The grade of an M-ideal

Peter Harmand and Dirk Werner

Abstract. We assign two parameters to an M-ideal J in a Banach space X, called the lower and upper grade, which are defined in terms of the size of balls contained in the set of best approximants from J. These quantities serve to measure how far J resembles an M-summand, and they enter into geometric descriptions in various ways. For instance they allow estimates of the interior of the metric complement of J and of lower bounds for Lipschitz projections onto M-ideals.

1. Introduction

In this paper we consider the notion of a grade of an M-ideal which serves to distinguish between M-ideals and M-summands in a quantitative way. Recall that a subspace J of a Banach space X is called an M-summand if J is the range of an M-projection, i.e., a linear projection Q : X → X satisfying

∥x∥ = max{∥Qx∥, ∥x − Qx∥} ∀x ∈ X.

J is an M-ideal if J^{⊥} := {x^{∗} ∈ X^{∗} | x^{∗}|_{J} = 0} is the range of an Lprojection, i.e., a linear projection P : X^{∗} → X^{∗} satisfying

∥x^{∗}∥ = ∥P x^{∗}∥ + ∥x^{∗} − P x^{∗}∥ ∀x^{∗} ∈ X^{∗}.

The kernel of this projection can be identified with J^{∗} so that J^{∗} is isometric with a subspace of X^{∗}. More precisely, the ‘orthogonal’ complement of J^{⊥} is

ker P = J^{#} := {x^{∗} ∈ X^{∗} | ∥x^{∗}∥ = ∥x^{∗}|_{J}∥}.  (1.1)

Clearly, M-summands are M-ideals, but the converse does not hold; Mideals which are not M-summands will be termed proper. For example, in X = C(K), K a compact Hausdorff space, the M-ideals are precisely the subspaces

J_{D} = {f ∈ C(K) | f|_{D} = 0}

for some closed subset D ⊂ K, and J_{D} is an M-summand if and only if D is clopen (i.e., closed and open). A detailed exposition of M-ideal theory can be found in [3].

In order to distinguish between M-ideals and M-summands, we attach a number g ∈ [0, 1] to an M-ideal J ⊂ X. Actually, we propose a ‘lower grade’ g_{∗} and an ‘upper grade’ g^{∗} for that purpose, with g^{∗} = 0 characterising Msummands. We present here, in a coherent form, results related to these notions, some of which can also be found in [3]. (In fact, we take the chance to correct an invalid statement from [3, p. 54], cf. (2.2) below.) In addition, we discuss pertinent examples.

We use standard notation such as B_{X} for the closed unit ball of X, ex C for the set of extreme points of C and L(X, Y) (resp. K(X, Y)) for the space of bounded (resp. compact) linear operators from X into Y. The symbol B_{X}(x, r) denotes the closed ball in X with centre x and radius r.

2. The lower grade

Let us first recall some notions from approximation theory. A subspace J of a Banach space X is called proximinal if

∀x ∈ X ∃y ∈ J ∥x − y∥ = d(x, J) := inf{∥x − ξ∥ | ξ ∈ J}.

The set of all such y is called the set of best approximants and denoted by P_{J}(x). Thus J is proximinal if and only if P_{J}(x) ≠ ∅ for all x ∈ X. The set-valued map P_{J} is called the metric projection. The metric complement J^{θ} is defined as

J^{θ} = {x ∈ X | ∥x∥ = d(x, J)} = {x ∈ X | 0 ∈ P_{J}(x)}.

It is a basic fact in M-ideal theory that M-ideals are proximinal [3, Proposition II.1.1].

The following definition comes from [2].

Definition 2.1 A closed convex bounded subset B of a Banach space Y is called a pseudoball of radius r if its diameter is 2r > 0 and if for each finite collection y_{1}, …, y_{n} of points with ∥y_{i}∥ < r there is y ∈ B such that

y + y_{i} ∈ B for i = 1, …, n.

If ρ = sup{s ≥ 0 | B contains a ball of radius s}, then the grade of B is defined to be

g(B) = 1 − \frac{ρ}{r}.

A pseudoball which is not a ball will be called proper. Singletons are considered as pseudoballs with radius 0 and grade 0.

Equivalently B is a pseudoball of radius r > 0 if and only if

⋂_{i} (y_{i} + B) ≠ ∅

for each finite family y_{1}, …, y_{n} satisfying ∥y_{i}∥ < r.

Note that B is a closed ball of radius r > 0 if and only if

⋂_{∥y∥<r}(y + B) ≠ ∅,

in which case the intersection consists of the centre of the ball. In a pseudoball there is only a ‘centre’ for any finite set of directions. Also, note that g(B) = 0 if and only if B is a ball. Thus, the pseudoball B_{2} should be considered as ‘more proper’ than B_{1} if g(B_{1}) ≤ g(B_{2}).

To see a simple example, let 0 ≤ s ≤ 1. It is easy to check that

B_{s} = {(s_{n}) ∈ c_{0} | |s_{n} − s| ≤ 1 ∀n ∈ N}  (2.1)

is a pseudoball in c_{0} with radius 1 and g(B_{s}) = s.

The next two results explain our interest in pseudoballs here.

Proposition 2.2

(a) Let J be an M-ideal in X, and let x ∈ X. Then P_{J}(x) is a pseudoball in J with radius d(x, J).

(b) If P_{J}(x) is a pseudoball of radius d(x, J) for all x ∈ X, then J is an M-ideal.

Proof. See [5] or [3, Prop. II.1.3]. □

Conversely, every pseudoball arises as in Proposition 2.2(a).

Theorem 2.3 Let B be a pseudoball of radius 1 in Y. Then there are a Banach space X containing Y with dim X/Y = 1 and x ∈ X with d(x, Y) = 1 such that Y is an M-ideal in X and P_{Y} (x) = B.

Proof. See [2] or [3, Th. II.3.10]. □

The following result has first appeared in [2].

Theorem 2.4 For a closed convex bounded subset B ⊂ Y and r ≥ 0, equivalence between (i) and (ii) holds:

(i) B is a pseudoball of radius r.

(ii) The weak^{∗} closure of B in Y^{∗∗} is a ball with radius r (and centre

y_{B}^{∗∗}, say). In this case

r · g(B) = d(y_{B}^{∗∗}, Y).

Proof. See [2] or [3, Th. II.1.6]. □

We shall now identify the centre y_{B}^{∗∗}.

Corollary 2.5 Let J be an M-ideal in X, and let P be the associated Mprojection from X^{∗∗} onto J^{⊥⊥}. Consider the pseudoball B = P_{J}(x) for some x ∈ X. Then P x = y_{B}^{∗∗}, the centre of \overline{B}^{w∗}. More precisely: the canonical isometry i_{X}^{∗∗} from J^{∗∗} onto J^{⊥⊥} maps y_{B}^{∗∗} onto P x. Moreover (if d(x, J) = 1)

g(P_{J}(x)) = d(P x, J) = inf_{y∈J} sup_{y^{∗}∈B_{J^{∗}}} |⟨y^{∗}, x − y⟩|.  (2.2)

Proof. We consider J^{∗} via unique Hahn-Banach extensions as a subspace of X^{∗}, cf. (1.1). We have for y^{∗} ∈ J^{∗}

y^{∗}(B_{J^{∗∗}}(y_{B}^{∗∗}, 1)) = B_{K}(⟨y_{B}^{∗∗}, y^{∗}⟩, ∥y^{∗}∥).

On the other hand we know from Theorem 2.4 (assuming without loss of generality d(x, J) = 1)

B_{K}(⟨y_{B}^{∗∗}, y^{∗}⟩, ∥y^{∗}∥) = \overline{y^{∗}(B)} ⊂ \overline{y^{∗}(B_{X}(x, 1))} = B_{K}(⟨y^{∗}, x⟩, ∥y^{∗}∥).

Hence ⟨y_{B}^{∗∗}, y^{∗}⟩ = ⟨x, y^{∗}⟩ for all y^{∗} ∈ J^{∗}, which gives y_{B}^{∗∗} = P x.

That g(P_{J}(x)) = d(P x, J) now follows from Theorem 2.4. Also, the M-projection P is the adjoint of the L-projection P_{∗} on X^{∗} with range J^{∗}. Therefore

sup_{y^{∗}∈B_{J^{∗}}} |⟨y^{∗}, x⟩| = sup_{x^{∗}∈B_{X^{∗}}} |⟨P_{∗}x^{∗}, x⟩| = sup_{x^{∗}∈B_{X^{∗}}} |⟨x^{∗}, P x⟩| = ∥P x∥ (2.3)

and

d(P x, J) = inf_{y∈J} ∥P x − y∥ = inf_{y∈J} ∥P (x − y)∥ = inf_{y∈J} sup_{y^{∗}∈B_{J^{∗}}} |⟨y^{∗}, x − y⟩|. □

The grade of the pseudoballs P_{J}(x) is the appropriate tool for investigating the (possibly empty) interior of the metric complement J^{θ} of an M-ideal J. The following proposition answers a question raised in [4] where it is conjectured that an M-ideal is an M-summand as soon as its metric complement has an interior point. Although this conjecture could have been refuted on the basis of the familiar M-ideals J_{D} in C(K) (cf. also Example 2.9), we now have a precise estimate of the interior of J^{θ} by this proposition.

Proposition 2.6 Let J be an M-ideal in X, and let D = {x ∈ X | ∥x∥ = d(x, J) = 1}.

(a) If B_{J}(y_{0}, 2r) ⊂ P_{J}(x_{0}) for some x_{0} ∈ D, then B_{X}(x_{0} − y_{0}, r) ⊂ J^{θ}.

(b) If B_{X}(x_{0}, r) ⊂ J^{θ} for some x_{0} ∈ D, then P_{J}(x_{0}) contains a ball of

radius 2r − ε for whatever ε > 0. Therefore, J^{θ} has empty interior if and only if

g_{∗}(J, X) := inf{g(P_{J}(x)) | d(x, J) = 1} = 1.

Proof. (a) We may assume that y_{0} = 0. Define

|x| = sup{|⟨x, y^{∗}⟩| | y^{∗} ∈ B_{J^{∗}}}.  (2.4)

This is a seminorm, and we have

∥x∥ = max{|x|, d(x, J)}.  (2.5)

In fact, we may write X^{∗} = J^{⊥} ⊕_{1} J^{∗} from which we easily conclude that ex B_{X^{∗}} = ex B_{J^{⊥}}∪ex B_{J^{∗}}. It is left to observe that ∥x∥ = sup{|⟨x, x^{∗}⟩| | x^{∗} ∈ ex B_{X^{∗}}}, |x| = sup{|⟨x, y^{∗}⟩| | y^{∗} ∈ ex B_{J^{∗}}}, and d(x, J) = sup{|⟨x, x^{∗}⟩| | x^{∗} ∈ ex B_{J^{⊥}}}.

Now let z ∈ X, ∥z − x_{0}∥ < r. We wish to show that |z| < d(z, J) (so that z ∈ J^{θ}), then (a) will follow from the closedness of J^{θ}. First note that |z − x_{0}| < r and d(z − x_{0}, J) < r by (2.5). Secondly, we have

|x_{0}| ≤ 1 − 2r.  (2.6)

To see this, choose, given ε > 0, y^{∗} ∈ S_{Y^{∗}} such that

|x_{0}| ≤ ⟨y^{∗}, x_{0}⟩ + ε.

Next choose y ∈ B_{J} such that

⟨y^{∗}, y⟩ ≤ −1 + ε.

Note 2ry ∈ P_{J}(x_{0}), i.e., ∥x_{0} − 2ry∥ = 1. The estimate

|x_{0}| ≤ ⟨y^{∗}, x_{0}⟩ + ε = ⟨y^{∗}, x_{0} − 2ry⟩ + 2r⟨y^{∗}, y⟩ + ε ≤ 1 + 2r(−1 + ε) + ε

now proves (2.6).

Altogether one obtains

|z| < |x_{0}| + r ≤ 1 − r = d(x_{0}, J) − r < d(z, J),

as desired.

(b) Let P be the M-projection from X^{∗∗} onto J^{⊥⊥} ≅ J^{∗∗}. In view of Theorem 2.4 and Corollary 2.5 we have to show that d(P x_{0}, J) ≤ 1 − 2r. This is implied by (cf. (2.2))

|⟨y^{∗}, x_{0}⟩| ≤ 1 − 2r ∀y^{∗} ∈ B_{J^{∗}}.

To prove this for some given y^{∗} ∈ B_{J^{∗}}, fix ε > 0 and choose y ∈ int B_{J} such that ⟨y^{∗}, y⟩ is real and so close to 1 that

|⟨y^{∗}, y⟩ ± α| ≤ 1 only if |α| ≤ ε.

In particular ⟨y^{∗}, y⟩ ≥ 1 − ε.

Since P_{J}(x_{0}) is a pseudoball (Proposition 2.2), we may find \overline{y} ∈ P_{J}(x_{0}) with \overline{y} ± y ∈ P_{J}(x_{0}). Let us first observe

|⟨y^{∗}, \overline{y}⟩ − ⟨y^{∗}, x_{0}⟩| ≤ ε.  (2.7)

In fact,

⟨y^{∗}, \overline{y}⟩ ± ⟨y^{∗}, y⟩ ∈ y^{∗}(P_{J}(x_{0})) ⊂ B_{K}(⟨y^{∗}, x_{0}⟩, 1)

(cf. Theorem 2.4) so that (2.7) follows. On the other hand, for z = (1 − r)x_{0} + r\overline{y} we have

y^{∗}(z) = y^{∗}(z + ry) − ry^{∗}(y) ≤ (1 − r) − r(1 − ε)  (2.8)

where we used ∥x_{0} − (z + ry)∥ ≤ r since \overline{y} + y ∈ P_{J}(x_{0})), consequently

∥z + ry∥ = d(z + ry, J) = d((1 − r)x_{0}, J) = 1 − r.

(2.7) and (2.8) together give our claim. □

The previous result suggests the following definition.

Definition 2.7 If J is an M-ideal in X, we call the number

g_{∗}(J, X) = inf{g(P_{J}(x)) | d(x, J) = 1}

the lower grade of J in X.

We shall see in Example 3.2 that the infimum in the above definition need not be attained.

In order to compute g_{∗}(J, X) the following lemma is useful. Recall that the Dixmier characteristic of a subspace V ⊂ X^{∗} is defined to be

r(V, X^{∗}) = sup\{ρ | ρB_{X^{∗}} ⊂ \overline{B_{V}} ^{w∗}\}.

Lemma 2.8 Let J be an M-ideal in X with codimension 1. Then

g_{∗}(J, X) = r(J^{∗}, X^{∗}).

Proof. We introduce the canonical operator

I_{X,J^{∗}} : X → J^{∗∗}, ⟨I_{X,J^{∗}}(x), y^{∗}⟩ = ⟨y^{∗}, x⟩.

The definition of the characteristic implies that I_{X,J^{∗}} is an (into-) isomorphism with ∥I_{X,J^{∗}}^{−1}∥ = r(J^{∗}, X^{∗})^{−1} if the characteristic is positive. Therefore,

r(J^{∗}, X^{∗}) = inf_{{x∈X \atop ∥x∥=1}} sup_{{y^{∗}∈J^{∗} \atop ∥y^{∗}∥≤1}} |⟨y^{∗}, x⟩|,  (2.9)

and this formula also holds in case r(J^{∗}, X^{∗}) = 0. In the notation of (2.4) we thus have r(J^{∗}, X^{∗}) = inf_{∥x∥=1} |x|, and (2.5) and (2.3) imply that in fact

r(J^{∗}, X^{∗}) = inf_{{∥x∥=1 \atop d(x,J)=1}} |x| = inf_{{∥x∥=1 \atop d(x,J)=1}} ∥P x∥.

On the other hand, g(P_{J}(x)) is independent of x ∈ X\J, provided d(x, J) = 1, since dim X\J = 1. Therefore, if ∥x∥ = d(x, J) = 1, then

g_{∗}(J, X) = g(P_{J}(x)) = d(P x, J) ≤ ∥P x∥

by (2.3) and so g_{∗}(J, X) ≤ r(J^{∗}, X^{∗}). To prove the converse inequality, let x as above and y ∈ J. Then ∥x − y∥ ≥ 1 and for ξ = (x − y)/∥x − y∥ we have

∥P ξ∥ = \frac{∥P x − y∥}{∥x − y∥} ≤ ∥P x − y∥,

which proves that r(J^{∗}, X^{∗}) ≤ g_{∗}(J, X). □

Example 2.9 For the M-ideals J_{D} ⊂ C(K) we have

g_{∗}(J_{D}, C(K)) = 0 if D has nonempty interior,
g_{∗}(J_{D}, C(K)) = 1 otherwise.

If D has nonempty interior, we may pick a continuous function f ≠ 0 with supp f ⊂ D. Hence J_{D} is an M-summand in Y := J_{D} ⊕ K{f} so that r(J_{D}^{∗}, Y^{∗}) = 0. On the other hand, if int D = ∅, then K\D is dense, and for every f ∈ C(K)\J_{D}, with Y as above, the canonical operator I_{Y,J_{D}^{∗}} is an isometry which implies that r(J_{D}^{∗}, Y^{∗}) = 1. An appeal to Lemma 2.8 yields the desired result.

Example 2.10 We will show next that for function algebras the same dichotomy as for C(K)-spaces arises, namely the lower grade of an M-ideal in a function algebra A ⊂ C(K) is either 0 or 1. Let us recall that precisely the subspaces J_{D} ∩ A = {f ∈ A | f|_{D} = 0}, with D a p-set, are the M-ideals of A; see [3, p. 15 and Th. V.4.2]. We may and shall assume that K is the Shilov boundary of A so that the set of p-points is dense in K. Then we have

g_{∗}(J_{D} ∩ A, A) = 0 if D has nonempty interior,
g_{∗}(J_{D} ∩ A, A) = 1 otherwise.

Let J = J_{D} ∩A. Suppose first that D has nonempty interior. Let U ⊂ D be open and t ∈ U be a p-point. Then there are functions f_{n} ∈ A such that f_{n}(t) = 1, ∥f_{n}∥ ≤ 2 and |f_{n}| ≤ 1/n off U, in particular off D (see [3, p. 15]). If μ ∈ J^{∗}, ∥μ∥ = 1, then μ can be represented by a measure of norm 1 that is supported on K\D. Therefore

∥I_{A,J^{∗}}(f_{n})∥ = sup_{μ∈B_{J^{∗}}} |⟨μ, f_{n}⟩| → 0,

and the canonical operator I_{A,J^{∗}} has no bounded inverse. Lemma 2.8 and (2.9) imply that g_{∗}(J, A) = 0.

If int D = ∅, then by the same token g_{∗}(J, A) = 1, since the functionals μ_{k} = δ_{k}|_{A}, k ∈ K\D, are in J^{∗} and have norm 1 and since K\D is dense:

∥f∥ = sup_{k∉D} |f(k)| = sup_{k∉D} |⟨μ_{k}, f⟩| ≤ sup_{μ∈B_{J^{∗}}} |⟨μ, f⟩| = ∥I_{A,J^{∗}}(f)∥.

Example 2.11 If J is an M-ideal in its bidual (see Chapter III in [3] for examples), then g_{∗}(J, J^{∗∗}) = 1; this follows from Lemma 2.8 and Goldstine’s theorem.

Example 2.12 If K(X, Y) is an M-ideal in L(X, Y) (see Chapter VI in [3]) and K(X, Y) ≠ L(X, Y), then g_{∗}(K(X, Y), L(X, Y)) = 1. To show this it is enough to prove that the unit ball of K(X, Y)^{∗} is weak^{∗} dense in the unit ball of L(X, Y)^{∗}, by Lemma 2.8. If this were not the case, then there would be some operator T ∈ L(X, Y), ψ ∈ B_{L(X,Y)^{∗}} and α ≥ 0 such that

⟨ψ, T ⟩ = 1 > α ≥ ⟨φ, T ⟩ ∀φ ∈ B_{K(X,Y)^{∗}}.

In particular, we have ∥T ∥ ≥ 1. But checking the above inequality for all functionals φ : S ↦ ⟨Sx, y^{∗}⟩, x ∈ B_{X}, y^{∗} ∈ B_{Y^{∗}}, yields ∥T ∥ ≤ α < 1: a contradiction.

We now turn to geometric descriptions of M-ideals, especially of those of codimension 1. If J is an M-ideal in X and dim X/J = 1, then from what was observed in the proof of Proposition 2.6 it follows that J = ker p with p ∈ ex B_{X^{∗}} and d(x, J) = |p(x)|. Moreover p attains its norm since J is proximinal.

The first result extends the idea of a cylindrically shaped unit ball from M-summands to M-ideals: the lid L of the cylinder is the translate of a pseudoball in J.

Proposition 2.13 Let J be an M-ideal in X, dim X/J = 1, and

L := {x ∈ X | ∥x∥ = 1 = p(x)}

with X^{∗} = J^{∗} ⊕_{1} Kp. Then

int B_{X} ⊂ co ⋃_{|θ|=1} θL ⊂ B_{X}.  (2.10)

Proof. The second inclusion is obvious. To prove the first for real scalars take x with ∥x∥ < 1 and put α := p(x). Then |α| < 1 and α = λ·1+(1−λ)·(−1) for a suitable λ ∈ (0, 1). We will get x as a convex combination of elements from L and −L by an application of the strict 2-ball property of M-ideals; see [3, Th. I.2.2]. To use this argument we have to translate x into J.

Choose x_{0} ∈ X with ∥x_{0}∥ = 1 and p(x_{0}) = −1. Then y := x + αx_{0} ∈ J = ker p. Since d(λx_{0}, J) = |p(λx_{0})| = λ and d(y + (1 − λ)x_{0}, J) = |p(y + (1 − λ)x_{0})| = 1 − λ each of the two balls

B_{X}(λx_{0}, λ) and B_{X}(y + (1 − λ)x_{0}, 1 − λ)  (2.11)

meets J. The inequality

∥λx_{0} − [y + (1 − λ)x_{0}]∥ = ∥αx_{0} − y∥ < 1

shows that the interiors of the two balls have nonempty intersection. Hence, by the strict 2-ball property

B_{X}(λx_{0}, λ) ∩ B_{X}(y + (1 − λ)x_{0}, 1 − λ) ∩ J ≠ ∅.  (2.12)

Putting P := P_{J}(x_{0}) = B_{X}(x_{0}, 1) ∩ J we see that

λP ∩ (y + (1 − λ)P) ≠ ∅.

So there are x_{1}, x_{2} ∈ P with λx_{1} = y+(1−λ)x_{2}, i.e., y = λx_{1}+(1−λ)(−x_{2}). With the definition of y and λ this yields the desired representation

x = λ(x_{1} − x_{0}) + (1 − λ)(−x_{2} + x_{0}) ∈ λL + (1 − λ)(−L).

For K = C choose |θ| = 1 such that p(θx) is real. The argument for real scalars then shows that θx ∈ co(L ∪ −L). □

It is clear from the above proof that we get equality in the right-hand inclusion of (2.10) if J even has the so-called strong 2-ball property, which means that to conclude (2.12) it is enough to assume that the two balls in (2.11) meet instead of requiring that the interiors of these balls have nonempty intersection.

Let us record two consequences of Proposition 2.13. We get a statement analogous to (2.10) for M-ideals of arbitrary codimensions if we replace L by the set D from Proposition 2.6.

Corollary 2.14 Let J be an M-ideal in X, and D = {x ∈ X | ∥x∥ = 1 = d(x, J)}. Then

int B_{X} ⊂ co D ⊂ B_{X}.

Proof. This follows from Proposition 2.13 by observing that every point in int B_{X} is in a space Y = J ⊕ K{x_{0}}, J is an M-ideal in Y, and θL ⊂ D. □

The sets θL serve as substitutes for B_{J} also in another way.

Corollary 2.15 With the assumption and notation from Proposition 2.13 we have for f ∈ J^{∗} and |θ| = 1

∥f∥ = sup{Re f(x) | x ∈ θL}.

Proof. From Proposition 2.13 we obtain for g ∈ X^{∗}

∥g∥ = sup{Re g(x) | x ∈ γL, |γ| = 1}.

Using this and ∥f + θ^{−1}p∥ = ∥f∥ + 1, we find, given ε > 0, an x = γy ∈ γL such that

Re(f + θ^{−1}p)(x) = Re[f(x) + θ^{−1}γ] > ∥f∥ + 1 − ε.

Since Re f(x) ≤ ∥f∥, this yields Re θ^{−1}γ ≥ 1 − ε, hence |θ − γ| < \sqrt{2ε}. So

Re f(θy) = Re f((θ − γ)y) + Re f(x) > −\sqrt{2ε}∥f∥ + ∥f∥ − ε.  □

The next result shows the limitations of the idea of a cylindrical unit ball: for M-ideals with lower grade 1 there is nothing above the lid and nothing below the bottom. The reader is advised to draw a two-dimensional picture to understand the reformulation of this claim in the second statement of the following proposition.

Proposition 2.16 Let J be an M-ideal in X. Then g_{∗}(J, X) = 1 if and only if for all x ∈ X, ∥x∥ > 1, there is y ∈ co(B_{X} ∪ {x}) ∩ J with ∥y∥ > 1.

Proof. “⇒”: By the definition of g_{∗} and Lemma 2.8, J is an M-ideal in Y = Y_{x} = J ⊕ K{x} and B_{J^{∗}} is σ(Y^{∗}, Y)-dense in B_{Y^{∗}}. Writing Y^{∗} = J^{∗} ⊕_{1} K{p}, we may assume that p(x) =: α > 0; the claim is trivial for x ∈ J (= ker p).

Choose g ∈ S_{Y^{∗}} with g(x) = ∥x∥ > 1. Because of \overline{B_{J^{∗}}} ^{w∗} = B_{Y^{∗}} and the w^{∗}-lower semicontinuity of the norm, there is

f ∈ S_{J^{∗}} with f(x) =: 1 + η > 1.

Applying Corollary 2.15 with θ = −1 to f we find

x_{0} ∈ S_{Y} with p(x_{0}) = −1 and Re f(x_{0}) > 1 − \frac{η}{α}.

For y = \frac{1}{α + 1} x + \frac{α}{α + 1} x_{0} ∈ co(B_{Y} ∪ \{x\}) ⊂ co(B_{X} ∪ \{x\}) we obtain

p(y) = \frac{1}{α + 1} α + \frac{α}{α + 1}(−1) = 0, i.e., y ∈ J

and

Re f(y) > \frac{1}{α + 1}(1 + η) + \frac{α}{α + 1}(1 − \frac{η}{α}) = 1, i.e., ∥y∥ > 1.

“⇐”: If g_{∗}(J, X) < 1, then, again by the definition of g_{∗} and Lemma 2.8, there is a z ∈ X such that, with Y = J ⊕ K{z}, B_{J^{∗}} is not σ(Y^{∗}, Y)-dense in B_{Y^{∗}}. Separating \overline{B_{J^{∗}}} ^{w∗} from p we find for a suitable x ∈ Y (⊂ X)

sup_{f∈B_{J^{∗}}} |f(x)| = 1 < p(x) ≤ ∥x∥.

For x_{0} ∈ B_{X} and f ∈ B_{J^{∗}} this yields

|f(λx + (1 − λ)x_{0})| ≤ 1,

therefore, if λx + (1 − λ)x_{0} ∈ J, then ∥λx + (1 − λ)x_{0}∥ ≤ 1. □

3. The upper grade

We will now attach another number to an M-ideal J in X.

Definition 3.1 If J is an M-ideal in X, we call the number

g^{∗}(J, X) = sup{g(P_{J}(x)) | d(x, J) = 1}

the upper grade of J in X.

As a result of Theorem 2.4 and Corollary 2.5, the condition g^{∗}(J, X) = 0 characterises M-summands among M-ideals.

We also note in the case dim X/J = 1 that g_{∗}(J, X) = g^{∗}(J, X) = g(P_{J}(x)) whenever d(x, J) = 1 and that g_{∗}(Y, X) = g^{∗}(Y, X) = g(B) in the context of Theorem 2.3.

If J is an M-ideal in X, then a fortiori it is an M-ideal in Y_{x} = J ⊕K{x} for each x ∈ X\J. If g(P_{J}(x)) = 0, then J is an M-summand in Y_{x}, and if g(P_{J}(x)) = 1, then J is far from being an M-summand. Thus, if g_{∗}(J, X) = 1, this indicates that J behaves like a ‘very’ proper M-ideal in every direction x, whereas g^{∗}(J, X) = 1 only yields the existence of one such direction.

Note that g_{∗}(J, X) ≤ g^{∗}(J, X) ≤ 1 so that g^{∗}(J, J^{∗∗}) = 1 if J is an Mideal in J^{∗∗}, by Example 2.11, and g^{∗}(K(X, Y), L(X, Y)) = 1 in the setting of Example 2.12. We also have

g^{∗}(J_{D}, C(K)) = 0 if D is clopen,
g^{∗}(J_{D}, C(K)) = 1 otherwise.

If D is clopen, then J_{D} is an M-summand, and the upper grade is 0. If D is not clopen, we will argue that the canonical operator I_{Y,J_{D}^{∗}}, for Y = J_{D} ⊕ K{1}, is an isometry. In fact, for f = g + λ1 ∈ Y we have, since \overline{K\D} ∩ D ≠ ∅, that sup_{t∈K\D} |f(t)| ≥ |λ| so that ∥f∥_{∞} = sup_{t∈K\D} |f(t)| which is enough to prove our claim.

The following example shows that neither the infimum in the definition of the lower grade nor the supremum in the definition of the upper grade need be attained.

Example 3.2 If 0 < s ≤ 1 let

X_{s} = (c, ∥ ∥_{s})  where  ∥(ξ_{n})∥_{s} = max{∥(ξ_{n})∥_{∞}, (1/s)|ξ_{∞}|}

and ξ_{∞} = lim ξ_{n}. Define for a sequence (s_{n}) in (0, 1)

X = (⊕ ∑ X_{s_{n}})_{c_{0}}  and  J = (⊕ ∑ c_{0})_{c_{0}}.

Then J is an M-ideal in X and

(a) if inf s_{n} = 0 then g_{∗}(J, X) = 0, but g_{∗}(J, Y) > 0 for all Y with

J ⊂ Y ⊂ X and dim Y/J = 1,

(b) if sup s_{n} = 1 then g^{∗}(J, X) = 1, but g^{∗}(J, Y) < 1 for all Y with

J ⊂ Y ⊂ X and dim Y/J = 1.

Proof. Note first that X_{s}^{∗} ≅ ℓ^{1}⊕_{1}K{p_{s}} with p_{s}((ξ_{n})) = (1/s)ξ_{∞}. Hence c_{0} is an M-ideal in X_{s}, and one deduces easily using the restricted 3-ball property [3, Th. I.2.2] that J is an M-ideal in X. Moreover, we have g_{∗}(c_{0}, X_{s}) = g^{∗}(c_{0}, X_{s}) = s.

(a) Using sequences supported only in the n-th coordinate space one can check that g_{∗}(J, X) = 0, since inf s_{n} = 0. Assume that there is an x ∈ X \ J such that J is an M-summand in Y = J ⊕ K{x}, i.e., x is M-orthogonal to J. The projection P_{n} from X onto the n-th coordinate space X_{s_{n}} is an M-projection, so by [1, Cor. 3] x_{n} = P_{n}x is M-orthogonal to J, hence in particular M-orthogonal to J_{n} = c_{0}. This gives x_{n} = 0, since c_{0} is not an M-summand in X_{s_{n}}, and thus x = 0: a contradiction.

(b) As in (a) it is easy to see that g^{∗}(J, X) = 1. We will show that for all x = (x_{n}) ∈ X with d(x, J) = 1 the pseudoball P_{J}(x) = B_{X}(x, 1) ∩ J has non-empty interior. Without restriction we may additionally assume that ∥x∥ = 1. By the definition of the space X there are a finite set E ⊂ N and some ε_{0} > 0 such that ∥x_{n}∥_{s_{n}} = 1 for n ∈ E and ∥x_{n}∥_{s_{n}} ≤ 1 − ε_{0} for n ∉ E. This gives, for n ∉ E, the inclusion ε_{0}B_{J_{n}} ⊂ B_{X_{sn}} (x_{n}, 1) ∩ J_{n} where again J_{n} is the c_{0}-space in the n-th coordinate. For n ∈ E we have d(x_{n}, J_{n}) ≤ 1. Using g^{∗}(J_{n}, X_{s_{n}}) = s_{n} one obtains from this for some suitable ε_{n} > 0 and some j_{n}^{0} ∈ J_{n} that B_{J_{n}}(j_{n}^{0}, ε_{n}) ⊂ B_{X_{sn}} (x_{n}, 1) ∩ J; note that d(x_{n}, J_{n}) < 1 or even d(x_{n}, J_{n}) = 0 does not spoil this. Defining j^{0} = (j_{n}^{0}) ∈ J with j_{n}^{0} = 0 for n ∉ E and ε = min{ε_{n} | n ∈ E or n = 0} one finds that B_{J}(j^{0}, ε) is contained in B_{X}(x, 1) ∩ J. □

The upper grade of an M-ideal tells us something about the existence of M-orthogonal directions for finite dimensional subspaces.

Proposition 3.3 Let J be an M-ideal in X and put α := g^{∗}(J, X). Then, given ε > 0 and a finite dimensional subspace E ⊂ J, one may find y_{0} ∈ J satisfying

\frac{α}{1 + ε} max\{∥e∥, |λ|\} ≤ ∥e + λy_{0}∥ ≤ (1 + ε) max\{∥e∥, |λ|\}  (3.1)

for all e ∈ E, λ ∈ K.

Proof. If α = 0 then the statement is trivially satisfied with y_{0} = 0. For α > 0 we may assume, because we have an ε > 0 at our disposal in (3.1), that there is an x ∈ X such that d(x, J) = 1 and α = g(P_{J}(x)). Then J is an M-ideal in Y = Y_{x} = J ⊕ K{x} and α = g_{∗}(J, Y) = g^{∗}(J, Y).

As in the proof of Lemma 2.8 the canonical operator

I_{Y,J^{∗}} : Y → J^{∗∗}, ⟨I_{Y,J^{∗}}(y), x^{∗}⟩ = ⟨x^{∗}, y⟩

is an (into-) isomorphism whose inverse has norm α^{−1}. To get started, consider the decomposition

Y^{∗∗} = J^{⊥⊥} ⊕_{∞} lin{x_{0}^{∗∗}}

in which we assume ∥x_{0}^{∗∗}∥ = 1. Let δ > 0 and choose with the help of the principle of local reflexivity an injective operator

T : lin(E ∪ {x_{0}^{∗∗}}) → Y

such that ∥T ∥ · ∥T^{−1}∥ ≤ 1 + δ and T e = e for all e ∈ E. For x_{0} := T x_{0}^{∗∗} we obtain

\frac{1}{1 + δ} max\{∥e∥, |λ|\} ≤ ∥e + λx_{0}∥ ≤ (1 + δ) max\{∥e∥, |λ|\}

for all e ∈ E, λ ∈ K.

We have to ‘push’ x_{0} into J. As a first step, we map it into J^{∗∗} by means of y_{0}^{∗∗} := I_{Y,J^{∗}}(x_{0}). It follows easily that

\frac{α}{1 + δ} max\{∥e∥, |λ|\} ≤ ∥e + λy_{0}^{∗∗}∥ ≤ (1 + δ) max\{∥e∥, |λ|\}

for all e ∈ E, λ ∈ K. As a second step, we again apply the principle of local reflexivity to obtain an injection

S : lin(E ∪ {y_{0}^{∗∗}}) → J

with ∥S∥ · ∥S^{−1}∥ ≤ 1 + δ which extends the identity on E.

With an appropriate choice of δ and y_{0} := S(y_{0}^{∗∗}) we finally achieve the desired result. □

If the upper grade of an M-ideal J in X equals 1, then it follows from the above proposition and an obvious induction process that J contains a sequence equivalent to the standard basis of c_{0}. The same conclusion holds if only g^{∗}(J, X) > 0. This can be deduced from Proposition 3.3 and the classical Bessaga-Pełczyński theorem, which characterises spaces containing c_{0} in terms of weakly unconditionally Cauchy series. However, in M-ideal theory the containment of c_{0} in proper M-ideals is usually – and more naturally – proved using the so-called intersection property; see [3, Th. II.4.7].

We have seen in (3.1) that the grade of an M-ideal J in X says something about the space J itself, irrespective of its position in X. Conversely, the geometry of the space J – more precisely, the set of pseudoballs in J – determines which grades are possible if J is an M-ideal in some superspace X. If J contains no proper pseudoball (e.g., if J is reflexive or, by what was remarked above, if J fails to contain a copy of c_{0}), then J is already an M-summand if it is an M-ideal; i.e., only g^{∗}(J, X) = 0 is possible.

If a space J has a proper pseudoball, one asks if it contains even pseudoballs B with empty interior, i.e., with g(B) = 1. This is the case for J = c_{0}, which is the only space so far, where we have exhibited in (2.1) a pseudoball of grade different from 0 and 1. The following result will help to decide this question.

Proposition 3.4 If a Banach space Y contains a pseudoball with empty interior, then

0 ∈ \overline{ex}^{w∗} B_{Y^{∗}}.

Proof. If B is a pseudoball with radius 1 and grade 1, then by Theorem 2.3 there is a Banach space X containing Y such that Y is an M-ideal in X and g^{∗}(Y, X) = 1.

Let y_{1}, …, y_{n} ∈ Y, ∥y_{i}∥ = 1, and δ > 0. We have to produce some p ∈ ex B_{Y^{∗}} such that

|p(y_{i})| ≤ δ ∀i = 1, …, n.

To this end consider E := lin{y_{1}, …, y_{n}} and ε = δ/3 > 0. Choose y_{0} according to Proposition 3.3. In particular (since α = 1 here) ∥y_{0}∥ ≥ 1/(1 + ε) so that there exists p ∈ ex B_{Y^{∗}} with p(y_{0}) ∈ R and p(y_{0}) > 1/(1 + ε) − ε. Hence for i = 1, …, n and suitable scalars θ_{i} with modulus 1

|p(y_{i})| = p(θ_{i}y_{i} + y_{0}) − p(y_{0}) ≤ ∥θ_{i}y_{i} + y_{0}∥ − p(y_{0}) ≤ 1 + ε − (\frac{1}{1 + ε} − ε) ≤ δ.

□

Example 3.5 The real ℓ^{1}-predual space Y = {y ∈ c | y_{1} +2y_{∞} = 0} (where y_{∞} := lim y_{n}) contains proper pseudoballs, but no pseudoballs with empty interior.

Proof. All the claims will follow once we identify the dual of

X = (c, ∥ ∥) where ∥x∥ = max{∥x∥_{∞}, |x_{1} + 2x_{∞}|}.

To this end put e_{k}^{∗}(x) = x_{k} and f_{∞}^{∗}(x) = x_{1} + 2x_{∞}. Then for n ∈ N and a_{k} ∈ R, k = 1, …, n or k = ∞,

∥∑_{k=1}^{n} a_{k}e_{k}^{∗} + a_{∞}f_{∞}^{∗}∥_{X^{∗}} = ∑_{k=1}^{n} |a_{k}| + |a_{∞}|.  (3.2)

The inequality ≤ in (3.2) is clear by the definition of the norm ∥ ∥. For the converse and sgn a_{1} = sgn a_{∞} consider x = (sgn a_{1}, …, sgn a_{n}, 0, …). Then ∥x∥ = 1 and

(∑_{k=1}^{n} a_{k}e_{k}^{∗} + a_{∞}f_{∞}^{∗})(x) = ∑_{k=1}^{n} |a_{k}| + |a_{∞}|.  (3.3)

If sgn a_{1} ≠ sgn a_{∞} and a_{1}, a_{∞} ≠ 0 use x = (sgn a_{1}, …, sgn a_{n}, sgn a_{∞}, …). Note that in this case sgn a_{1}+2 sgn a_{∞} = sgn a_{∞}, so that ∥x∥ = 1, and again (3.3) holds. Also the remaining case, a_{1} = 0 or a_{∞} = 0, is easily handled in a similar way. By (3.2) X^{∗} is isometrically isomorphic to ℓ^{1}; note that ∥ ∥ is an equivalent norm on c. Also ex B_{X^{∗}} = {±e_{k}^{∗} | k ∈ N} ∪ {±f_{∞}^{∗}}.

Now, Y = ker f_{∞}^{∗} is an M-ideal in X, Y^{∗} ≅ \overline{lin}\{e_{k}^{∗} | k ∈ N\} ≅ ℓ^{1}, and ex B_{Y^{∗}} = {±e_{k}^{∗} | k ∈ N}. Since Y^{∗} is σ(X^{∗}, X)-dense in X^{∗}, Y is a proper M-ideal in X, hence Y contains a proper pseudoball. However, the σ(Y^{∗}, Y)-closure of ex B_{Y^{∗}} equals ex B_{Y^{∗}} ∪ {±1/2e_{1}^{∗}}, so that Y can’t contain a pseudoball with empty interior by Proposition 3.4. □

With the notation of the above proof define x ∈ X by x_{1} = −1 and x_{k} = 1 for k ≥ 2. Then ∥x∥ = 1 and d(x, Y) = |f_{∞}^{∗}(x)| = 1. The pseudoball

B = P_{Y} (x) = {y ∈ Y | |y_{1} − (−1)| ≤ 1 and |y_{k} − 1| ≤ 1 for k ≥ 2}

has grade 1/3: B_{Y} (y, 2/3) ⊂ B for y with y_{1} = −4/3 and y_{k} = 2/3 for k ≥ 2; and, as one easily shows, B contains no ball of Y with radius larger than 2/3. Hence g_{∗}(Y, X) = g^{∗}(Y, X) = 1/3.

We don’t know if Y contains pseudoballs with grades larger than 1/3. We also don’t know if there is a quantitative version of Proposition 3.4 relating the supremum of the grades of pseudoballs in Y with the minimum of the norms of elements in \overline{ex}^{w∗} B_{Y} (which is 1/2 in the above example).

With the help of the notion of an upper grade we will now prove:

Proposition 3.6 Suppose J is an M-ideal in X. If there exists a Lipschitz projection π from X onto J, then its Lipschitz constant L is at least 2 · g^{∗}(J, X).

Proof. Let x ∈ X, d(x, J) = 1, and consider B = P_{J}(x) = B_{X}(x, 1) ∩ J. Then

B = π(B) ⊂ π(B_{X}(x, 1)) ⊂ B_{J}(π(x), L).

Now we take σ(J^{∗∗}, J^{∗})-closures and use Proposition 2.2, Theorem 2.4, and Corollary 2.5 to obtain

B_{J^{∗∗}}(P x, 1) ⊂ B_{J^{∗∗}}(π(x), L)

where P is the M-projection from X^{∗∗} onto J^{⊥⊥} ≅ J^{∗∗}. Hence

2 P x − y ∈ B_{J^{∗∗}}(π(x), L) ∀y ∈ B

so that by Theorem 2.4 and Corollary 2.5

L ≥ 2 · ∥P x − (y + π(x))/2∥ ≥ 2 · d(P x, J) = 2 · g(B).  □

Proposition 3.6 in particular yields lower bounds for projection constants. For example, it is clear that c_{0} is the range of a norm-2 linear projection on c, but since g^{∗}(c_{0}, c) = 1 there is no projection onto c_{0} with smaller norm. (This is no doubt a well-known fact.) More interesting is the case of c_{0} ⊂ ℓ^{∞}. Here, no linear continuous projections exist, but Lindenstrauss [6] has produced a nonlinear 2-Lipschitz projection from ℓR^{∞} onto c_{0}. Again we conclude that the Lipschitz constant 2 is optimal.

References

[1] E. Behrends: M-complements of M-ideals, Rev. Roum. Math. Pures Appl. 29, 537–541, 1984.

[2] E. Behrends and P. Harmand: Banach spaces which are proper M-ideals, Studia Math. 81, 159–169, 1985.

[3] P. Harmand, D. Werner, and W. Werner: M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics 1547, Springer, Berlin-Heidelberg-New York, 1993.

[4] R. B. Holmes, B. Scranton, and J. Ward: Approximation from the space of compact operators and other M-ideals, Duke Math. J. 42, 259–269, 1975.

[5] Å. Lima: On M-ideals and best approximation, Indiana Univ. Math. J. 31, 27–36, 1982.

[6] J. Lindenstrauss: On nonlinear projections in Banach spaces, Michigan Math. J. 11, 263–287, 1964.

Fachbereich Mathematik, Universität Oldenburg, Postfach 25 03, D-26 111 Oldenburg, Germany; e-mail: harmand@math.uni-oldenburg.de

I. Mathematisches Institut, Freie Universität Berlin, Arnimallee 2–6, D-14 195 Berlin, Germany; e-mail: werner@math.fu-berlin.de