Operators on C(ω

153 (1997)

Operators on C(ω^α) which do not preserve C(ω^α)

by

Dale E. A l s p a c h (Stillwater, Okla.)

Abstract. It is shown that if α, ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from C(ω^ωα) onto itself such that if Y is a subspace of C(ω^ωα) which is isomorphic to C(ω^ωα), then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from C(ω^ωα) onto itself there is a subspace of C(ω^ωα) which is isomorphic to C(ω^ωα) on which the operator is an isomorphism.

In an earlier paper [A2] we proved that there is an operator on C(ω^ω2) which is not an isomorphism on any subspace which is isomorphic to C(ω^ω2) but the operator is onto C(ω^ω2). This is in contrast with the situation for C(ω) and C(ω^ω) where there are no surjective operators which do not pre- serve isomorphically a copy of the space, [P], [A1]. Bourgain [B] proved a very general result which gives an estimate on the size of the ordinal β such that any operator on C(ω^ωα) which is surjective must be an isomorphism on a subspace isomorphic to C(ω^ωβ). Recently Gasparis [G], [G1] has gen- eralized the example in [A2] to the case of operators on C(ω^ωα+1) to show that there are surjective operators on these spaces which do not preserve a copy of C(ω^ωα+1). For most ordinals α this is very far from the estimate given by Bourgain.

Bourgain used the Szlenk index and a combinatorial argument in the proof of his result. Implicit in his proof is the notion of γ-families of sets which was independently developed by Wolfe [W], and in [A2]. The existence of γ-families with associated measures is an indication of the amount of topological disjointness in a subset of C(K)^∗ whereas the Szlenk index only indicates disjointness. Bourgain essentially shows that a large Szlenk index forces the existence of a γ-family of sets with the size of γ dependent on

1991 Mathematics Subject Classification: Primary 46B03; Secondary 06A07, 03E10.

Key words and phrases: ordinal index, Szlenk index, Banach space of continuous func- tions.

[81]

the Szlenk index. The existence of a γ-family is equivalent to a condition on an ordinal index which we have named the Wolfe index. Thus from this view point Bourgain proves that the Szlenk index gives some lower bound on the Wolfe index. In some cases he infers that the two indices are of roughly equivalent size. In this paper we give a very general construction of examples of the type in [A2] and [G] and show that there are many more ordinals for which the Szlenk and Wolfe index are very different.

We will use notation similar to that in [A2]. In particular, if γ is an ordinal, C(γ) is the space of continuous functions on the ordinals less than or equal to γ with the order topology, which we denote by [1, γ]. If K is a topological space, K^(β) is the β-derived set of K. If L ⊂ C(K)^∗, then L^(β) will be the β-derived set of L with respect to the w^∗-topology. If K is a countable compact Hausdorff space, then K is homeomorphic to [1, ω^βn], where the cardinality of K^(β) is n, for some n ∈ N (cf. [MS]). It was shown in [BP] that C(ω^ωα) is isomorphic to C(ω^βn) if and only if ω^α≤ β < ω^α+1. Thus from the point of view of the isomorphic theory of Banach spaces, the spaces C(ω^ωα), α < ω₁, are a complete set of representatives of the C(K)-spaces for C(K) separable and K countable.

1. A topological construction. In order to define the operator we need to develop a method of constructing special sets of measures on ω^ωα which are homeomorphic to ω^ωα but which have supports which are almost disjoint but are not topologically well separated. In [A2] we used the porcupine topology, [BD], to effect the construction. Here we use a similar construction but with somewhat different notation. The operators that we construct are of the same form as that in our earlier work. Namely, we produce a compact Hausdorff space K and a w^∗-closed subset L of C(K)^∗ and we define an operator from C(K) into C(L) by evaluation. In this paper we need to iterate the construction of [A2]. To this end we introduce a general procedure for extending a pair (K, L) by a sequence of spaces K_n, where K and K_n are compact Hausdorff spaces, each K_n has a distinguished point k_n,0 and L is a set of purely atomic finitely supported probability measures on K.

For each k ∈ K, we let L(k) = {l ∈ L : l(k) 6= 0} ∪ {∅} and S(k, L) be the one point compactification of P

n

P

l∈L(k)K_n \ {k_n,0}, where we use P

i∈IW_i to denote the disjoint sum of topological spaces W_i with the topology generated by sets of the formS

i∈IGi with Gi open for each i ∈ I.

We denote the points of S(k, L) as 4-tuples (k, l, n, j) where l ∈ L(k) and j ∈ K_n. The point added will be denoted by (k, ∅) although it is also (k, l, n, k_n,0) for any l ∈ L(k) and n. Note that if L(k) = {∅}, then S(k, L(k)) = {(k, ∅)}.

We want to define a topology on the disjoint union of the sets S(k, L).

Intuitively, we want to glue S(k, L) to K at the point k by identifying (k, ∅) with k. We also want to extend the measures L by sets of measures L_n on

Kn and make a copy of Ln for each l ∈ L(k), n ∈ N. More formally, we make the following definition.

Definition 1.1. Suppose that K and K_n, n ∈ N, are compact Hausdorff spaces, Kn has a distinguished point kn,0 and L, Ln are sets of purely atomic disjointly supported probability measures on K, K_n, respectively, with δ_kn,0 ∈ L_n, for each n. Define (K, L) ⊗ {(K_n, L_n) : n ∈ N} to be the pair (K⁰, L⁰) where K⁰ is the compact Hausdorff space and L⁰ is the set of atomic probability measures on K⁰ described below. K⁰ is the set of 4-tuples (k, l, n, jn), k ∈ K, l ∈ L(k), n ∈ N and jn ∈ Kn, with the topology generated by sets of the form

[

k∈K

G_k∪ [

k∈G

{(k, l, n, j_n) : k ∈ K, l ∈ L(k), n ∈ N, j_n∈ K_n}

\ [

(k,l,n)∈F

{k} × {l} × {n} × F_k,l,n,

where Gk is an open subset of

{(k, l, n, j_n) : l ∈ L(k), n ∈ N, j_n∈ K_n\ {k_n,0}} = S(k, L) \ {(k, ∅)}

for each k, G is an open set in K, F is a finite set of triples (k, l, n) with k ∈ K, n ∈ N and l ∈ L(k), and F_k,l,nis a compact subset of K_n\{k_n,0}. For each k ∈ K we identify all of the points (k, l, n, k_n,0) such that l ∈ L(k), n ∈ N with the point (k, ∅). (Formally, K⁰is a set of equivalence classes of 4-tuples, but only the elements with fourth entry k_n,0 are in non-trivial classes.) Let φ be the map from K into K⁰defined by φ(k) = (k, ∅) and let Φ be the map from M(K) into M(K⁰) which is induced by φ. Let

L⁰=n X

k∈supp l

l(k) X

jn∈Kn

l_n(j_n)δ_(k,l,n,jn): l ∈ L, n ∈ N, l_n ∈ L_n o

.

In keeping with our identification, X

k∈supp l

l(k)δ_(k,l,n,kn,0) = X

k∈supp l

l(k)δ_(k,∅) = Φ(l)

for each l ∈ L, n ∈ N, and Φ(l) ∈ L⁰ because X

k∈supp l

l(k) X

jn∈Kn

δ_kn,0(j_n)δ_(k,l,n,jn) = X

k∈supp l

l(k)δ_(k,l,n,kn,0)

and we have assumed that δk_n,0 ∈ Ln, for each n.

The next lemma lists some properties of the construction.

Lemma 1.1. Suppose that K and K_n, n ∈ N, are compact Hausdorff spaces, Kn has a distinguished point kn,0 and L, Ln, n ∈ N, are sets of purely atomic finitely supported probability measures on K, K_n, respectively,

with δk_n,0 ∈ Ln, for each n, as above. Then if (K⁰, L⁰) = (K, L)⊗{(Kn, Ln) : n ∈ N},

(1) K⁰ is a compact Hausdorff space and φ is a homeomorphism of K into K⁰.

(2) A net (k_d, l_d, n_d, j_d)_d∈D in K⁰\φ(K) converges to (k, l, n, j) for some j 6= kn,0 if and only if there exists d0 ∈ D such that kd = k, ld = l, and n = n_d for all d ≥ d₀ and (j_d)_d∈D converges to j.

(3) A net (k_d, l_d, n_d, j_d)_d∈D in K⁰ \ φ(K) converges to (k, l, n, k_n,0) = (k, ∅) if and only if the following hold:

(a) (k_d)_d∈D converges to k.

(b) If D₁ = {d : k_d = k} is cofinal in D, then for each l and n, D1,l,n = {d ∈ D1 : ld = l, nd = n} is not cofinal in D or (j_d⁰)_d∈D converges to k_n,0, where j_d⁰ = j_d if d ∈ D_1,l,n and j⁰_d= k_n,0 otherwise.

(4) The map

l → Φ(l) = X

k∈supp l

l(k)δ_(k,∅)

for l ∈ L is a homeomorphism of L into L⁰ in the weak^∗ topology.

(5) Each l⁰∈ L⁰ is atomic and has finite support.

(6) If L, L_n, n ∈ N, are compact in the weak^∗ topology, then L⁰ is compact in the weak^∗ topology.

(7) If (ld) is a convergent net in Ln with limit l0 and l ∈ L, then

 X

k∈supp l

l(k) X

jn∈Kn

l_d(j_n)δ_(k,l,n,jn)



d

converges to

X

k∈supp l

l(k) X

jn∈Kn

l₀(j_n)δ_(k,l,n,jn)

for each l ∈ L.

P r o o f. We have given a basis for the topology on K⁰ in the definition above. In order to verify the first property we first observe that {(k, ∅) : k ∈ K} is homeomorphic to K. Notice that the basis for the topology of K⁰ given in the definition above defines the topology on {(k, ∅) : k ∈ K} to be the topology {φ(G) : G is open in K}. Thus φ is a homeomorphism. If O is an open cover of K⁰ by basic open sets, then there is a finite subset O⁰ of O which covers φ(K). K⁰\S

{Gi : Gi ∈ O⁰} is contained in a finite union of closed subsets of the form {k} × {l} × {n} × F_k,l,n, where F_k,l,n is a compact subset of Kn\ {kn,0}. The topology on {k} × {l} × {n} × Fk,l,n is the topology induced by identifying this with F_k,l,nin K_n. Therefore a finite

number of additional sets from O will cover each {k} × {l} × {n} × Fk,l,n. This proves the first assertion.

For the second, notice that if G is an open set contained in K_n\ {k_n,0} and j ∈ G, then {k} × {l} × {n} × G is an open neighborhood of (k, l, n, j).

Thus the net must eventually be in {k} × {l} × {n} × G, and (2) follows.

Define a map ζ from K⁰ onto {(k, ∅) : k ∈ K} by ζ(k, l, n, j) = k. Clearly, ζ is continuous and this gives (3)(a), if the net converges. If D1= {d : kd= k}

were cofinal in D, D_1,l,n were cofinal in D and (j_d⁰)_d∈D had a convergent subnet with limit j 6= k_n,0, then there would be an open set G containing j and contained in K \ {kn,0}. However, {k} × {l} × {n} × G would be an open set in K⁰ containing (k, l, n, j) and thus (k, l, n, j_d⁰)_d∈D would converge to (k, l, n, j), which is impossible. Thus (3)(b) holds. Conversely, if we are given a net satisfying (3)(a) and (b) and G⁰is an open set containing (k, l, n, kn₀), then G⁰ contains a neighborhood of (k, l, n, k_n,0) of the form

H = [

k⁰∈G

{(k⁰, l⁰, n⁰, j_n0) : k⁰∈ K, l⁰∈ L(k), n⁰∈ N, j_n0 ∈ K_n0}

\ [

(k⁰,l⁰,n⁰)∈F

{k⁰} × {l⁰} × {n⁰} × F_k0,l⁰,n⁰.

Because (k_d)_d∈D converges to k, there is a d₀∈ D such that (k_d, l_d, n_d, j_d) ∈ H ∪ S

(k⁰,l⁰,n⁰)∈F{k⁰} × {l⁰} × {n⁰} × Fk⁰,l⁰,n⁰ for all d ≥ d0. Because F is finite, we may assume, by choosing another d₀ and passing to a subset of G if necessary, that F = {(k, l⁰, n⁰) : (l⁰, n⁰) ∈ F⁰} for some finite set F⁰. By (b) we know that for each (l⁰, n⁰) ∈ F⁰ there is a d_l0,n⁰ such that if (k_d, l_d, n_d, j_d) = (k, l⁰, n⁰, j_d) and d ≥ d_l⁰_,n⁰, then (k_d, l_d, n_d, j_d) 6∈ {k} × {l⁰} × {n⁰} × Fk,l⁰,n⁰. If d ≥ dl⁰,n⁰, for all (l⁰, n⁰) ∈ F⁰, and d ≥ d0, then (k_d, l_d, n_d, j_d) ∈ H.

Because φ is a homeomorphism (4) is immediate. (5) is obvious from the definition and the fact that (k, l, n, j) = (k⁰, l⁰, n⁰, j⁰) if and only if k = k⁰, l = l⁰, n = n⁰ and j = j⁰, or j = j⁰ = k_n,0 and k = k⁰. To see that L⁰ is compact if L, L_n, n ∈ N, are, let (l_d⁰)_d∈D be a net in L⁰, where

l⁰_d= X

k∈supp l_d

X

j∈K_n(d)

l_d(k)l_n(d)⁰⁰ (j)δ_{(k,ld,n(d),j)}

for each d ∈ D. Because L and the Lnare compact, we may assume by pass- ing to a subnet that the nets (l_d)_d∈D and (l⁰⁰_n(d))_d∈D converge to l and l⁰⁰, respectively. Here we are thinking of (l⁰⁰_n(d)) as a net in S

n∈NΦ_n(L_n), where Φ_n is the map induced by the natural embedding φ_n of K_n into the one point compactification of S

n∈NKn\ {kn,0}. Because Φ is w^∗-continuous, (Φ(l_d))_d∈D converges to Φ(l). If ε > 0 and k ∈ supp l, let G_k be an open set containing k and such that l(Gk) < l(k) + ε. We may assume that the sets G_k are disjoint. We must consider two cases. First suppose that (l_d)

has a constant subnet. Then l_d⁰ =P

k∈supp l

P

j∈K_n(d)l(k)l_n(d)⁰⁰ (j)δ_{(k,ld,n(d),j)} for the elements in the subnet and the limit of the subnet is P

k∈supp ll(k)P

n∈N,j∈K_nl⁰⁰(j)δ_(k,l,n,j). If there is no constant subnet, then any convergent subnet of points (k_d, l_d, n_d, j_d)_d∈D will have a limit of the form (k, ∅), where k is the limit of the first coordinates in the subnet, by (2) and (3). We claim that there is a convergent subnet with limit P

k∈supp ll(k)δ_(k,∅). Indeed, because (l_d) converges to l, there exists a d₀ such that ld(Gk) > l(Gk) − ε for all k ∈ supp l, d ≥ d0. This implies that

l⁰_d [

r∈G_k

{(r, m, n, t) : r ∈ K, m ∈ L(k), n ∈ N, t ∈ K_n}

 [

(r,m,n)∈F

{r} × {m} × {n} × Fr,m,n



> l(G_k) − ε − X

(r,m,n)∈F

l⁰_d({r} × {m} × {n} × F_r,m,n).

Notice that l⁰_d({r} × {m} × {n} × Fr,m,n) = 0 if ld 6= m or r 6∈ supp ld. Because F is finite, and we have assumed that there is no constant subnet of (ld), by choosing another d1 ≥ d0 we will have l_d⁰({r} × {m} × {n} × Fr,m,n) = 0 for all (r, m, n) ∈ F and all d ≥ d1. Because l⁰_d is a probability measure and ε > 0 is arbitrary, (l_d⁰) converges to Φ(l) =P

k∈supp ll(k)δ_(k,∅). Thus L⁰ is compact.

(7) is immediate from the definition.

Our next lemma will allow us to compute topological information about the spaces K⁰ and L⁰ from the component pieces provided the pieces are properly attached.

Lemma 1.2. Let K, L, Kn, Ln, n ∈ N, K⁰ and L⁰ be as in the previous lemma. In addition assume that K is homeomorphic to [1, ω^ωαm], K_n is homeomorphic to [1, ω^ωβ(n)m(n)], L (with the w^∗-topology) is homeomorphic to [1, ω^ωγp], and Ln is homeomorphic to [1, ω^ωγ(n)p(n)]. Moreover , assume that

K^(ωαm)= {k₀}, L^(ωγp)= {δ_k0}, K_n^{(ωβ(n)m(n))} = {k_n,0}, L^(ω_n^γ(n)p(n))= {δ_kn,0} for all n. Let

ω^BM = sup{ω^β(n)m(n) : n ∈ N} and ω^ΓP = sup{ω^γ(n)p(n) : n ∈ N}.

Then

(1) K^{0(ωBM )} ⊂ φ(K) and if S

{supp l : l ∈ L} = K, then K⁰ is homeo- morphic to [1, ω^{ωBM +ωαm}] and K^{0(ωBM +ωαm)}= {φ(k₀)}.

(2) L^{0(ωΓP )} = Φ(L), L⁰ (with the w^∗-topology) is homeomorphic to [1, ω^{ωΓP +ωγp}] and L^{0(ωΓP +ωγp)}= {δ_φ(k0)}.

(3) If for each l ∈ L, there is a subset H_l of K such that l(H_l) ≥ ε, and (Hl)l∈L are disjoint, and for each n ∈ N, l⁰⁰ ∈ Ln, there is a subset H_n,l⁰⁰ of K_n \ {k_n,0} such that l⁰⁰(H_n,l⁰⁰) ≥ ε and (H_n,l⁰⁰)_l⁰⁰_∈Ln are disjoint for each n, then there are disjoint subsets H_l⁰0 of K⁰ for each l⁰ ∈ L⁰ such that l⁰(H_l⁰0) ≥ ε. Moreover , if l ∈ L, then we can define H_Φ(l)⁰ = φ(Hl) and if

l⁰= X

k∈supp l

l(k) X

j_n∈K_n

ln(jn)δ_(k,l,n,jn)

for some l ∈ L, n ∈ N, l_n∈ L_n\ {δ_kn,0}, then we can define H_l⁰0 = [

k∈supp l

{k} × {l} × {n} × H_ln.

P r o o f. First observe that because K, L, and K_n are countable and the measures in L are finitely supported, K⁰ is countable. If n ∈ N, j ∈ Kn and j 6= k_n,0, then for any k ∈ K, l ∈ L, with l(k) 6= 0, {(k, l, n, j⁰) : j⁰ 6= k_n,0} is an open neighborhood of (k, l, n, j) in K⁰ homeomorphic to Kn\ {kn,0}.

Thus (k, l, n, j) is in the same derived sets of K⁰ as of K_n. In particular, Kn^{(ωβ(n)m(n))}= {kn,0} and thus K^{0(ωBM )} ⊂ φ(K). IfS

{supp l : l ∈ L} = K, then for each k ∈ K, {(k, l, n, j) : j ∈ K_n} ⊂ K⁰for some l ∈ L and therefore (k, l, n, kn,0) ∈ K^{0(ωβM )}. If k is an isolated point in K, then φ(k) is the limit only of sequences which are eventually in

{(k, l, n, j) : l ∈ L, l(k) 6= 0, n ∈ N, j ∈ K_n}.

Hence (k, ∅) = (k, l, n, kn,0) 6∈ K^{0(ωBM +1)}. Because φ is a homeomorphism, it follows that K^{0(ωBM )}= φ(K⁽⁰⁾). Similarly, K^{0(ωBM +%)}= φ(K^(%)) for all %.

In particular, K^{0(ωBM +ωαm)}= {φ(k₀)}.

Observe that it follows from Lemma 1.1 that L⁰is countable and compact because K, K_n, L, and L_nare, and thus it is sufficient to consider the derived sets. If l ∈ L, n ∈ N, then {P

k∈supp ll(k)P

j∈supp lnln(j)δ_(k,l,n,j) : ln ∈ Ln} is homeomorphic (by the obvious map) to L_n and

n X

k∈supp l

l(k) X

j∈supp ln

l_n(j)δ_(k,l,n,j) : l_n ∈ L_n

o_(ω^γ(n)_p(n))

= {Φ(l)}.

Therefore Φ(L) ⊂T

n∈NL^{0(ωγ(n)p(n))} = L^{0(ωΓP )}. If l_n ∈ L_n\ {δ_kn,0}, then X

k∈supp l

l(k) X

j∈supp ln

l_n(j)δ_(k,l,n,j)− X

k∈supp l

l(k)l_n(k_n,0)δ_k,∅

is non-zero and is supported in the open set S

k∈supp l{(k, l, n, j) : j ∈

Kn\ {kn,0}} and only elements of L⁰ of the form X

k∈supp l

l(k) X

j∈supp l⁰_n

l⁰_n(j)δ_(k,l,n,j)

with l_n⁰ ∈ L_n\ {δ_kn,0} are supported in this set. Therefore Φ(L) = L^{0(ωΓP )}. Because Φ is a homeomorphism, it follows that Φ(L)^(%) = L^{0(ωBP +%)} for all %, proving the second assertion.

For each l⁰ 6∈ Φ(L), we have defined H_l⁰0 to be a subset of K⁰\ φ(K).

These sets are clearly disjoint. Also if l⁰= X

k∈supp l

l(k) X

j∈supp l_n

l_n(j)δ_(k,l,n,j),

then

l⁰(H_l⁰) = X

k∈supp l

l(k)l_n(H_ln) = l_n(H_ln).

Because Φ is induced by the homeomorphism φ, it follows that H_Φ(l)⁰ = φ(Hl), l ∈ L, is a family of disjoint subsets of φ(K) with Φ(l)(H_Φ(l)⁰ ) = l(Hl).

In order to prove that the evaluation map from C(K) into C(L) is sur- jective we will need to show that L is equivalent to the usual unit vector basis of l₁. The elements of L are not perturbations of disjointly supported elements and thus the proof uses some special properties of the construction.

We introduce a natural ordering on the elements of L which reflects these properties of the construction.

Definition 1.2. Suppose M is a family of measures on a measurable space (Ω, B) and for each µ ∈ M there is a set Hµ∈ B such that Hµ∩ Hµ⁰

= ∅ if µ 6= µ⁰, and µ(H_µ) 6= 0. Then µ ⁰ µ⁰ if and only if there is a scalar a ∈ (0, |µ(H_µ)/(2µ⁰(H_µ⁰))|] such that µ_|^S_{Hµ0:µ06=µ} = aµ⁰_|^S_{H

µ0:µ⁰6=µ} and

|µ⁰|(Hµ) = 0. Define µ  ν if and only if there is a finite sequence (µi) in M such that µ = µ₀⁰µ₁⁰. . . ⁰µ_k= ν.

Notice that µ  µ is impossible and the relation  is transitive by definition. Thus we can define a partial order on M by µ  µ⁰ if and only if µ = µ⁰ or µ  µ⁰. Although the relation is really on the pairs (µ, Hµ), we will write it as though it were on the measures. This will not present any difficulty because the sets H_µ will be fixed during the construction.

The relation above occurs naturally in the construction of the pairs (K, L). For (K⁰, L⁰) = (K, L) ⊗ {(Kn, Ln) : n ∈ N} as in Lemma 1.1, each l⁰ ∈ L⁰ which is of the formP

k∈supp ll(k)P

j_n∈K_nl_n(j_n)δ_(k,l,n,jn) for some l ∈ L, n ∈ N, ln ∈ Ln, satisfies l⁰_|K0\(supp l)×{l}×{n}×K_n = ln(kn,0)l. If we

have the sets (H_l⁰0)l⁰∈L⁰, defined as in Lemma 1.2(3), and for l⁰⁰ ∈ Ln, we have supp l⁰⁰⊆ H_n,l⁰⁰ ∪ {k_n,0}, then l^00l.

The next lemma is similar to Proposition IV.13 of [G] or Proposition 4.4 in [G1]. It will be used to show that the sets of measures L that we construct actually are equivalent to the basis of l₁.

Lemma 1.3. Suppose that M is a set of mutually singular probability measures on a measurable space (K, B), ε > 0, and that (µ_n) is a sequence of (finite) convex combinations of the measures in M and (A_n) is a sequence of disjoint measurable sets. Let ⁰ and  be defined as above for M ={µn} and H_µn = A_n. Suppose that (µ_n, A_n)^∞_n=1 satisfy the following:

(1) For each n ∈ N, µ_n(A_n) ≥ ε.

(2) For each n ∈ N, either there is a unique n⁰∈ N such that µn⁰µn⁰

or for all n⁰ 6= n, µ_n(A_n⁰) = 0.

(3) For all n 6= m if it is not the case that µn µm, then µn(Am) = 0.

Then kP

cnµnk ≥ (2ε/3)P

|cn| for any sequence of scalars (cn).

P r o o f. Because we only use information about the measures on the sets A_n, without loss of generality we may assume that µ_n ∈ co{m ∈ M : m(Ak) > 0, for some k}∪{0}. The relation  defines a partial order on {µn}.

Observe that if µ ⁰ ν and µ = P

j∈Fb^µ_jm_j and ν = P

j∈Gb^ν_jm_j, where mj ∈ M and b^µ_j, b^ν_j are non-zero for all j, then F ⊃ G. Therefore, since each µ_n is a finite convex combination, for any n(0) ∈ N there is a unique finite maximal sequence (µ_n(i))^k_i=0 such that µ_n(0) ⁰ µ_n(1) ⁰ . . . ⁰ µ_n(k). Let (cn) be a finite sequence of scalars and let

F = {n : ∃n⁰ such that c_n0 6= 0 and µ_n0  µ_n}.

Clearly, F is a finite set. Partition F into sets (F_j)^J_j=0 such that for each j < J and n ∈ F_j there is an n⁰ ∈ F_j+1 such that µ_n ⁰ µ_n⁰ and for all n⁰6= n, n⁰∈ Fj, µn⁰ and µn are incomparable. If µn⁰µn⁰, let an,n⁰ denote the scalar such that µ_n|^S_{Ak:k6=n}= a_n,n⁰µ_n0|S

{A_k:k6=n}. For notational con- venience, let an,n⁰ = 0 if it is not the case that µn ⁰ µn⁰. A simple induction argument using (2) and (3) shows that

X

c_nµ_n =

XJ

j=0

X

n(j)∈Fj

c_n(j)µ_n(j)

=

XJ

j=0

X

n(j)∈F_j

c_n(j)µ_n(j)|^S_{A

n:µ_n(j)µn}

=

XJ

j=0

X

n(j)∈Fj

X

n:µ_nµ_n(j)

cnµ_n|An(j) .

Another induction argument and the definition of the scalars an,n⁰= 0 give the following inequality:

X

cnµn

≥

XJ

j=0

X

n(j)∈F_j

c_n(j)+ X

n(j−1)∈F_j−1

an(j−1),n(j)(c_n(j−1)

+ X

n(j−2)∈F_j−2

an(j−2),n(j−1)(c_n(j−2)+ . . .

+ X

n(0)∈F0

a_n(0),n(1)c_n(0))) 

µn(j)(A_n(j)).

Now we split off 1/3 of each term and shift the index on these pieces to combine with the subsequent related term:

XJ

j=0

X

n(j)∈Fj

c_n(j)+ X

n(j−1)∈Fj−1

an(j−1),n(j)

 c_n(j−1)

+ X

n(j−2)∈F_j−2

an(j−2),n(j−1)



c_n(j−2)+ . . .

+ X

n(0)∈F0

a_n(0),n(1)c_n(0)



µ_n(j)(A_n(j))

= XJ

j=0

X

n(j)∈F_j

2 3 

c_n(j)+ X

n(j−1)∈F_j−1

an(j−1),n(j)

 c_n(j−1)

+ X

n(j−2)∈Fj−2

an(j−2),n(j−1)



c_n(j−2)+ . . .

+ X

n(0)∈F0

a_n(0),n(1)c_n(0)

µ_n(j)(A_n(j))

+1 3

X

µ_n(j−1)⁰µ_n(j)

c_n(j−1)+ X

n(j−2)∈Fj−2

an(j−2),n(j−1)

 c_n(j−2)

+ X

n(j−3)∈Fj−3

an(j−3),n(j−2)



c_n(j−3)+ . . .

+ X

n(0)∈F0

a_n(0),n(1)c_n(0)



µ_n(j−1)(A_n(j−1))

 .

The condition µ_n(j−1) ⁰ µ_n(j) is equivalent to an(j−1),n(j) 6= 0 and by the definition of ⁰, 2an(j−1),n(j)µ_n(j)(A_n(j)) ≤ µ_n(j−1)(A_n(j−1)). Therefore

by the triangle inequality,

X cnµn

≥

XJ

j=0

X

n(j)∈Fj

2

3|c_n(j)|µ_n(j)(A_n(j)) ≥ XJ

j=0

X

n(j)∈Fj

2

3|c_n(j)|ε.

2. Construction of the operators. The aim of this section is to pro- duce pairs (K_α, L_α)_α<ω1 by transfinite induction so that K_α is a countable compact Hausdorff space and Lα is a w^∗-closed subset of the probability measures in C(K_α)^∗ which is equivalent to the basis of l₁.

Fix an ordinal ζ < ω1. Let ζn ↑ ω^ζ and for each n ∈ N let Sn= [1, ω^ζn] with the order topology and let T_n = {¹₂(δ_β + δ_ωζn) : β ≤ ω^ζn}. Let the distinguished point of S_n be ω^ζn. Let S₀= [1, 1] and T₀= {δ₁}. Define

(K1, L1) = (S0, T0) ⊗ {(Sn, Tn) : n ∈ N}.

It is easy to see that up to a homeomorphism of [1, ω^ωζ] we could have defined K₁ = [1, ω^ωζ] and L₁ = {¹₂(δ_β + δ_ωωζ) : β ≤ ω^ωζ}. We take the distinguished point k₁ of K₁ to be φ(1) where 1 ∈ S₀. Now suppose that we have defined Kγ and Lγ for all γ < α. Let kγ denote the distinguished point of K_γ. There are two cases. First assume that α = α⁰+ 1 for some α⁰. Define

(K_α, L_α) = (K₁, L₁) ⊗ (K_α0, L_α0).

Let the distinguished point be k_α= φ(k₁). (More formally we should have a sequence of spaces {(Kα_n, Lα_n) : n ∈ N} on the right of ⊗, but we can take (K_α1, L_α1) = (K_α⁰, L_α⁰) and (K_αn, L_αn) = (S₀, T₀) for n > 1. These spaces (S0, T0) have no effect on (Kα, Lα).) If α is a limit ordinal, let (αn) be an increasing sequence of ordinals with limit α. Let

(K_α, L_α) = (S₀, T₀) ⊗ {(K_αn, L_αn) : n ∈ N}.

Let φ(1), for 1 ∈ S₀, be the distinguished point of K_α. The definition for α a limit ordinal depends on the sequence (αn). However, the properties of the space are not dependent on the sequence and we will assume that whenever we use a sequence approaching α, it is the same one. This completes the definition of the pairs (Kα, Lα). Notice that we actually have such a trans- finite family of spaces for each ζ < ω₁. The choice of ζ will be made in the proof of Theorem 3.5.

Now we must consider the properties of these pairs. First we compute the topological information by using Lemma 1.2. As noted above, K1and L1are homeomorphic to [1, ω^ωζ]. Notice that we have the following relations. If K_α⁰ and Lα⁰ are homeomorphic to [1, ω^ωζβ], then Kα⁰+1 and Lα⁰+1 are homeo- morphic to [1, ω^ωζ(β+1)], by Lemma 1.2(1) and (2). If (αn) is an increasing sequence of ordinals with limit α and K_αn and L_αn are homeomorphic to