (iv). [] /1

In the sequel, we shall often make use of the fact that A is injective, a property that follows from (iv), which, however, can be proved in a more direct way (see [58]).

3.4 Corollary. Suppose that the Banach algebra 92 can be written as 92=II~e@9.[0 where Q(a)< Ila[[ for every a e 9.1o\{0 }. Then Z(92) is trivial.

Proof. An immediate consequence of the fact that for all Banach spaces X, any Z~ Z(X), [I T II = 0(T) and of (iv) in the above.

A consequence of this corollary is that, under the above circumstances, 92 has the so called strong Banach-Stone property. See 1,3] for details. The next theorem gives the characterization of inner M-ideals to be used in what follows.


3.5 Theorem. Let 9.1 be a unital Banach algebra and 3 ~_93 a subspace. Then the following are equivalent:

(i) 3 is a left inner M-ideal. (ii) 3 is a left ideal and an M-ideal containing a right 1-approximate unit for 3.

(iii) 3 is a left ideal and contains a right approximate unit (p~) that satisfies

lim sup I[sp~+t(e-p~)ll <1 Vs, t~B~. #l

This equivalence remains true after exchanging "left" and "right" accordingly. If, in addition, 3 is a two-sided inner M-ideal, then the net (p~) in (iii) can be chosen as a two-sided approximate unit and to satisfy

lim sup II sp~ + t ( e - p~)ll _-< 1 ~t

and

l imsup JIp~s +(e-p~)t[] < 1 ~t

for all s, t ~ B~ simultaneously.

Proof. (iii)=:-(ii) The condition posed on (p~) implies lim sup [IP~[I < 1 and so, by 2.4, we may suppose [[P~II < 1 for all ~. It has to be checked that 3 indeed is an M-ideal: We shall apply 2.9. With this goal in mind, suppose that f l .2.a~Bj,

e XeBx and e > 0 are arbitrarily given. Select an index ~ such that [IJiP~-Ji[[ < ~ i

e =1,2,3 as well as Iljip~+x(e-p~)l[ < 1 + ~ for i= 1,2,3, and put j=xp~. Then J e 3 ,

IlJ~+ x - i l l < liJ~p=-j~ll + IIj~p~+x(e-p~)[I < l + e ,

and 3 must be an M-ideal. (ii)=~(i) By 2.4, 3 "• contains a right unit w.r.t, the first Arens multiplication

which we denote by p. Since 3 "• is a left ideal (2.3), the mapping R~ is a projection onto this space. But M-projections are unique among norm one projections with equal range (2.6), and so 3 is a right inner M-ideal.

(i)=>(iii) Suppose that R~ : 9.I** ~ 3 l• is an M-projection for some p e 9.(**. Let F__qg.I**, G_~92" and H___~ run through the respective set of finite dimensional subspaces and order the set

B= {(F, G,H,e) IFC= 9.I**, G~_9~*,H~_9.1,e>O}

by

(F, G,H,~)~((F, d , ~ , ~ . ~ FC=g, G~_d, HC___~I,~> ~.

Identify the space H ~ 9,1 with the operator space {L h [ h ~ H} and choose for every /~eB an operator T~ that satisfies for conditions (i)-(iv) listed in 2.1 for the subspaces and the number e determined by ft. Let p~: = Tp(p). [Note that Tp(a**) is eventually defined for all a**eg.l**.] We have by condition (iv) and since L** _- Li,~x) for either of the Arens multiplications (2.3 (iii)) that for every a e

lim liap ~ T~(ap~ = lim HLoT~(p~ - T~L*~*(p~ =0 , r r

214 W. Werner

By property (i) and the above

lim [[ap~ tim Ilap ~ T~(ap~ = 0

for all a e 91, and accordingly, pO is a right approximate unit. Similarly, we conclude that for s, t ~ B~ by property (iii) of 2.1

lim sup II sp~ + t (e- p~)II = lim sup II Tdsp) + T~(t)- T~(tp) ll

= lirn sup II Ta(sP+ t(e-p))l[

< 1 .

To finish the proof, we have to force the p~ into ~. To this end, we take a net (qa) from Bj with a(91"*, 91")- lim q~ = p. Note that we may use the same index set.

Since by (ii) of 2.1, w * - l i m p ~ = p , and since p~-qa~91, we must have #

w - lim (p~- qp) = 0. In passing to an appropriate II �9 II-convergent net of convex #

combinations we obtain /%

It" [1 - l im Z t , . , (p~, , . - qa,, =) = O, i = l

together with l immax I[xp~,.=-xl1:0. We now put p~= fashion, �9 i ~ N=

lim [[xp,-xl] < lim xp~- ~, ti,.xp~,,. + ti,~xp~,,.--x i = 1 i = 1

< lim max I[xp~,.,-xl[ ~ N ~

= 0 .

t,.~qa,,. In this i = 1

Similarly, for all s, t e B~, lim II sp~ + t(e - p~)ll ~ 1, and (iii) follows.

We are still left with the case of a two-sided inner M-ideal ~. Recall (fourth Multihn(9/ )c~Multinn(91 )with remark after Definition 3.2) that there is pe ~2 ** r, 1 **

/~p91 = Rpg/ = 3 . TO refine the properties of (p~) in this situation, we use the index set

B*={(F,G, BL, HR, e)IpeF~_91**, G~_91", HL.RC_91, e>0}

ordered in the now obvious way, and identify the spaces HL ~_ 9i and HR_-C 91 with the operator spaces {Lh I h e HL} and {Rh I h ~ HR}. Keeping these minor changes in mind, we may proceed as above. []

As the ease of the disk algebra defined on the full unit disk already shows, it is not to be expected that for subalgebras ~ and inner M-ideals ~ the space ~ c ~ again is an M-ideal, much less an inner one. [To be more concrete, let J = { f e C{D)[ f (0)=0}. Since 5o ~chA(D), JnA(D) is impossibly an M-ideal in A(D).] However, we have

Inner M-ideals in Banaeh algebras 215

3.6 Proposition. Suppose that 93 is a unital Banach algebra and let ~ be a subalgebra of 9.l with e e ~B. I f pe ~• is idempotent, it gives rise to a right inner M-ideal 3 in 93 by virtue of ~**p = 3 • if and only if ~**p = 3~ ~- for some M-ideal ~o in ~ . I f one of these conditions is satisfied then 3o = 3 c ~ , and 3 is nontrivial if and only if 3 r ~ is. In this case, one can find a left approximate unit (p~) for 3 in Ba, ~ with the property that, for all s, t ~ B~,

lira sup [Ip~s + ( e - p~)tll 5 l .

A similar statement holds for left and two-sided inner M-ideals.

Proof. Plainly, since p e ~_L~, we have for all ~v ~ S~ and any Q ~ R that POP) < Q iff P0Pla) -<- g which leads to {~p ~ S~ I POP) < Q} = r-X {~p ~ S~ [ p(~p) < Q}. Here r: S ~ S ~ denotes the restriction map. An examination of this equation quickly reveals that {~p c S~{ p(tp)-< e} is w*-closed if and only if the set {~p e S~I p(q~) < ~} has the same property, which in turn implies that p as a function on S~ is lower semicontinuous iff p shows the same behaviour when considered as a function on SB. Hence, according to 3.3(ii), 9.1**p=~ l i for some M-ideal 3 of 93 is equivalent to @•177 • for some M-ideal 3o of ~B. But then 3 and 3o are connected by 30 = {b c ~3 1 bp = b} = 3 ~ 3 . Next, since e ~ ~, we conclude that 3 4:93 for an ideal 3 of 9A is equivalent to 3n~4=@. Since 3 = {0} iff p = 0 we have 3c'~3 = {0} iff 3 = {0}. This proves that 3 is not trivial iff 3o is not. Let us finally construct the required net (p,): To this aim, start with (p o) in Bz as delivered by 3.5, and select furthermore q~ ~ Bn~z converging also to p in the a(93"*, 93*)-topology. The claim then follows by an application of the same blocking technique as used in the proof of 3.5. []

In what follows, we will study particular examples of inner M-ideals. We start this enterprise in this section with the description of a rather general situation in which inner M-ideals appear quite naturally. We first need a lemma.

3.7 Lemma. For every Banach algebra 93,

1 • 1 2 ** Multi~(93) =CMult~(gJ ).

This statement persists when "left" is changed to "right" and "2" to "1".

Proof. We will apply 2.5. To this end, choose f** e Mult[,~(93) •177 and let a**, b** e 93** with II a** + 2b** II <- r for all 2 with IAI ~ II f** II. Use the principle of local reflexivity (2.1) to find nets (a~) and (b~) with Ila~+&b~ll < r+e= for all I,~l

]]f**[I, (~) tending to zero, w * - lim a~=a as well as w * - lim b~=b. If (fp) at ~l

denotes a net in Mult~(93) with the property that Ilf~Jl<llf**l[ and w*- lim fp --- f**, then, by the w*-continuity of the mapping Rb, (2.3), we have that

# tla~+f**b~[[ < lim sup Ila~+fBb~l[ <r+e~ for all ~. By the w*-continuity of the

# naap L~,, (2.3), Ila**+f**.bJl< lira sup tJa,+f**.b, tl ~r , and we are done. The

Proof for the "r, 1" case is similar.

3~ Proposition. Let 93 be a unital Banach algebra and ~ be a subalgebra of Mutt~(93) such that e ~ ~B. Suppose further that 3 is an M-ideal in ~ .

216 W. Werner

(i) The space 92a: = li--n {jao IJ e ~, ao e 92} is a left inner M-ideal, and there is a left approximate unit (p~) in B 3 for 923 with l imsup IIpats+(e-p~)tll <-1 for all s, t r B~.

(ii) We have 923= f a e 9.I]fim p~,a=a t as well as ~=~Bc~92~. Moreover, 923 is k I - - . /

trivial if and only if ~ has this property.

Also here we are in a position where the cases for"left" and, respectively, "right" can be treated in an completely analogous way. For some more concrete examples for this type of subspaces see the next section, [8, 58] or [6~].

Proof. (i) Since ~B is itself a function algebra, it is not very difficult to see that ~-L=p.~B-t• for some p ~ 5 -L• We have p e Mulfin,(92) • and so, by the above lemma, ~ 2 ** peMulti~n(~ ). Since ~ is unital and 3-L• -L we have p~9.I~ • Furthermore, pjao =jao for all j e ~ and each ao ~ 92 and hence, pa = a for all a ~ 923. By weak*-continuity of the second Arens multiplication, we have for each FE92~ • with F = w * - l i m f ~ for some net fate9.I z that p . F = w * - l i m p f ~ = F .

at at

This shows that p is a left unit for 92~• with respect to the second Arens product. Since 92~• is a right ideal, it follows that 923 must be a right inner M-ideal. The net (p,) is obtained just as in the proof of 3.5 with the aid of a net (q,) in Bz converging to p in the ~r(92'*, 92*)-topology.

(ii) The first equation is immediate from (i), and the second is clear by virtue of

~ = { b ~ l l i m p~b=b} = ~ .

Finally, ~ = ~ if and only if p = e which in turn is equivalent to 923 = 92. Similarly, 3 = {0} is the same as p = 0 and 923 = {0}. []

Proof of 2.10. Put 9A = L(X) in the above theorem.Then ~B _~ Multl,n(92), and, by the above, we find a net (P~) in B o with

lira sup IIP, T1 +(Id-Pat)T2ll < 1

for all I'1,2 e Bt.(x). This easily implies that

lira sup I[P:xl + ( I d - Pat)x2 I I < 1 ~t

for all xl, 2 e Bx. Similarly, for all x ~ Xa, Patx converges to x in norm, and since the range of each P , is contained in Xs, one can proceed exactly as in the proof of 3.5 to settle the first par t of the claim. The assertion about the form of X 3 follows from [38, Theorem9.2]. []

By now, it is unclear whether one of the above M-ideals happens to be non trivial whenever Mult(X) has this property. It can be shown that X** always contains a non trivial M-ideal as soon as Mult(X) is known to be non trivial [62, Korollar 4.1 5].

3.9 Corollary. ([33]). Let 92 be a function algebra on a compact space K. A subspace is an M-ideal of 9.I i f and only if 3 is the annihilator of a p-set of 92.

Proof We shall use the fact [27, Theorem 12.7] that a closed subset D of K is a p-set iff for all/~ ~ M(K), # ~ ~I l implies/~w e 92• Suppose that D has this property.


Then Po(/~): = #1o leads to a well defined L-projection P on 92*= C(K)/91 • with (P91") = 3~. Conversely, let 3 be an M-ideal of 91. MultI,,(C(K))= C(K) permits us to apply 3.5, and there must be a closed set D in K with JD = C(K)z and ~ = Jon91. By the same result, we obtain an approximate unit (p~) for the M-ideal Jo of C(K) contained in Baoc~91. It is easy to check that (Po) can be characterized by the condition

Ve>0VU open ~D3~ oeAVct>otol(e-p~)lr\vl<e,

which is in light of [27, 1 2.2 Lemma] all we have to know in order to show that D is a p-set for 92. []

4 M-ideals in spaces of bounded operators

The results of Sect. 3 can be used to decide for which classes of Banach spaces all M-ideals of the Banach algebra L(X) are inner. The following theorem lists all the cases in which this is known to be true up to now.

4.1 Theorem (I-8,16,26]). Suppose that X is a Banach space over the complex numbers. Then all M-ideals in L(X) are inner in case - X is uniformly smooth or uniformly convex - X equals Co L where L is locally compact and countably paracompact - X equals s

I f the scalars are real, all M-ideals are inner in L(C(K)).

In this section, we shall add the cases that X is a function algebra, that X = LI(#), or that X is the predual of one such space. With the exception of [8-], where the author uses entirely different methods, all results mentioned above use the approach which goes back to [55] and is based on 3.3(ii):

4.2 Theorem. Let fi >= 0 and denote by R(6) the rectangle in the complex plane with vertices at the points -6+_i6 and 1+6+i6. I f ~ is an M-ideal in 92 and P: 9t** ~ ~ •177 is the M-projec tion that pertains to ~ then there is a net ( P~) in B~ such that w * - lim P~=P(e) as well as a net ~ in • with lim e~=0 and V(T~)~_R(e~).

Ot

Sometimes, operators with the property that V(T) is "almost real" in the above sense are perturbations of elements in Mult(X) or CI(X). Our tool to detect such a phenomenon is given by the next proposition.

4.3 Proposition. Suppose that T ~ L(X) with V( T) c= R( 6). I f J c= X is an M-ideal then d(Tx, J)<=61le[I for all x e J .

Proof. Let x e Sj and x* e Sj be arbitrary and pick an element p eJ~ • the complementary//-summand of jx• such that (p, x*)e II(X*). Then we have for all 7 e F that yx*(x+~p)= 1 as well as llx+TPl[ = 1 and so, (x+~p, 7x*)eII(X*). Using 2.5, it follows that for all ~ e F

~x* Tx + x* T**p ~ V(T*) c= V(T)_~ R(6).

But this is only possible when [x*Tx[<=6. It follows that d(Tx, J) ~'sup{lx*Txllx*~S~} ~llxll. []

In the case 6 = 0, this result has been known (and refound several times, see [1 1, 20, 48, 50-]).

218 W. Werner

4.4 Lemma. Let 92 be a function algebra and e > O. I f T ~ L(92) has the property that V(T)~_R(e) then there is an a~92 such that II T-Mai l < 2~.

Proof. Put a : = T(1) and let k~ch92. Then {k} is a p-set [56, Theorem 7.18], and ~ :=ker t~k is an M-ideal (by 3.9). Since for any xe92 ~:=~k(x)e--x is in ~k, we have by 4.3

II ~ II > d(Tx, 3k) = I~kCT~)l = 16k(ax-- Tx)l.

Since I1~11 <211xll, this yields l iax- Tx]l = sup [Sk(aX-- Tx)l_-<2~llxll. [] k e c h g J

4.5 Theorem. Let 92.1 be a function algebra on some compact Hausdorff space K. Then every M-ideal in L(92) is inner. Furthermore, the M-ideals of L(9.1) are precisely the subspace of the form

L(gor)~ = lin {M~ T I T ~ L(9.I), a e 5) = { T e L(92) limsup 11T*•k][ = 0},

where ~ is an M-ideal of 92 and D is the p-set corresponding to ~3.

Proof. That all subspaces of the above form are in fact M-ideals was shown in 3.9. We are thus left with showing that all M-ideals do look this way. So, let ~ be an M-ideal in L(92) and denote by P its defining M-projection P:/_.(92)** ~ • 1 7 7 Start with a net T~ converging in the weak*-topology to P(e) and with the property that V(T~) c= R(e~) for some net (~) of positive numbers tending to zero. By 4.4, we may suppose without loss of generality that T~ = M,~ for some a, ~ 92. But then it follows that P(e) = w* - lim Ma~ ~ Multl,~L(92) •177 whence by 3.7, P(e) ~ Multi,.L(92~' 2 **). It

follows that P(e) is an M-projection. By injectivity of the map A (3.3), we must have P = L~,~9, and so, 3 is inner. Since P(e) ~ 92• when 9.I is identified with a subalgebra of Mult[.~L(92), we may apply 3.6 to see that P(e) gives rise t O an inner M-ideal .3 in

as well. Next, observe that we are in a situation treated by 3.8. Consequently,

3 = L(92)j =1~ {MoT I T e L(92), a ~ 3} ,

since the inner projection pertaining to/-'(92)a is also P(e). We are thus left with showing that

L(92)~= {T~L(92)l imsup IIT*6kll=0}.

To this end, we observe that we have for an operator of the form M~T withje3 ,

lim sup II(MjT)*~kll = lim sup II/(k)T*~kll -<- II TII lim sup Ij(k)l = 0 . k"*D k ~ D k--*D

By 3.8, we find L(9.1)~_~ {T~L(92)limsupk_.D IIT*6kll=0}"

On the other hand, for any Te L(92) with lim sup I[ T*fik I1 = 0 we may for all e > 0 k ~ D

select an open neighbourhood U o f / ) such that IIT*6kll <e for all k~ U. As a e

consequence of [27, 12.21 there is ~ p e ~ such that I(e-u I < ~ and hence, regarding elements of L(92) as vector valued functions on K,

l I M i T - TII =max{ IIM,-~,T,t~I[, IIM,-~T,~wtl} <e .

It now follows that lira M ~ T = T and TeL(92)~. [] ~t


We next treat the case of the M-ideals in the space L(X) where X denotes a predual of a function algebra.

4.6 Theorem. Suppose that X is a preduai of a function algebra 9.[ on some compact space K.

O) I f 3 is an M-ideal in L(X) then it is right inner and, furthermore, there is a p-set D s K for 9.I such that

3 = 3 ' ~ = {TeL(X)limsupk~o I[T**fk[l=0}.

(ii) In the special case that X=LI(g), for each closed subset D of K, where C(K) = D(I~)*, 3 ~m is an M-ideal.

Proof Denote by Adj the antihomomorphism that maps an operator T~ L(X) to T*. Since Adj** is an antihomomorphism from L(X)** equipped with the first Arens product to L(~I)** when the latter is furnished with the second Arens product (see 2.3), we may identify L(X)** with a subalgebra of L(gA)**. Let P:L(X)**~3 •177 be the M-projection given by 3- As in the proof of 4.4, we start with a net (T~) in Bux ) converging to P(e) in the w*-topology and satisfying V(T,) c__. B(~,) for some net (~,) of real numbers converging to zero. We clearly have that (T~) converges also in the a(gA**, oA*)-topology to P(e). Moreover, by 4.3, we may disturb T~* slightly in norm to obtain a net M, , ~ ~gA) still converging to P(e). Following the lines of 4.4's proof, we see that P(e)eMultlh~(L(gA)**) which amounts to P(e)~ Mult~ih~(L(X)**). Hence, 3 must be right inner. Applying 3.6, we get an M-ideal 31 in L(9.I) with 31c~L(X)=3, and an application of 4.4 now yields (i).

To prove (ii), let X = LI(#) and choose a closed subset D of K. We must show that

{TeL(L1(,)) limsup [[T**rk[ l =0} = ~,o,

is an M-ideal. For this purpose, let $1,2, a ~ Bj(o~ and T e BLtLq,) ~ be given. Pick an open neighbourhood U of D such that for all teUllS**rti[<e as well as a continuous function ~p with 0__< lp__< 1, ~Plo = 0 and tpK~v = 1. Putting S = MhT we have S ~to~ because M h is w*-continuous (2.7). Since, in addition,

[IS/+ T-S[[ = max {[[(Si+(1 - h)T)lv[[, [[(Si+ (1 - h)T)lK\V[] } __< 1 +e .

the claim follows from 2.9. []

We do not know how the sets appearing in part (i) of the above theorem may be characterized. Note that 4.6 applies in particular to the space IA/H ~. Here, it can be shown that all p-sets D in H ~~ =(L~/H~) * give rise to M-ideals in L(L1/H~). Let us COme to the last result in this section. Also here, the nature of the closed sets D iavolved is somewhat obscure.

4.7 Theorem. I f X is a (complex) Ll-predual then all M-ideals of L(X) are left inner and possess the form

3=L(X)n3 ("~, ~vhere D denotes a closed subset in the compact space K with X** = C(K).

4 The proofof this theorem can be given in analogy to the first part of the proof of .6. The only ingredient we should add is

220 W. Werner

4.8 Lemma. I f an operator T on LI(#) has the property that x*TxeR(e) for all (x*, x)~ H(LI(p)). Then there is an operator S ~ C l(Ll(lO) such that [I T -S l l <2~.

Proof. By 4.4 we can find an operator S* which is contained in Z(LI(#) *) such that IIT*-S*H _<_2e. But S* is w*-continuous (2.7), hence the result. []

5 K(X) as an M-ideal in L(X)

In this final section we turn our interest to the space of compact operators on X. It will become clear, that this space must be inner once it is known that it is an M-ideal. This will permit an application of 3.5 in order to give a characterization for Banach space X for which K(X) is an M-ideal in L(X). To get started with, let us recall the following interesting consequences that the property "K(X) is an M-ideal of L{X)" has, which was observed by Harmand and Lima [31]:

5.1 Proposition. I f K(X) is an M-ideal in L(X) then there is a net (T~) in Br~x) such that (T~) converges to Idx and (T~*) to Id x. in the respective strong operator topologies. A similar result holds for A(X).

Proof. Let P* :L(X)**->K(X) l i be an M-projection and choose a net (T~) in Bgtx) converging in the w*-topology to P*(Id). Then, for all ~p e K(X) l•

lim ~p(T~- Id) = lim Ip(T~)- P~(Id) = lim ~p(T~- POd)) = 0.

Since the functionals x**| 6 L(X)* are normed by K(X), they must belong to K(X) • [this follows quickly from the fact that K(X) is an M-ideal] and so, lim (x** | T*) = (x**| Idx.> which in light of 2.11 settles our claim. The

proof is similar in the case of A(X). []

In the above reasoning we followed the ideas of the proof the referee gave in order to prove 5.1. of [31], which is slightly stronger than the above proposition and which, however, will also follow from the next theorem.

5.2 Theorem. Let X be a Banach space. (i) I f the space K(X) is an M-ideal in L(X) then it is a two-sided inner M-ideal,

and, in addition, there is a net T~ ~ Bgtx ) such that for all x e X and x*~ X*

lim (x*| T~) = (x*| Id) f l

which has the property that for all S, T e Bux )

lira sup I[ T~S + ( I d - T~)TIt -_< 1 ~t

and

lira sup I[ S T~ + T ( Id - T~)ll =_6 i . fit

(ii) Conversely, whenever a net (T~) in Bx(x) exists which converges to the identity with respect to the weak operator topology and which fulfills the first of the above norm conditions, then K(X) is a two-sided inner M-ideaL Analogous statements hold for A(X).


Proof Suppose that K(X) is an M-ideal. Then by the above proposi t ion, there is a net (T,) such that T~ ~ Idx and T~* ~ Idx. strongly. It follows f rom the p roo f of 2.11 that (T~) mus t be a two-sided approx imate unit for K(X). This allows us to apply 3.5 to finally prove the claim made in (i). The p roof of par t (ii) is an immedia te consequence of 3.5 and 2.11. [ ]

5.3 Corollary. A Banach spaces X satisfies the relation

~ X ) * = ( X * @ , X * * ) e ~{X@ Or*) ~

if and only if there is a net of finite dimensional operators (F=) converging strongly to the identity on X such that lim sup IISF~+ T(Id--F=)l[ < 1 for all S, T~BL(x).

Proof One direct ion is trivial in light of the above result. I f there exists a net with this proper ty , then the approx imab le opera tors are an M-ideal in L(X). By [31], X* has the R a d o n N y k o d y m proper ty which together with the M A P of X yields (this is wel l -known and lastly goes back to Grothendieck , see e.g. [47]) A(X)* =(X(~,X*)*=X*@~X**, whence the result. [ ]

Let us r emark tha t the sequence of coordinate project ions (P,) on the space r p, 1 < p < ~ , which immedia te ly gives the metric app rox ima t ion proper ty , does not have the above proper ty . No te however, tha t things become more t ransparen t when the direct sum X ~ X together with some na tura l n o r m is considered (see [46] or [49]).

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Note added in proof. Recently N. Kalton essentially improved 5.2, thereby considerably enlarging the class of known examples of spaces X for which K(X) is an M-ideal in L(X).

innerm-ideals in banach algebras

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