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ON THE COMPACT OPERATORS CASE OF THE

BISHOP–PHELPS–BOLLOB ´

AS PROPERTY FOR NUMERICAL

RADIUS

DOMINGO GARC´

IA, MANUEL MAESTRE, MIGUEL MART´

IN, AND ´

OSCAR ROLD ´

AN

Abstract. We study the Bishop-Phelps-Bollob´as property for numerical radius re-

stricted to the case of compact operators (BPBp-nu for compact operators in short).

We show that C0pLqspaces have the BPBp-nu for compact operators for every Haus-

dorﬀ topological locally compact space L. To this end, on the one hand, we provide

some techniques allowing to pass the BPBp-nu for compact operators from subspaces to

the whole space and, on the other hand, we prove some strong approximation property

of C0pLqspaces and their duals. Besides, we also show that real Hilbert spaces and

isometric preduals of `1have the BPBp-nu for compact operators.

1. Introduction, notation, and known results

First we ﬁx some notation in order to be able to describe our aims and results with

precision. Given a Banach space Xover the ﬁeld Kof real or complex numbers, we

denote by X˚,BX, and SX, its topological dual, its closed unit ball, and its unit sphere,

respectively. If Yis another Banach space, LpX, Y qrepresents the space of all bounded and

linear operators from Xto Y, and we denote by KpX, Y qthe space of compact operators

from Xto Y. When Y“X, we shall simply write LpXq “ LpX, X qand KpXq “ KpX, Xq.

Given a locally compact Hausdorﬀ topological space L,C0pLqis the Banach space of all

scalar-valued continuous functions on Lvanishing at inﬁnity.

Given an operator TPLpXq, its numerical radius is deﬁned as

νpTq:“sup t|x˚pTpxqq|:px, x˚q P ΠpXqu ,

where ΠpXq:“ tpx, x˚q P SXˆSX˚:x˚pxq “ 1u. It is immediate that νpTq ď }T}for

every TPLpXqand that νis a seminorm on LpXq. Very often, νis actually a norm

on LpXqequivalent to the usual operator norm. The numerical index of the space X

measures this fact and it is given by

npXq:“inftνpTq:TPLpXq,}T} “ 1u

“maxtkě0: k}T} ď νpTq,@TPLpXqu.

Date: February 21, 2021.

2020 Mathematics Subject Classiﬁcation. Primary: 46B04; Secondary: 46B20, 46B25, 46B28.

Key words and phrases. Banach space; compact operator; Bishop-Phelps-Bollob´as property; numerical

radius attaining operator; approximation property.

The ﬁrst and second authors were supported by MINECO and FEDER project MTM2017-83262-C2-

1-P and by Prometeo PROMETEO/2017/102. The third author was supported by projects PGC2018-

093794-B-I00 (MCIU/AEI/FEDER, UE), A-FQM-484-UGR18 (Universidad de Granada and Junta de

Analuc´ıa/FEDER, UE), and FQM-185 (Junta de Andaluc´ıa/FEDER, UE). The fourth author was sup-

ported by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades, grant FPU17/02023, and by

MINECO and FEDER project MTM2017-83262-C2-1-P.

1

arXiv:2102.10937v1 [math.FA] 22 Feb 2021

2 D. GARC´

IA, M. MAESTRE, M. MART´

IN, AND O. ROLD´

AN

It is clear that 0 ďnpXq ď 1 and that npXq ą 0 if and only if the numerical radius is a

norm on LpXqequivalent to the operator norm. As in this paper we will mainly deal with

compact operators, we will also need the following concept from [11]. Given a Banach

space X, the compact numerical index of Xis

nKpXq:“inftνpTq:TPKpXq,}T} “ 1u

“maxtkě0: k}T} ď νpTq,@TPKpXqu.

We refer the reader to [11], [18], [19], [21, Subsection 1.1], and references therein for more

information and background.

An operator SPLpX, Y qis said to attain its norm whenever there exists some xPSX

such that }S}“}Spxq}. An operator TPLpXqis said to attain its numerical radius

whenever there exists some px, x˚q P ΠpXqsuch that νpTq“|x˚pTpxqq|. The sets of norm

attaining operators from Xto Yand of numerical radius attaining operators on Xwill

be denoted, respectively, by NApX, Y qand NRApXq.

In 1961, Bishop and Phelps [8] proved that the set NApX, Kqof norm attaining func-

tionals on a Banach space Xis always dense in X˚. However, this result has been shown

to fail for general operators between Banach spaces, as Lindenstrauss [22] proved in 1963.

We refer the reader to the survey [1] for more information and background on the density

of norm attaining operators, and to [23] for the compact operators version.

In 1970, Bollob´as [9] gave a reﬁnement of the Bishop-Phelps Theorem, showing that

you can approximate simultaneously a functional and a point where it almost attains

its norm by a norm-attaining funcional and a point where the new functional attains

its norm, respectively. In order to extend Bollob´as’ result to norm attaining operators

between Banach spaces, Acosta, Aron, Garc´ıa and Maestre [3] introduced in 2008 the

Bishop-Phelps-Bollob´as property as follows.

Deﬁnition 1.1 ([3]).A pair of Banach spaces pX, Y qhas the Bishop-Phelps-Bollob´as

property (BPBp for short) if given εP p0,1q, there exists ηpεqPp0,1qsuch that whenever

TPLpX, Y qand x0PSXsatisfy }T} “ 1 and }Tpx0q} ą 1´ηpεq, there are SPLpX, Y q

and x1PSXsuch that

}S}“}Spx0q} “ 1,}x0´x1} ă ε, }S´T} ă ε.

If the above property holds when we restrict the operators Tand Sto be compact, we

say that the pair pX, Y qhas the Bishop-Phelps-Bollob´as property for compact operators

(BPBp for compact operators for short).

With the above notation, the result by Bollob´as just says that the pair pX, Kqhas the

Bishop-Phelps-Bollob´as property for every Banach space X. In the paper [3] a variety of

pairs of spaces satisfying the BPBp are provided, together with examples of pairs pX, Y q

of Banach spaces failing the BPBp for which NApX, Y qis dense in LpX, Y q. We refer the

reader to the survey [2] and the paper [6] for more information and background on the

BPBp.

Motivated by this property, Guirao and Kozhushkina [16] introduced in 2013 the Bishop-

Phelps-Bollob´as property for numerical radius as follows.

Deﬁnition 1.2 ([16]).A Banach space Xis said to have the Bishop-Phelps-Bollob´as

property for numerical radius (BPBp-nu for short) if for every 0 ăεă1, there exists

ηpεq P p0,1qsuch that whenever TPLpXqand px, x˚q P ΠpXqsatisfy νpTq “ 1 and

ON THE BPBP-NU FOR COMPACT OPERATORS 3

|x˚pTpxqq| ą 1´ηpεq, there exist SPLpXqand py, y˚q P ΠpXqsuch that

νpSq“|y˚pSpyqq| “ 1,}T´S} ă ε, }x´y} ă ε, }x˚´y˚} ă ε.

Since then, several works have been done in order to study what spaces satisfy that

property. We summarize next some of the most important results on the matter:

(1) The spaces c0and `1have the BPBp-nu [16].

(2) L1pRqhas the BPBp-nu [14].

(3) Finite-dimensional spaces have the BPBp-nu [20].

(4) The real or complex space Lppµqhas the BPBp-nu for every measure µwhen

1ďpă 8 ([20, Example 8] except for the real case p“2, which is covered in [21,

Corollary 3.3]).

(5) Any uniformly convex and uniformly smooth Banach space Xwith npXq ą 0 has

the BPBp-nu [20].

(6) Every separable inﬁnite-dimensional Banach space can be renormed to fail the

BPBp-nu [20], even though the set of numerical radius attaining operators is always

dense in spaces with the Radon-Nikod´ym property.

(7) The real space CpKqhas the BPBp-nu under some extra conditions on the compact

Hausdorﬀ space K(for example, when Kis metrizable) [7]. Let us comment that

it remains unknown if the result is true for all compact Hausdorﬀ spaces or what

happens in the complex case.

We refer the interested reader to the cited papers [7,14,16,20,21] and the papers

[5,12] and references therein for more information and background.

In 2018, Dantas, Garc´ıa, Maestre and Mart´ın [13] studied the BPBp for compact oper-

ators. They presented some abstract techniques (based on results about norm attaining

compact operators by Johnson and Wolfe [17]) which allow to carry the BPBp for com-

pact operators from sequence spaces (such as c0and `p) to function spaces (as C0pLqand

Lppµq). As one of the main results, it is shown in [13] that the BPBp for compact opera-

tors of the pair pc0, Y qis suﬃcient to get the BPBp for compact operators of all the pairs

pC0pLq, Y qregardless of the locally compact Hausdorﬀ topological space L.

Our aim in this paper is to study the following property, inspired both by the BPBp

for compact operators and by the BPBp for numerical radius.

Deﬁnition 1.3. A Banach space Xis said to have the BPBp-nu for compact operators

if for every 0 ăεă1, there exists ηpεqPp0,1qsuch that whenever TPKpXqand

px, x˚q P ΠpXqsatisfy νpTq “ 1 and |x˚pTpxqq| ą 1´ηpεq, there exist SPKpXqand

py, y˚q P ΠpXqsuch that

νpSq“|y˚pSpyqq| “ 1,}T´S} ă ε, }x´y} ă ε, }x˚´y˚} ă ε.

The ﬁrst work where a somewhat similar property was introduced is [5], where the

BPBp-nu for subspaces of LpXqwas deﬁned and studied in the case of L1pµq, with µa

ﬁnite measure. Let us provide a list of spaces that are known to have the BPBp-nu for

compact operators.

Examples 1.4. The following spaces have the BPBp-nu for compact operators:

(a) Finite dimensional spaces [20, Proposition 2].

(b) c0and `1(adapting the proofs given in [16, Corollaries 3.3 and 4.2]).

4 D. GARC´

IA, M. MAESTRE, M. MART´

IN, AND O. ROLD´

AN

(c) L1pµqfor every measure µ(using [5, Corollary 2.1] for ﬁnite measures and adapting

[20, Theorem 9] to compact operators for the general case).

Adapting the results from [20] and [21], one also has that the Lppµqspaces have the

BPBp-nu for compact operators when 1 ăpă 8. However, the adaptation to the

compact operators case of the proofs in [20] and [21] needs to introduce some terminology.

Therefore, we enounce the result here but we pospone the proof to Subsection 1.1.

Proposition 1.5. Lppµqhas the BPBp-nu for compact operators, for every measure µ

and 1ăpă 8.

Our main objective in this paper is to prove the following result, which is not covered

by Examples 1.4.

Theorem 1.6. If Lis a locally compact Hausdorﬀ space, then C0pLqhas the BPBp-nu

for compact operators.

As a consequence, we get that L8pµqspaces have the BPBp-nu for compact operators,

completing Example 1.4.c and Proposition 1.5.

Corollary 1.7. L8pµqhas the BPBp-nu for compact operators for every measure µ.

Let us recall that it is shown in [7] that the real space CpKqhas the BPBp-nu for some

compact Hausdorﬀ spaces K, but the general case, as well as the complex case, remain

open as far as we know. However, Theorem 1.6 gives a complete answer in the case of

compact operators.

To get the proof of Theorem 1.6, we need two kind of ingredients. On the one hand,

we provide in Section 2some abstract results that will allow us to carry the BPBp-nu for

compact operators from sequence spaces into function spaces, in some cases. The more

general result of this kind is Lemma 2.1, which will be the ﬁrst ingredient for the proof of

Theorem 1.6. It is somehow an extension of [13, Lemma 2.1] but it needs more restrictive

hypothesis in order to deal with the numerical radius instead of with the norm of the

operators. We also provide in that section some useful particular cases of Lemma 2.1

which allow to show, for instance, that every predual of `1has the BPBp-nu for compact

operators, see Corollary 2.6. The second ingredient for the proof of Theorem 1.6 is some

strong approximation property of C0pLqspaces and their duals which will be provided in

Section 3(see Theorem 3.4) and which will allow us to apply Lemma 2.1 in this case, thus

providing the proof of Theorem 1.6. Let us also comment that Theorem 3.4 gives a much

stronger approximation property of C0pLqand its dual space than [13, Lemma 3.4].

1.1. Lpspaces have the BPBp-nu for compact operators.

In this subsection, we will adapt the concepts and results from [20] and [21] to the

compact operators case to show that Lppµqspaces have the BPBp-nu for compact operators

for 1 ăpă 8, providing thus a proof of Proposition 1.5.

In [20, Deﬁnition 5] a weaker version of the BPBp-nu, the weak BPBp-nu, was intro-

duced and studied. We present here the compact operators version of that property.

Deﬁnition 1.8. A Banach space Xis said to have the weak BPBp-nu for compact oper-

ators if for every 0 ăεă1, there exists ηpεqPp0,1qsuch that whenever TPKpXqand

ON THE BPBP-NU FOR COMPACT OPERATORS 5

px, x˚q P ΠpXqsatisfy νpTq “ 1 and |x˚pTpxqq| ą 1´ηpεq, there exist SPKpXqand

py, y˚q P ΠpXqsuch that

νpSq“|y˚pSpyqq|,}T´S} ă ε, }x´y} ă ε, }x˚´y˚} ă ε.

Note that this is a similar property to the BPBp-nu for compact operators, but without

asking for the condition νpSq “ 1 (see Deﬁnition 1.3).

It is shown in [20, Proposition 4] that uniformly convex and uniformly smooth Banach

spaces have the weak BPBp-nu. This result also holds for the compact operators version

by an easy adaptation of the proof.

Proposition 1.9. If a Banach space is uniformly convex and uniformly smooth, then it

has the weak BPBp-nu for compact operators.

Proof. We can follow the proof of [20, Proposition 4], just keeping in mind that if the

original operator T0is compact, then the rest of operators Tnfrom that proof are also

compact, and so, Sis compact too.

Later, in [20, Proposition 6], it is proven that in Banach spaces with positive numerical

index, the BPBp-nu and the weak BPBp-nu are equivalent. This property is also true for

the compact operators versions of the properties if we use the compact numerical index.

Proposition 1.10. Let Xbe a Banach space such that nKpXq ą 0. Then Xhas the

BPBp-nu if, and only if, it has the weak BPBp-nu.

Proof. It suﬃces to follow the proof from [20, Proposition 6] but with both Tand Sbeing

now compact operators, and using nKpXqinstead of npXq.

As a consequence of these two results, similarly to what is done in [20], we get that all

Lppµqspaces have the BPBp-nu for compact operators when 1 ăpă 8 in the complex

case and when 1 ăpă 8,p‰2 in the real case. This is so because, on the one hand, in

the real case,

nKpLppµqq ě npLppµqq ą 0`1ăpă 8, p ‰2˘

by [24] and, on the other hand, nKpXq ě 1{eą0 for every complex Banach space (see

[18, Eq. (1) in p. 156], for instance).

This provides the proof of Proposition 1.5 for all values of pin p1,`8q in the complex

case and for all values of pin p1,`8q except for p“2 in the real case.

Our next aim is to show that real Hilbert spaces also have the BPBp-nu for compact

operators, by adapting the ideas from [21].

First, given a real Banach space X, we consider the following subset of KpXq:

ZKpXq:“ TPKpXq:νpTq “ 0(

which is the set of all skew-hermitian compact operators on X. Observe that

ZKpXq “ KpXq X ZpXq,

where ZpXqis the Lie-algebra of all skew-hermitian operators on X(see [21, p. 1004] for

instance). Adapting the concept of second numerical index given in [21], we deﬁne the

6 D. GARC´

IA, M. MAESTRE, M. MART´

IN, AND O. ROLD´

AN

second numerical index for compact operators of a Banach space Xas the constant

n1

KpXq:“inf νpTq:TPKpXq,}T`ZKpXq} “ 1(

“max Mě0: M}T`ZKpXq} ď νpTqfor all TPKpXq(,

where }T`ZKpXq} is the quotient norm in KpXq{ZKpXq.

The next result is a version for compact operators of [21, Theorem 3.2].

Proposition 1.11. Let Xbe a real Banach space with n1

KpXq ą 0. Then, the BPBp-nu

for compact operators and the weak BPBp-nu for compact operators are equivalent in X.

Proof. It suﬃces to adapt the steps from the proof of [21, Theorem 3.2] to the case of

compact operators. That is: all the involved operators T,S,S1and S2are now compact,

the ZpXqset is replaced by ZKpXq, and the index n1pXqis replaced by n1

KpXq.

We are going to see next that the second numerical index for compact operators of a

real Hilbert space equals one.

Proposition 1.12. Let Hbe a real Hilbert space. Then, n1

KpHq “ 1.

The proof of this result will be an adaptation of the one of [21, Theorem 2.3]. Recall

that in a real Hilbert space endowed with an inner product p¨|¨q,H˚identiﬁes with Hby

the isometric isomorphism xÞÝÑ p¨|xq. Therefore, ΠpXq “ tpx, xq P HˆH:xPSHu,

and so, for every TPLpHq, one has νpTq “ supt|pT x|xq|:xPSHu. We ﬁrst need to give

the compact operators version of [21, Lemma 2.4] whose proof is an obvious adaptation

of the proof of that result.

Lemma 1.13. Let Hbe a real Hilbert space.

(a) ZKpHq“tTPKpHq:T“ ´T˚u.

(b) If TPKpHqis selfadjoint (i.e. T“T˚), then }T} “ νpTq.

We are now ready to present the pending proof of Proposition 1.12.

Proof of Proposition 1.12.It suﬃces to adapt the proof of [21, Theorem 2.3] to the com-

pact operators case, that is: the involved operators Tand Sare now compact, and the

set ZpXqis replaced by ZKpXq.

As a consequence of Propositions 1.9,1.11, and 1.12, we get the following result which

provides the proof of the pending part of Proposition 1.5.

Corollary 1.14. If His a real Hilbert space, then it has the BPBp-nu for compact oper-

ators.

2. First ingredient: the tools

In this section, we will provide an abstract result that will allow us later to carry

the BPBp-nu for compact operators from some sequence spaces to function spaces. The

most general version that we are able to prove is the following, which is inspired in [13,

Lemma 2.1], but it needs more requirements. We need some notation ﬁrst. An absolute

norm | ¨ |ais a norm in R2such that |p1,0q|a“ |p0,1q|a“1 and |ps, tq|a“ |p|s|,|t|q|afor

every ps, tq P R2. Given a Banach space X, we say that a projection Pon Xis an absolute

projection if there is an absolute norm |¨|asuch that }x} “ ˇˇp}Ppxq},}x´Ppxq}qˇˇafor

ON THE BPBP-NU FOR COMPACT OPERATORS 7

every xPX. Examples of absolute projections are the M- and L-projections and, more

in general, the `p-projections. We refer the reader to [13] for the use of absolute norms

with the Bishop-Phelps-Bollob´as type properties and to the references therein for more

information on absolute norms.

Lemma 2.1. Let Xbe a Banach space satisfying that nKpXq ą 0. Suppose that there is

a mapping η:p0,1q ÝÑ p0,1qsuch that given δą0,x˚

1, . . . , x˚

nPBX˚and x1, . . . , x`P

BX, we can ﬁnd norm one operators r

P:XÝÑ r

PpXq,i:r

PpXq ÝÑ Xsuch that for

P:“i˝r

P:XÝÑ X, the following conditions are satisﬁed:

(i) }P˚px˚

jq ´ x˚

j} ă δ, for j“1, . . . , n.

(ii) }Ppxjq ´ xj} ă δ, for j“1, . . . , `.

(iii) r

P˝i“Id r

PpXq.

(iv) r

PpXqsatisﬁes the Bishop-Phelps-Bollob´as property for numerical radius for com-

pact operators with the mapping η.

(v) Either Pis an absolute projection and iis the natural inclusion, or nKpr

PpXqq “

nKpXq “ 1.

Then, Xsatisﬁes the BPBp-nu for compact operators.

Let us comment on the diﬀerences between the lemma above and [13, Lemma 2.1].

First, condition (ii) is more restrictive here than in that lemma, where it only dealt with

one point. Second, the requirements of item (v) on the compact numerical index or on the

absoluteness of the projections did not appear in [13, Lemma 2.1], but they are needed

here as numerical radius does not behave well in general with respect to extensions of

operators.

Proof. Given εP p0,1q, let ε0pεqbe the unique number with 0 ăε0pεq ă 1 such that

ε0pεqˆ2

3`1

p1´ε0pεqq nKpXq˙“ε,

which, in particular, satisﬁes that ε0pεq ă ε. From now on, we shall simply write ε0instead

of ε0pεq. We deﬁne next

(1) η1pεq:“min #ε2

0pnKpXqq2

72 ,`η`ε0

3˘˘2pnKpXqq2

72 +`εP p0,1q˘,

where ηis the function appearing in the hypotheses of the lemma. We ﬁx TPKpXqwith

νpTq “ 1 (thus, }T} ď 1

nKpXq) and px1, x˚

1q P ΠpXqsuch that

|x˚

1pTpx1qq| ą 1´η1pεq.

Since T˚pBX˚qis relatively compact, we can ﬁnd x˚

2, . . . , x˚

nPBX˚such that

min

2ďjďn}T˚px˚q ´ x˚

j} ă η1pεqfor all x˚PBX˚.

Similarly, since TpBXqis relatively compact, we can ﬁnd x2, . . . , x`PBXsuch that

min

2ďjď`}Tpxq ´ xj} ă η1pεqfor all xPBX.

Let r

P:XÝÑ r

PpXq,i:r

PpXq ÝÑ Xand P:“i˝r

P:XÝÑ Xsatisfying the conditions

(i)-(v) for x1, . . . , x`PBX,x˚

1, . . . , x˚

nPBX˚and δ“η1pεq.

8 D. GARC´

IA, M. MAESTRE, M. MART´

IN, AND O. ROLD´

AN

Now, for every x˚PBX˚, we have

}T˚px˚q ´ P˚pT˚px˚qq}

ďmin

2ďjďn }T˚px˚q ´ x˚

j}`}x˚

j´P˚px˚

jq} ` }P˚px˚

jq ´ P˚pT˚px˚qq}(ă3η1pεq,

and hence, }T´T P } “ }T˚´P˚T˚} ď 3η1pεq. On the other hand, for each xPBX, we

have

}Tpxq ´ PpTpxqq} ď min

2ďjď`t}Tpxq ´ xj}`}xj´Ppxjq} ` }Ppxjq ´ PpTpxqq}u

ă3η1pεq,

and then, }T´P T } ď 3η1pεq. Therefore,

}PTP ´T}ď}PTP ´P T }`}P T ´T} ď }T P ´T}`}P T ´T} ď 6η1pεq.

Consider pr

Ppx1q, i˚px˚

1qq P r

PpXq ˆ p r

PpXqq˚. Note that it is not true in general that

pr

Ppx1q, i˚px˚

1qq P Πpr

PpXqq, but we have that }r

Ppx1q} ď 1, }i˚px˚

1q} ď 1, and also, that

x˚

1pipr

Ppx1qqq “ x˚

1px1q

loomoon

“1

´x˚

1pipr

Ppx1qq ´ x1q

loooooooooomoooooooooon

}P x1´x1}ăη1pεq

ùñ Repx˚

1pipr

Ppx1qqqq ě 1´η1pεq.

By the Bishop-Phelps-Bollob´as Theorem (see [10, Corollary 2.4.b] for this version), there

exist py, y˚q P Πpr

PpXqq satisfying that

max !}y´r

Ppx1q},}y˚´i˚px˚

1q})ďa2η1pεq ď ε0

3.

Next, we observe that the following two inequalities hold:

}r

P˚py˚q ´ x˚

1} ď } r

P˚py˚q ´ r

P˚pi˚px˚

1qq} ` } r

P˚pi˚px˚

1qq ´ x˚

1}

ďa2η1pεq ` η1pεq ď 2

3ε0.

(2)

(3) }ipyq ´ x1} ď }ipyq ´ ipr

Ppx1qq} ` }ipr

Ppx1qq ´ x1} ď a2η1pεq ` η1pεq ď 2

3ε0.

Let T1:“r

P˝T˝i:r

PpXq ÝÑ r

PpXq.

Claim. We have that

|y˚pT1yq| ą 1´η´ε0

3¯and |y˚pT1yq| ą 1´ε0.

Indeed, from equations (2) and (3), we obtain that

ˇˇx˚pTpx1qq´ r

P˚py˚pTpipyqqqqˇˇ

ď |x˚

1pTpx1qq ´ x˚

1pTpipyqqq| ` |x˚

1pTpipyqqq ´ r

P˚py˚pTpipyqqqq|

ď }T}}x1´ipyq} ` }T}}x˚

1´r

P˚py˚q}

ď2}T}´a2η1pεq ` η1pεq¯.

Now, we can estimate |y˚pT1pyqq| as follows:

|y˚pT1pyqq| “ ˇˇr

P˚py˚pTpipyqqqqˇˇ

ěˇˇx˚

1pTpx1qqˇˇ´ˇˇx˚

1pTpx1qq ´ r

P˚py˚pTpipyqqqqˇˇ

ě1´η1pεq ´ 2}T}a2η1pεq ´ 2}T}η1pεq.

ON THE BPBP-NU FOR COMPACT OPERATORS 9

From here, using the deﬁnition of η1pεqgiven in Eq. (1) and the fact that }T} ď 1{nKpXq,

we get both assertions of the claim.

In particular, we get that νpT1q ě 1´ε0ą0. On the other hand, we also have that

νpT1q ď 1. Indeed, if there were some pq, q˚q P Πpr

PpXqq with |q˚pT1pqqq| ą 1, we would

get

|q˚pT1pqqq| “ |q˚pr

PpTpipqqqqq| “ |p r

P˚pq˚qqpTpipqqqq| ą 1,

but νpTq “ 1, and

pr

P˚pq˚qqpipqqq “ q˚pr

Ppipqqqq “ q˚pqq “ 1.

Thus pipqq,r

P˚pq˚qq P ΠpXq, and that is a contradiction.

We deﬁne now the operator r

T:“T1

νpT1q. Clearly, r

Tis a compact operator such that

νpr

Tq “ 1. From the claim, we get that

ˇˇy˚pr

Tpyqqˇˇ“1

νpT1q|y˚pT1pyqq| ě |y˚pT1pyqq| ą 1´η´ε0

3¯.

Now, since r

PpXqhas the BPBp-nu for compact operators with the mapping η, there exist

a compact operator r

S:r

PpXq ÝÑ r

PpXqwith νpr

Sq “ 1 and pz, z˚q P Πpr

PpXqq such that

νpr

Sq “ ˇˇz˚pr

Spzqqˇˇ“1,}z´y} ă ε0

3,}z˚´y˚} ă ε0

3,}r

S´r

T} ă ε0

3.

Let t“ipzq P BXand t˚“r

P˚pz˚q P BX˚. We have that

t˚ptq “ z˚pr

Ppipzqqq “ z˚pzq “ 1.

Thus pt, t˚q P ΠpXq, and also, by (2) and (3),

}t´x1} ď }t´ipyq} ` }ipyq ´ x1}“}ipzq ´ ipyq} ` }ipyq ´ x1} ă ε0

3`2ε0

3“ε0ďε,

}t˚´x˚

1} ď } r

P˚pz˚q ´ r

P˚py˚q} ` } r

P˚py˚q ´ x˚

1} ă ε0

3`2ε0

3“ε0ďε.

We deﬁne S“i˝r

S˝r

P:XÝÑ X, which is a compact operator. It is clear that νpSq ě 1

since

|t˚pSptqq| “ |z˚pr

Ppipr

Spr

Ppipzqqqqqq| “ |z˚pr

Spzqq| “ 1.

Also,

}S´T}“}i˝r

S˝r

P´T}

ď }i˝r

S˝r

P´i˝r

T˝r

P}`}i˝r

T˝r

P´PTP}`}PTP ´T}

“ }i˝r

S˝r

P´i˝r

T˝r

P} ` ››››

PTP

νpT1q´PTP››››` }PTP ´T}

ď } r

S´r

T}`}T} ¨ ˇˇˇˇ

1

νpT1q´1ˇˇˇˇ` }PTP ´T}

and, since }T} ď 1

nKpXq, 1 ´ε0ďνpT1q ď 1, and 6η1pεq ď ε0

3, we continue as:

ďε0

3`ε0

p1´ε0qnKpXq`6η1pεq ď ε0ˆ2

3`1

p1´ε0qnKpXq˙ăε.

We ﬁnish the proof if we prove that νpSq ď 1. We consider the following cases:

10 D. GARC´

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‚Case 1: if r

Pis an absolute projection and iis the natural inclusion, as a conse-

quence of [11, Lemma 3.3], we get that

νpSq “ νpi˝r

S˝r

Pq “ νpr

Sq “ 1.

‚Case 2: if nKpXq “ nKpr

PpXqq “ 1, then

νpSq“}S}ď}r

S} “ νpr

Sq “ 1.

Hence, the result follows in the two cases.

We will now provide some applications and consequences of the previous lemma. Given

a continuous projection P:XÝÑ X, if we set r

P:XÝÑ r

PpXq “ PpXq Ă X(that is,

r

Pis just the operator Pwith a restricted codomain) and i:PpXq ÝÑ Xis the natural

inclusion then, trivially, we have that P“i˝r

Pand that r

P˝i“Id r

PpXq. This easy

observation allows to get the following particular case of Lemma 2.1.

Proposition 2.2. Let Xbe a Banach space with nKpXq ą 0. Suppose that there exists a

net tPαuαPΛof norm-one projections on Xsatisfying that tPαpxqu ÝÑ xfor all xPXand

tP˚

αpx˚qu ÝÑ x˚for all x˚PX˚, and that there exists a function η:p0,1q ÝÑ p0,1qsuch

that all the spaces PαpXqwith αPΛhave the BPBp-nu for compact operators with the

function η. Suppose, moreover, that for each αPΛ, at least one of the following conditions

is satisﬁed:

(1) the projection Pαis absolute,

(2) nKpPαpXqq “ nKpXq “ 1.

Then, the space Xhas the BPBp-nu for compact operators.

We may now obtain the following consequence of the above result. Given a Banach

space Xand mPN, the space `m

8pXqrepresents the `8-sum of mcopies of X, and we will

write `8pXqfor the `8-sum of countably inﬁnitely many copies of X. Similarly, c0pXqis

the c0-sum of countably inﬁnitely many copies of X. When X“K, we just write `m

8for

`m

8pKq.

Corollary 2.3. Let Xbe a Banach space with nKpXq ą 0. Then, the following statements

are equivalent:

(i) The space c0pXqhas the BPBp-nu for compact operators.

(ii) There is a function η:p0,1q ÝÑ p0,1qsuch that all the spaces `n

8pXq, with nPN,

have the BPBp-nu for compact operators with the function η.

Moreover, if Xis ﬁnite dimensional, these properties hold whenever c0pXqor `8pXqhave

the BPBp-nu.

Proof. That (ii) implies (i) is a consequence of Proposition 2.2 since for every nPN, the

operator on c0pXqwhich is the identity on the ﬁrst ncoordinates and 0 elsewhere is an

absolute projection whose image is isometrically isomorphic to `n

8pXq.

(i) implies (ii) is a consequence of [12, Proposition 4.3], as one can easily see `n

8pXq

as an `8-summand of c0pXq. Let us comment that the function ηvalid for all `n

8pXqis

the function valid for c0pXqand this actually follows from the proof of [12, Theorem 4.1]

(from which [12, Proposition 4.3] actually follows).

ON THE BPBP-NU FOR COMPACT OPERATORS 11

Finally, when Xhas ﬁnite dimension, if c0pXqor `8pXqhas the BPBp-nu, then con-

dition (ii) holds by using [12, Theorem 4.1] and the fact that `n

8pXqis ﬁnite-dimensional

and so, every operator from `n

8pXqto itself is compact.

As stated in Examples 1.4, that c0and the spaces `n

8for nPNhave the BPBp-nu for

compact operators is a consequence of [16, Corollary 4.2] and [20, Proposition 2]. Actually,

the fact that all the space `n

8have the BPBp-nu with the same function ηfollows from

[16, Corollary 4.2] and (the proof of) [12, Theorem 4.1]. However, let us note that we can

also get this result as a consequence of our previous corollary.

Corollary 2.4. There is a function η:p0,1q ÝÑ p0,1qsuch that the space c0and the

spaces `n

8with nPN, have the BPBp-nu for compact operators with the function η.

Additionally, [12, Proposition 4.3] also implies that whenever `n

8pXqhas the BPBp-nu

for compact operators for some nPN, then so does X, although the converse remains

unknown in general (even for n“2).

Another consequence of Proposition 2.2 is the following:

Corollary 2.5. Let Xbe a Banach space with nKpXq ą 0. Suppose that there exists a

net tPαuαPΛof norm-one projections on Xsuch that αĺβimplies PαpXq Ă PβpXq, that

tP˚

αpx˚qu ÝÑ x˚for all x˚PX˚, and that there exists a function η:p0,1q ÝÑ p0,1qsuch

that all the spaces PαpXqwith αPΛhave the BPBp-nu for compact operators with the

function η. Suppose, moreover, that for each αPΛ, at least one of the following conditions

is satisﬁed:

(1) the projection Pαis absolute,

(2) nKpPαpXqq “ nKpXq “ 1.

Then, the space Xhas the BPBp-nu for compact operators.

Proof. Observe that in order to apply Proposition 2.2 we only need that tPαxu ÝÑ xin

norm for all xPX. But this is proved in [13, Corollary 2.4], so we are done.

The previous result can be used to prove that all the preduals of `1have the BPBp-nu

for compact operators.

Corollary 2.6. Let Xbe a Banach space such that X˚is isometrically isomorphic to `1.

Then Xhas the BPBp-nu for compact operators.

Proof. By using a deep result due to Gasparis [15], it is shown in the proof of [13, Theorem

3.6] that there exists a sequence of norm-one projections Pn:XÝÑ Xsatisﬁying that

Pn`1Pn“Pn(and so, PnpXq Ă Pn`1pXq), that PnpXqis isometrically isomorphic to `n

8,

and also that P˚

npx˚q ÝÑ x˚for all x˚PX˚(this claim holds since the sets Yndeﬁned on

that proof satisfy that their union is dense in X˚“`1).

Next, as PnpXqis isometrically isomorphic to `n

8, on the one hand we have that all

the spaces PnpXqhave the BPBp-nu for compact operators with the same function ηas

a consequence of Corollary 2.4. On the other hand, npXq “ npPnpXqq “ 1 for all nPN

(see [18], for instance) so, in particular, nKpXq “ nKpPnpXqq “ 1 for all nPN. Finally,

Corollary 2.5 provide the desired result.

12 D. GARC´

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AN

3. Second ingredient: a strong approximation property of C0pLqspaces and

their duals

The aim of this section is to provide some strong approximation property of C0pLq

spaces and their duals which allow to use Lemma 2.1 (actually, Proposition 2.2) to give a

proof of Theorem 1.6. We need a number of technical lemmas.

Lemma 3.1. Let Lbe a locally compact space, let tK1, . . . , KMube a family of pairwise

disjoint non-empty compact subsets of L, and let KĂLbe a compact set with

M

Ť

m“1

KmĂ

K. If tU1, . . . , URuis a family of relatively compact open subsets of Lcovering Ksuch that

for each mthere is an rpmqwith KmĂUrpmq,m“1, . . . , M , then there exists an open

reﬁnement tZ1, . . . , ZSu,MďSďR`Mwith Z1, . . . , ZMpairwise disjoint, satisfying:

(1) For m“1, . . . , M,KmĂZm, and KmXZs“ H for all sP t1, . . . , Suztmu.

(2) For all s0ąM, there exists zs0PZs0zˆŤ

s‰s0

Zs˙.

Proof. As tK1, . . . , KMuare pairwise disjoint, there exist tV1,...VMupairwise disjoint

open subsets of Lwith KmĂVmĂUrpmq,m“1, . . . , M .

The family "V1, . . . , VM, U1zˆM

Ť

m“1

Km˙, . . . , URzˆM

Ť

m“1

Km˙*is another cover of K

by open subsets of Lsubordinated to tUruR

r“1. We deﬁne Zm:“Vmfor m“1, . . . , M,

and Wr:“UrzˆM

Ť

m“1

Km˙for r“1, . . . , R.

If W1ĂV1Y. . . YVM, then tV1, . . . , VM, W2, . . . , WRuis again a cover of K. If that

happens again and again until WR, we have that tZ1, . . . , ZMuis the cover we were looking

for. In other case, let r1ě1 be the ﬁrst natural number such that there exists wr1P

Wr1zˆM

Ť

m“1

Vm˙, and denote ZM`1:“Wr1. The family tV1, . . . , VM, Wr1, Wr1`1, . . . , WRu

is a cover of Kby open sets, and then, so is the family

V1, . . . , VM, Wr1, Wr1`1ztwr1u, . . . , WRztwr1u(.

Consider now r2ąr1the ﬁrst natural number such that there exists wr2PWr2ztwr1uand

wr2RV1Y. . . YVMYWr1. Let ZM`2:“Wr2ztwr1uand proceed as before. In at most R

steps, we get tZ1, . . . , ZSu,MďSďR`M, such that

‚KmĂZmfor m“1,...M.

‚ˆM

Ť

m“1

Km˙XZs“ H for sąM.

‚For all s0ąM, there exists wrs0´MPZs0zˆŤ

s‰s0

Zs˙.

We next provide a result showing the existence of certain partitions of the unity. We

separate the non-compact case (Lemma 3.2) and the compact case (Lemma 3.3) for the

sake of clarity. We start with the non-compact case.

Lemma 3.2. Let Lbe a non-compact locally compact space. Let KĂLbe a compact set

and tK1, . . . , KMua family of pairwise disjoint non-empty compact subsets of K. Given a

ON THE BPBP-NU FOR COMPACT OPERATORS 13

family tU1, . . . , URuof relatively compact open subsets of Lthat cover K, let tZ1, . . . , ZSu

be a family of open subsets of Lcovering Ksuch that they satisfy the thesis of Lemma 3.1,

and denote by ZS`1the set LzˆS

Ť

s“1

Zs˙. Then, there exists a partition of the unity

subordinated to tZsuS`1

s“1,tϕsuS`1

s“1, such that:

(1) tϕ1, . . . , ϕMuhave disjoint support.

(2) ϕmpKmq ” 1, for m“1, . . . , M .

(3) For all MăsďS`1, there exists zsPZssuch that ϕspzsq “ 1.

(4) For s“1, . . . , S `1,supppϕsq Ă Zs.

(5) pϕ1` ¨ ¨ ¨ ` ϕSqpxq “ 1, for all xPK.

Proof. By hypothesis, there exists some zS`1PLzˆS

Ť

s“1

Zs˙, since for all s,ZsĂ

R

Ť

r“1

Ur,

which is a compact set. Now, we follow the argument from the proof of [25, Theorem

2.13], but adapted to our case.

As KĂZ1Y. . . YZS, for each xPK, there exists a neighbourhood of x,Yx, with

compact closure YxĂZsfor some s. Consider x1, . . . , xpsuch that KĂYx1Y. . . YYxp.

For each 1 ďsďS, let Hsbe the union of those Yxjwhich lie in Zs, and if Măs0ďS,

we take Hs0Y tzs0u, with zs0PZs0zˆŤ

s‰s0

Zs˙. Note that the sets H1, . . . , HMand

HM`1YtzM`1u, . . . , HSYtzSuare non-empty. By Urysohn’s Lemma, there are continuous

functions gs:LÝÑ r0,1ssuch that gspHsq ” 1 and gsˇˇLzZs”0, for 1 ďsďM, and

gs0pHs0Y tz0uq ” 1 and gs0ˇˇLzZs0

”0 for Măs0ďS. Deﬁne

ϕ1:“g1,

ϕ2:“ p1´g1qg2,

.

.

.

ϕS:“ p1´g1qp1´g2q¨¨¨p1´gS´1qgS

Clearly, supppϕsq Ă Zsfor all s“1, . . . , S, and we have that

ϕ1` ¨ ¨ ¨ ` ϕS“1´ p1´g1q¨¨¨p1´gSq.

Since KĂH1Y. . . YHS, for each xPK, there exists s“spxqwith gspxq “ 1, and also,

for all s“1, . . . , M , we have that

txPL:ϕspxq ‰ 0uĂtxPL:gspxq ‰ 0u Ă Zs.

Therefore, the functions tϕ1, . . . , ϕMuhave disjoint support, and ϕ1` ¨ ¨ ¨ ` ϕS”1 on K.

We deﬁne ϕS`1:“1´ pϕ1` ¨ ¨ ¨ ` ϕSq “ p1´g1q¨¨¨p1´gSq. Moreover, KmĂZmfor

m“1, . . . , M , and KmXZs“ H for m‰s,m“1, . . . , M ,s“1, . . . , S. Hence,

ϕmpxq “

S

ÿ

s“1

ϕspxq “ 1,@xPKm, m “1, . . . , M.

14 D. GARC´

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On the other hand, if Măs0ďS, let zs0PZs0zˆŤ

s‰s0

Zs˙. We have that

ϕs0pzs0q “

S

ÿ

s“1

ϕspzs0q “ 1,

and

zS`1R

S

ď

s“1

Zs,thus ϕS`1pzS`1q “ 1.

The next result is the version of the previous lemma for compact topological spaces.

Lemma 3.3. Let Lbe a compact space. Let tK1, . . . , KMube a family of pairwise disjoint

non-empty compact subsets of L. Given a family tU1, . . . , URuof relatively compact open

subsets of Lthat cover it, let tZ1, . . . , ZSube a family of open subsets of Lcovering K

such that they satisfy the thesis of Lemma 3.1. Then, there exists a partition of the unity

subordinated to tZsuS

s“1,tϕsuS

s“1, such that:

(1) tϕ1, . . . , ϕMuhave disjoint support.

(2) ϕmpKmq ” 1, for m“1, . . . , M .

(3) For all MăsďS, there exists zsPZssuch that ϕspzsq “ 1.

(4) For s“1, . . . , S,supppϕsq Ă Zs.

(5) pϕ1` ¨ ¨ ¨ ` ϕSqpxq “ 1, for all xPK.

Proof. We can follow the proof of Lemma 3.2 taking K“Land adapting the steps from

that proof, keeping in mind that now ZS`1“ H (and hence there is not such a point

zS`1), and that the mapping ϕS`1is identically 0, and hence, it can be omitted.

The following result provides the promised approximation property of C0pLqspaces and

their duals.

Theorem 3.4. Let Lbe a locally compact space. Given tf1, . . . , f`u Ă C0pLqsuch that

}fj} ď 1for j“1, . . . , `, and given tµ1, . . . , µnu Ă C0pLq˚with }µj} ď 1for j“1, . . . , n,

for each εą0there exists a norm one projection P:C0pLq ÝÑ C0pLqsatisfying:

(1) }P˚pµjq ´ µj} ă ε, for j“1, . . . , n,

(2) }Ppfjq ´ fj} ă ε, for j“1, . . . , `,

(3) PpC0pLqq is isometrically isomorphic to `p

8for some pPN.

Let us comment that this result extends [13, Lemma 3.4] (which, actually, was itself an

extension of [4, Proposition 3.2] and [17, Proposition 3.2]). The main diﬀerence is that

here we are able to deal with an arbitrary number of functions of C0pLqin (2), while in

that lemma only one function is controlled, and besides, this was done with the help of

an inclusion operator which is not the canonical one. However, this diﬀerence is crucial in

order to apply Lemma 2.1 (or even its consequence Proposition 2.2).

The following observation on the theorem is worth mentioning.

Remark 3.5. Let us observe that by just conveniently ordering the obtained projections in

Theorem 3.4, we actually get the following: given a Hausdorﬀ locally compact topological

space L, there is a net tPαuαPΛof norm-one projections on C0pLq, converging in the strong

operator topology to the identity operator, such that tP˚

αuαPΛconverges in the strong

ON THE BPBP-NU FOR COMPACT OPERATORS 15

operator topology to the identity on C0pLq˚, and such that PαpC0pLqq is isometrically

isomorphic to a ﬁnite-dimensional `8space.

Proof of Theorem 3.4.We will assume ﬁrst that Lis not compact. Since fjPC0pLq,

j“1, . . . , `, there exists a compact set K0ĂLsuch that

sup

j“1,...,`

t|fjpxq|:xPLzK0u ă ε

4.

For each xPK0, there exists a relatively compact open subset Uxof Lcontaining xand

such that

|fjpxq ´ fjpyq| ă ε

2for yPUxand j“1, . . . , `.

Therefore, tUxuxPK0is a cover of K0, and so, there exist a ﬁnite subcover tU1, . . . , UR´1u

such that K0ĎU1Y. . . YUR´1, and if x, y PUrfor some r, then |fjpxq ´ fjpyq| ă ε

2, for

j“1, . . . , `.

We deﬁne µ:“řn

j“1|µj| P C0pLq˚. Since for each jP t1, . . . , nuµjis absolutely

continuous with respect to µ, by the Radon-Nikod´ym Theorem, there exists gjPL1pµq

such that µj“gjµ, that is,

µjpfq:“żL

fdµj“żL

fpxqgjpxqdµpxqfor all fPC0pLq.

Since the set of simple functions is dense in L1pµq, we may choose a set of simple functions

tsj:j“1, . . . , nusuch that }gj´sj}1ăε

4for j“1, . . . , n.

Next, we consider a family tAmuM

m“1of pairwise disjoint measurable sets with µpAmq ą 0

for all m, such that each Amis contained in one of the elements of the following cover

of L:tU1, . . . , UR´1, LzK0u, and also tαm,j :m“1, . . . , M, j “1, . . . , nusuch that sj“

řM

m“1αm,j χAm. This cover satisﬁes that if x, y PLzK0, or if x, y PUr, then |fjpxq ´

fjpyq| ă ε

2for all j“1, . . . , ` and all r“1, . . . , R ´1. Let Cąmaxt|αm,j |:m“

1, . . . , M, j “1, . . . , nu.

Since µis regular, for each 1 ďmďM, we can ﬁnd a compact set KmĂAmsuch that

µpAmzKmq ă ε

4MC and µpKmq ą 0 for all m“1, . . . , M .

Let K“K0YK1Y. . . YKM. As KzˆR´1

Ť

r“1

Ur˙is a compact subset of L, we can

cover it with ﬁnitely many relatively compact open subsets of LzK0that we will denote

UR, UR`1, . . . , UP. If we now apply Lemmas 3.1 and 3.2 to the family tU1, . . . , UPuand

the compacts tK1, . . . , KMuand K, we obtain a reﬁnement of relatively compact open

subsets of L,tZ1, . . . , ZSuwith KmĂZmfor m“1, . . . , M and tZ1,...ZMupairwise

disjoint, and deﬁning ZS`1to be the set LzˆS

Ť

s“1

Zs˙, we also have a partition of the unity

subordinated to tZsuS

s“1,tϕsuS`1

s“1, such that:

(i) tϕ1, . . . , ϕMuhave disjoint support.

(ii) ϕmpKmq ” 1 for m“1, . . . , M .

(iii) For all MăsďS`1, there exists zsPZssuch that ϕspzsq “ 1.

(iv) For s“1, . . . , S `1, supppϕsq Ă Zs.

(v) pϕ1`. . . `ϕSqpKq ” 1.

16 D. GARC´

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Now, we deﬁne P:C0pLq ÝÑ C0pLqby

Ppfq:“

M

ÿ

m“1

1

µpKmqˆżKm

fdµ˙ϕm`

S`1

ÿ

s“M`1

fpzsqϕs,for all fPC0pLq.

Let us ﬁrst check that (2) holds, that is, that }Ppfjq ´ fj} ă εfor all j“1, . . . , `. Let

xPL. We will distinguish two cases:

‚Case 1: if xP

M

Ť

m“1

Zm, then there exists exactly one m0such that xPZm0. Then,

for each j“1, . . . , `, we have:

|Ppfjqpxq ´ fjpxq| “ ˇˇˇˇˇPpfjqpxq ´

M

ÿ

m“1

fjpxqϕmpxq ´

S`1

ÿ

s“M`1

fjpxqϕspxqˇˇˇˇˇ

ďˇˇˇˇˇ

1

µpKm0q˜żKm0

fjpyqdµpyq¸´fjpxqˇˇˇˇˇϕm0pxq

loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon

(I)

`

`

S`1

ÿ

s“M`1

|fjpxq ´ fjpzsq|ϕspxq

looooooooooooooooomooooooooooooooooon

(II)

.

For (I), we have

(I) “ˇˇˇˇˇ

1

µpKm0q˜żKm0

pfjpyq ´ fjpxqq dµpyq¸ˇˇˇˇˇϕm0pxq ď 1

µpKm0qżKm0

ε

2dµpyq “ ε

2.

Now, for (II), let sP tM`1, . . . , S `1u. Note that if xRZs, then ϕspxq “ 0,

and if xPZs, we have that |fjpxq ´ fjpzsq| ă ε

2and řS`1

s“M`1ϕspxq ď 1, and so,

(II) ăε

2. Therefore, |Ppfjqpxq ´ fjpxq| ă εfor all xP

M

Ť

m“1

Zm, for all j“1, . . . , `.

‚Case 2: if xR

M

Ť

m“1

Zm, then for each j“1, . . . , `, we have

|Ppfjqpxq ´ fjpxq| “ ˇˇˇˇˇ

S`1

ÿ

s“M`1

pfjpxq ´ fjpzsqqϕspxqˇˇˇˇˇăε

2

as in item (II) of the previous case.

Summarizing, we get }Ppfjq ´ fj} ă εfor all j“1, . . . , `, getting thus (2).

Now we check (1), that is, that }P˚pµjq ´ µj} ă εfor all j“1, . . . , n. Indeed, ﬁrst

observe that if νis a regular Borel (real or complex) measure on L, its associated x˚

νP

C0pLq˚is deﬁned as

x˚

νpfq:“żL

fpxqdνpxq,@fPC0pLq,

ON THE BPBP-NU FOR COMPACT OPERATORS 17

and we identify x˚

ν”ν. In our case, we have that

P˚pνqpfq “ żL

Ppfqpxqdνpxq

“żL˜M

ÿ

m“1

1

µpKmqˆżKm

fdµ˙ϕmpxq¸dνpxq ` żL˜S`1

ÿ

s“M`1

fpzsqϕspxq¸dνpxq

“

M

ÿ

m“1

1

µpKmqˆżKm

fdµ˙żL

ϕmpxqdνpxq `

S`1

ÿ

s“M`1

fpzsqżL

ϕspxqdνpxq.

In particular, if supppνq Ă

M

Ť

m“1

Km, then by Lemma 3.1.(1)

S`1

ÿ

s“M`1

fpzsqżL

ϕspxqdνpxq ” 0,@fPC0pLq.

Let now νj:“tjµ, where tj:“řM

m“1αm,j χKm, for all j“1, . . . , n, that is,

νjpfq “ żL

fpxq˜M

ÿ

m“1

αm,j χKmpxq¸dµpxq,@fPC0pLq.

It holds that P˚pνjq “ νjfor j“1, . . . , n. Indeed, as supppνjq Ă

M

Ť

m“1

Km, we have

P˚pνjqpfq “

M

ÿ

m“1

1

µpKmqˆżKm

fdµ˙żL

ϕmpxq˜M

ÿ

l“1

αl,j χKlpxq¸dµpxq

“

M

ÿ

m“1

1

µpKmqˆżKm

fdµ˙żL

αm,j χKmpxqdµpxq

loooooooooooomoooooooooooon

αm,j µpKmq

“żL

fpxq˜M

ÿ

m“1

αm,j χKmpxq¸dµpxq “ νjpfq

for all fPC0pLqand all j“1, . . . , n.

Now, we know that }P˚} “ }P} ď 1 and, since Ppϕjq “ ϕjfor j“1, . . . , n, we get that

}P˚} “ 1. Therefore, since P˚pνjq “ νj, we get

}P˚pµjq ´ µj} ď }P˚pµj´νjq} ` }νj´µj}

ď }P˚}¨}µj´νj}`}µj´νj} ď 2}µj´νj}.

But we have

}µj´νj}“}gjµ´tjµ} ď }gjµ´sjµ}`}sjµ´tjµ}

“ }gj´sj}1` }sj´tj}1ăε

4`ε

4“ε

2,

18 D. GARC´

IA, M. MAESTRE, M. MART´

IN, AND O. ROLD´

AN

since

}sj´tj}1“żLˇˇˇˇˇ

M

ÿ

m“1

αm,j χAm´

M

ÿ

m“1

αm,j χKmˇˇˇˇˇdµ

ď

M

ÿ

m“1

|αm,j |

loomoon

ďC

µpAmzKmq ă MCε

4MC “ε

4,

for all j“1, . . . , n. Hence,

}P˚pµjq ´ µj} ď 2}µj´νj} ă 2ε

2“εfor j“1, . . . , n.

Let us ﬁnish the proof by checking (3). As µpKmq ą 0, we have Km‰ H,m“1, . . . , M.

Hence, we have that zsPZsfor s“1, . . . , S `1 and that zs0RŤ

s‰s0

Zsfor all s0“

1, . . . , S `1. By the deﬁnition of P, we have that PpC0pLqq “ spantϕs:s“1, . . . , S `1u

and we will be done by proving the following equality:

››a1ϕ1` ¨ ¨ ¨ ` aS`1ϕS`1››8“maxt|a1|,...,|aS`1|u “ }a}8

for every a“ pa1, . . . , aS`1q. Indeed, for xPL

ˇˇa1ϕ1pxq ` ¨ ¨ ¨ ` aS`1ϕS`1pxqˇˇď }a}8

S`1

ÿ

s“1

ϕspxq“}a}8.

But for each s,ˇˇa1ϕ1pzsq ` ¨ ¨ ¨ ` aS`1ϕS`1pzsqˇˇ“ |as|,

and then, ››a1ϕ1`. . . `aS`1ϕS`1››8ě }a}8.

Hence, the mapping ρ:`S`1

8ÝÑ C0pLqgiven by

pa1, . . . , aS`1q ÞÝÑ a1ϕ1`. . . `aS`1ϕS`1

is an isometry, and therefore, PpC0pLqq is isometrically isomorphic to `S`1

8.

Now, for the case when Lis compact, by taking K0“Land using Lemma 3.3 instead

of Lemma 3.2, a similar proof is valid, except that now all the elements depending on S`1

will vanish in the proof: here we get ZS`1“ H (hence zS`1does not exist), ϕS`1”0

(and hence it can be omitted), and so, the vector awill only have Scomponents; therefore

PpC0pLqq is isometrically isomorphic to `S

8in this case.

We are now ready to prove the main result of the paper.

Proof of Theorem 1.6.Let f1, . . . , f`PBC0pLq,µ1, . . . , µnPBpC0pLqq˚and εą0 be

given. Let P:C0pLq ÝÑ C0pLqbe the projection from Theorem 3.4, which satisﬁes that

PpC0pLqq is isometrically isomorphic to `p

8for some pPN. Let r

P:XÝÑ r

PpC0pLqq be

the operator such that r

Ppfq “ Ppfqfor all fPC0pLq, and let i:r

PpC0pLqq ÝÑ C0pLqbe

the natural inclusion. Let ηbe the mapping with which all `n

8spaces has the BPBp-nu for

compact operators (see Corollary 2.4). Since npC0pLqq “ 1 and np`n

pq “ 1 for all nPN(see

[21, Proposition 1.11] for instance), in particular, nkpPpC0pLqqq “ nkpC0pLqq “ 1. There-

fore, we are in the conditions to apply Lemma 2.1 and get that C0pLqhas the BPBp-nu

for compact operators, as desired.

ON THE BPBP-NU FOR COMPACT OPERATORS 19

Alternatively, by Remark 3.5, we may prove Theorem 1.6 applying Proposition 2.2

instead of Lemma 2.1.

Acknowledgment. The authors would like to thank Bill Johnson for kindly answering

several inquiries.

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(Domingo Garc´ıa) Departamento de An´

alisis Matem´

atico, Universidad de Valencia, Doctor

Moliner 50, 46100 Burjasot (Valencia), Spain ORCID: 0000-0002-2193-3497

Email address:domingo.garcia@uv.es

(Manuel Maestre) Departamento de An´

alisis Matem´

atico, Universidad de Valencia, Doctor

Moliner 50, 46100 Burjasot (Valencia), Spain ORCID: 0000-0001-5291-6705

Email address:manuel.maestre@uv.es

(Miguel Mart´ın) Departamento de An´

alisis Matem´

atico, Facultad de Ciencias, Universidad

de Granada, 18071 Granada, Spain. ORCID: 0000-0003-4502-798X

Email address:mmartins@ugr.es

(´

Oscar Rold´an) Departamento de An´

alisis Matem´

atico, Universidad de Valencia, Doctor

Moliner 50, 46100 Burjasot (Valencia), Spain. ORCID: 0000-0002-1966-1330

Email address:oscar.roldan@uv.es