The Daugavet equation for operators not fixing a copy of C(S)

Lutz Weis and Dirk Werner

Abstract. We prove the norm identity ∥Id + T ∥ = 1 + ∥T ∥, which is known as the Daugavet equation, for operators T on C(S) not fixing a copy of C(S), where S is a compact metric space without isolated points.

1. Introduction

An operator T: X → X on a Banach space is said to satisfy the Daugavet equation if

∥Id + T ∥ = 1 + ∥T ∥;  (1.1)

this terminology is derived from Daugavet’s theorem that a compact operator on C[0, 1] satisfies (1.1). Many authors have established the Daugavet equation for various classes of operators, e.g., the weakly compact ones, on various spaces; we refer to [1], [2], [4], [7], [10], [11], [13] and the references in these papers for more information.

The most far-reaching result for operators on L^{1}[0, 1] is due to Plichko and Popov [7, Th. 9.2 and Th. 9.8] who prove that an operator on L^{1}[0, 1] which does not fix a copy of L^{1}[0, 1] satisfies the Daugavet equation. (As usual, T: X → X fixes a copy of a Banach space E if there is a subspace F ⊂ X isomorphic to E such that T |_{F} is an (into-) isomorphism.) In this paper we shall establish the corresponding result for operators on spaces of continuous functions.

Theorem 1.1 Let (S, d) be a compact metric space without isolated points. If T: C(S) → C(S) does not fix a copy of C(S), then T satisfies the Daugavet equation.

In the next section we shall provide a direct proof of this theorem which is measure theoretic in spirit, and in Section 3 we shall give another argument which relies on a deep result due to Rosenthal [8]. Though this proof is much shorter it is less revealing than the one in Section 2; therefore we found it worthwhile to present both arguments.

Either of our approaches depends on the analysis of the representing kernel of an operator T: C(S) → C(S); this is the family of Borel measures on S defined by μ_{s} = T^{∗}δ_{s}, s ∈ S. In [10] (see also [11]) a necessary and sufficient condition on the kernel of T was established in order that T satisfy the Daugavet equation. Here we record a special case which suffices for our needs.

Lemma 1.2 If the representing kernel (μ_{s})_{s∈S} of a bounded linear operator T: C(S) → C(S) satisfies

inf_{s∈U} |μ_{s}({s})| = 0  (1.2)

for all nonvoid open subsets U of S, then T satisfies the Daugavet equation.

Our results are valid for real and complex spaces; we denote the scalar field by 𝕂.

2. Proof of Theorem 1.1

We fix a diffuse probability measure λ on S whose support is S. To show the existence of such a measure, consider a countable basis of open sets O_{n} ⊂ S. Since each O_{n} is perfect, we know from a theorem of Bessaga and Pełczyński (see [6, p. 52]) that there are diffuse probability measures λ_{n} with supp λ_{n} ⊂ O_{n}; it remains to define λ = ∑ 2^{−n}λ_{n}.

Now let T: C(S) → C(S) be a bounded linear operator and (μ_{s})_{s∈S} its representing kernel. It is enough to show that T fixes a copy of C(S) if (1.2) of Lemma 1.2 fails for some open set U. Thus we assume, for some open U ⊂ S and α > 0, that

|μ_{s}({s})| ≥ α  ∀s ∈ U.  (2.1)

We shall also assume without loss of generality that ∥T ∥ = 1.

Operators not fixing a copy of C(S) 3

We shall decompose the μ_{s} into their atomic and diffuse parts. It is instrumental for our argument that this can be accomplished in a measurable fashion, as was proved by Kalton [5, Th. 2.10]. More precisely we can write

μ_{s} = ∑_{n=1}^{∞} a_{n}(s)δ_{σ_{n}(s)} + ν_{s}

where

• each a_{n}: S → 𝕂 is measurable for the completion Σ_{λ} of the Borel sets of S with respect to λ,

• each σ_{n}: S → S is Σ_{λ}-Borel-measurable,

• each ν_{s} is diffuse, and s ↦ |ν_{s}| is Σ_{λ}-Borel-measurable (we are considering the weak^{∗} Borel sets of the unit ball of C(S)^{∗}),

• |a_{1}(s)| ≥ |a_{2}(s)| ≥ … for all s ∈ S,

• σ_{n}(s) ≠ σ_{m}(s) for all s ∈ S whenever n ≠ m,

• ∑_{n=1}^{∞} |a_{n}(s)| ≤ 1 for all s ∈ S.

Put β = λ(U) > 0. Applying the Egorov and Lusin theorems, we may find a compact subset S_{1} ⊂ S with λ(S_{1}) ≥ 1 − β/2 such that

• each a_{n}|_{S_{1}} is continuous on S_{1},
• each σ_{n}|_{S_{1}} is continuous on S_{1},
• ∑ |a_{n}(s)| converges uniformly on S_{1},

• lim_{n→∞} sup_{s∈S_{1}} |ν_{s}|(B(t, 1/n)) → 0 for all t ∈ S; (2.2)

here B(t, r) denotes the closed ball with centre t and radius r. To prove (2.2) fix a countable dense subset Q of S. Let U_{n} be the countable set of all finite covers of S by means of open balls of radius 1/n with centres in Q. Then

φ_{n}(s) = inf_{C∈U_{n}} sup_{B∈C} |ν_{s}|(B)

defines a sequence of Σ_{λ}-measurable functions, and we have φ_{n} → 0 pointwise, since the ν_{s} are diffuse and S is compact. By Egorov’s theorem (φ_{n}) tends to 0 uniformly on a large subset of S. This easily implies our claim.

After discarding relatively isolated points of S_{1}, if necessary, we come up with a perfect subset S_{1} with the above properties; note that U ∩ S_{1} ≠ ∅ since this intersection has positive λ-measure. Next we pick N ∈ ℕ such that

∑_{n>N} |a_{n}(s)| ≤ α/3  ∀s ∈ S_{1}.  (2.3)

Let s ∈ U ∩ S_{1}. Then

α ≤ |μ_{s}({s})| = |∑_{n=1}^{∞} a_{n}(s)δ_{σ_{n}(s)}({s})| .

Consequently σ_{k}(s) = s for some k; in addition, we must have

|a_{k}(s)| ≥ α  (2.4)

for this k so that k ≤ N. This shows that

U ∩ S_{1} = ⋃_{l=1}^{N}{s ∈ U ∩ S_{1}: σ_{l}(s) = s}.

Each of these N sets is relatively closed in U ∩ S_{1}, so one of them, with index k say, contains a relatively interior point s_{0}. Therefore, there exists some δ > 0 with the following properties:

For B_{l} = σ_{l}(S_{1} ∩ B(s_{0}, δ)), l = 1, …, N, we have

(a) B_{k} = S_{1} ∩ B(s_{0}, δ) ⊂ S_{1} ∩ U, and σ_{k}(s) = s for all s ∈ B_{k},

(b) B_{l} ∩ B(s_{0}, δ) = ∅ for l ≠ k (since s_{0} = σ_{k}(s_{0}) ≠ σ_{l}(s_{0}) and these functions are continuous on S_{1}),

(c) |ν_{s}|(B(s_{0}, δ)) ≤ α/3 for all s ∈ S_{1} (see (2.2)).

Now let V = S_{1} ∩ B(s_{0}, δ/2). This is a compact perfect subset of S and thus C(V) is isomorphic to C(S) by Milutin’s theorem [12, Th. III.D.19]. Also let W = {s ∈ S: d(s, s_{0}) ≥ δ}. Again, this is a compact subset of S, V ∩ W = ∅, and we have from (a) and (b) above

(a′) σ_{k}(s) = s for all s ∈ V, (b′) B_{l} ⊂ W for all l = 1, …, N, l ≠ k.

By the Borsuk-Dugundji theorem (see e.g. [12, Cor. III.D.17]) there exists an isometric linear extension operator L: C(V ∪ W) → C(S). Let E = {f ∈ C(V ∪ W): f|_{W} = 0} and F = L(E); thus F ≅ E ≅ C(V), and F is isomorphic to C(S).

We finally prove that T |_{F} is an isomorphism. In fact, let us show that

∥T f∥ ≥ \frac{α}{3} ∥f∥ ∀f ∈ F.

Suppose f ∈ F with ∥f∥ = 1. Then ∥f|_{V} ∥ = 1 and f|_{W} = 0. Pick s ∈ V with |f(s)| = 1. Then we have (recall that V ⊂ S_{1} ∩ U)

∥T f∥ ≥ |T f(s)|
= |∑_{n=1}^{∞} a_{n}(s)f(σ_{n}(s)) + ∫_{S} f dν_{s}|
≥ |a_{k}(s)| − ∑_{n>N} |a_{n}(s)| − ∫_{S} |f| d|ν_{s}|
(by (a′) and (b′))
≥ α − \frac{α}{3} − |ν_{s}|(B(s_{0}, δ))
(by (2.4), (2.3) and since f|_{W} = 0)
≥ \frac{α}{3}  (by (c)).

This completes the proof of the theorem. □

3. Operators whose adjoints have separable ranges

In this section we discuss another, formally shorter, approach to Theorem 1.1 via the following proposition.

Proposition 3.1 Suppose S is a compact Hausdorff space without isolated points. If T: C(S) → C(S) is a bounded linear operator such that ran(T^{∗}) is separable, then T satisfies the Daugavet equation.

Proof. Let (μ_{s})_{s∈S} be the representing kernel of T, and let {μ_{s_{n}}: n ∈ ℕ} be dense in {μ_{s}: s ∈ S}. We define μ = ∑ 2^{−n}|μ_{s_{n}}|; then all the μ_{s} are absolutely continuous with respect to μ. Hence μ_{s}({s}) ≠ 0 only if |μ|({s}) ≠ 0. Therefore, {s ∈ S: μ_{s}({s}) ≠ 0} is countable, and, since S is perfect, its complement is dense by Baire’s theorem. This implies that the condition of Lemma 1.2 is satisfied, and the proposition is proved. □

Proposition 3.1 obviously covers the case of compact operators, operators factoring through a space with a separable dual, and, in the case of metric S, weakly compact operators; for the latter observe that T^{∗∗}(C(S)^{∗∗}) ⊂ C(S) so that T^{∗∗} and, consequently, T^{∗} have separable ranges. But even for nonmetrizable perfect S the Daugavet equation for weakly compact operators on C(S) can be derived from the argument of Proposition 3.1; one only has to recall that by a theorem of Bartle, Dunford and Schwartz [3, p. 306] also in this case all the μ_{s} are absolutely continuous with respect to some finite measure μ. This seems to yield the easiest proof of the Daugavet equation for these classes of operators.

If S is a metric space then Proposition 3.1 clearly follows from Theorem 1.1. On the other hand, Rosenthal [8] (see also [9]) has shown that an operator T: C(S) → C(S), S an uncountable compact metric space, with ran(T^{∗}) nonseparable fixes a copy of C(S). Therefore Proposition 3.1 implies Theorem 1.1 so that both these results are in fact equivalent. Moreover we note that in this setting T does not fix a copy of C(S) if and only if T does not fix a copy of ℓ^{1} if and only if T is weakly sequentially precompact, i.e., if (f_{n}) is a bounded sequence in C(S), then (T f_{n}) admits a weak Cauchy subsequence.

We finally extend Proposition 3.1 to the class of nicely embedded Banach spaces introduced in [11] so that we obtain a unified approach to several results of that paper.

Let S be a Hausdorff topological space, and let C^{b}(S) be the sup-normed Banach space of all bounded continuous scalar-valued functions. The functional f ↦ f(s) on C^{b}(S) is denoted by δ_{s}. We say that a linear map J: X → C^{b}(S) on a Banach space X is a nice embedding and that X is nicely embedded into C^{b}(S) if J is an isometry such that for all s ∈ S the following properties hold:

(N1) For p_{s} := J^{∗}(δ_{s}) ∈ X^{∗} we have ∥p_{s}∥ = 1.

(N2) lin{p_{s}} is an L-summand in X^{∗}. The latter condition means that there are projections Π_{s} from X^{∗} onto lin{p_{s}} such that

∥x^{∗}∥ = ∥Π_{s}(x^{∗})∥ + ∥x^{∗} − Π_{s}(x^{∗})∥ ∀x^{∗} ∈ X^{∗}.

We will also need the equivalence relation

s ∼ t if and only if Π_{s} = Π_{t}

on S. Then s and t are equivalent if and only if p_{s} and p_{t} are linearly dependent, which implies by (N1) that p_{t} = λp_{s} for some scalar of modulus 1. The equivalence classes of this relation are obviously closed.

We will consider the following nondiscreteness condition.

(N3) None of the equivalence classes Q_{s} = {t ∈ S: s ∼ t} contains an interior point.

If the set {p_{s}: s ∈ S} is linearly independent, this simply means: (N3′) S does not contain an isolated point. By (N2), the p_{s} are linearly independent as soon as they are pairwise linearly independent.

Proposition 3.2 Let S be a Baire topological Hausdorff space and suppose that X is nicely embedded into C^{b}(S) so that additionally (N3) holds. If T: X → X is a bounded linear operator with ran(T^{∗}) separable, then T satisfies the Daugavet equation.

Proof. We stick to the above notation and put q_{s} = T^{∗}(p_{s}). By [11, Prop. 2.1] it is sufficient to prove that

S′ = {t ∈ S: Π_{t}(q_{s}) = 0 ∀s}

is dense in S. If {q_{s_{n}}: n ∈ ℕ} denotes a dense subset of {q_{s}: s ∈ S}, then t ∉ S′ if and only if Π_{t}(q_{s_{n}}) ≠ 0 for some n. This implies that S \ S′ consists of at most countably many equivalence classes for the equivalence relation ∼ (cf. [11, Lemma 2.3]), and by (N3) and Baire’s theorem S′ must be dense.

□

It is proved in [11] that the following classes of Banach spaces satisfy the assumptions of Proposition 3.2:

• X is a function algebra whose Choquet boundary does not contain isolated points,

• X is a real L^{1}-predual space so that the set ex B_{X^{∗}} of extreme functionals does not contain isolated points (for the weak^{∗} topology),

• X is a complex L^{1}-predual space for which the quotient space ex B_{X^{∗}}/∼ does not contain isolated points, where ∼ means linear dependence,

• X is a space of type C_{Λ} for certain subsets Λ of abelian discrete groups.

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Mathematisches Institut I, Universität Karlsruhe, D-76 128 Karlsruhe, Germany; e-mail: Lutz.Weis@math.uni-karlsruhe.de I. Mathematisches Institut, Freie Universität Berlin, Arnimallee 2–6, D-14 195 Berlin, Germany; e-mail: werner@math.fu-berlin.de