The alternative Dunford-Pettis property on projective tensor products

Full text

Math. Z. (2006) 252: 883–897
DOI 10.1007/s00209-005-0894-6
Mathematische Zeitschrift
Antonio M. Peralta ·Ignacio Villanueva
The alternative Dunford-Pettis property on
projective tensor products
Received: 13 June 2005 / Published online: 26 January 2006
© Springer-Verlag 2006
1 Introduction
In 1953 A. Grothendieck introduced the property known as Dunford-Pettis prop-
erty [18]. A Banach space Xhas the Dunford-Pettis property (DPP in the sequel)
if whenever (xn)and (ρn)are weakly null sequences in Xand X∗,respectively,
we have ρn(xn)→0. It is due to Grothendieck that every C(K)-space satisfies the
DPP. Historically, were Dunford and Pettis who first proved that L1(µ) satisfies
DPP. Since its introduction, the DPP has been intensively studied and developed
in many classes of Banach spaces.
In the last twenty years the problem of determine when the projective tensor
product of two Banach spaces satisfies the Dunford-Pettis property has focussed
the attention of several researchers.
Since the DPP is inherited by complemented subspaces, it follows that Xand Y
satisfy the DPP whenever Xˆ
⊗πYhas this property. However, M. Talagrand showed
in [26] that this necessary condition is not always sufficient by finding a Banach
space Xsuch that X∗has the Schur property and X∗ˆ
⊗πL1[0,1]does not satisfy
the DPP. In 1987, R. Ryan proved that Xˆ
⊗πYsatisfy the DPP and does not contain
a subspace isomorphic to 1if and only if Xand Yhave both properties.
First author partially supported by I+D MCYT project no. MTM2005-02541, and Junta de
Andalucía grant FQM 0199 and second author partially supported by I+D MCYT project no.
MTM2004-01308
A. M. Peralta (B
)
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071
Granada, Spain
E-mail: aperalta@ugr.es
I. Villanueva
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de
Madrid, Madrid 28040, Spain
E-mail: ignaciov@mat.ucm.es

884 A. M. Peralta, I. Villanueva
In [4], the authors showed that, for infinite compact Hausdorff spaces K1and
K2, the projective tensor product of C(K1)and C(K2)has the DPP if and only if
K1and K2are scattered, equivalently, C(K1)and C(K2)do not contain 1.This
result was generalised to the more general setting of C*-algebras and JB*-triples,
by showing that the projective tensor product of two Banach spaces Xand Ysat-
isfies the DPP and property (V)of Pelczynzki if and only if Xand Yhave both
properties and do not contain 1. As a consequence it is shown that the projective
tensor product of two infinite dimensional C*-algebras or JB*-triples Eand F(see
definition below) satisfies the DPP if and only if Eand Fsatisfy the DPP and do
not contain 1[3].
In 1997, W. Freedman introduced a strictly weaker version of the DPP, called
the alternative Dunford-Pettis property (see [15]). A Banach space Xhas the alter-
native Dunford-Pettis property (DP1 in the sequel) if whenever xn→xweakly in
X,with xn=x=1,and ρn→0 weakly in X∗,wehaveρn(xn)→0. Freed-
man showed in the same paper that DPP and DP1 are equivalent for von Neumann
algebras. L. J. Bunce and the first author of the present note prove in [7] that the
same result remains true for general C*-algebras. In the setting of JBW*-triples,
it was shown in [1] that a JBW*-triple Whas the DP1 if and only if it satisfies the
DPP or the Kadec-Klee property (KKP in the sequel).
Recently, F. Cabello, D. Pérez-García and the second author of the present note
prove in [9] that when the projective tensor product of two infinite dimensional
C(K)fails the DPP it also fails a weaker property by showing that it contains a
complemented copy of 2.
The main goal of this note is to study the DP1 on projective tensor products
of C*-algebras and JB*-triples. The main difference between DPP and DP1 is that
DPP is and isomorphic property (that is, it is preserved by isomorphisms) while
the DP1 is an isometric property (preserved by surjective linear isometries and not,
in general, by isomorphisms) (see [15, Example 1.6]). In the case of DP1, this will
add the difficulty of working with isometric conditions on the projective tensor
product of two Banach spaces.
In Section 2, we generalise some results obtained in [9] by replacing the C(K)
spaces with more general Banach spaces. More concretely, if Eand Fare Banach
spaces such that Econtains c0and Fcontains a C(K)space containing 1,then
Eˆ
⊗πFcontains a complemented copy of 2. As a consequence, we show that
the projective tensor product of two C*-algebras or JB*-triples contains a comple-
mented copy of 2whenever it fails the DPP.
In Section 3, we completely describe those C*-algebras whose projective ten-
sor product satisfies the DP1. The main tool to solve this problem is Theorem 3.5
whereweprovethatifEand Fare two Banach spaces such that Econtains an
isometric copy of c0and Fcontains and isometric copy of C[0,1],thenEˆ
⊗πF
does not have DP1. This result allows us to show that if Eand Fare JB∗-triples
such that Eis not reflexive and Fcontains 1,thenEˆ
⊗πFdoes not have DP1.
As a consequence, Corollary 3.7 shows that the projective tensor product of two
infinite dimensional C*-algebras Aand Bsatisfies DP1 if and only if Aand Bhave
DP1 and contain no copy of 1, equivalently, Aˆ
⊗πBsatisfies DPP.
Notation: Let Xbe a Banach space. Through the paper we denote by X∗the
dual space of X.IfYis another Banach space, L(X,Y)(respectively, K(X,Y))
will stand for the Banach space of all bounded linear operators (respectively,

DP1 on tensor products 885
compact linear operators) from Xto Y. We usually write L(X)and K(X)
instead of L(X,X)and K(X,X), respectively. The projective tensor product of X
and Ywill be denoted by Xˆ
⊗πY.
AJB*-triple is a complex Banach space Eequipped with a continuous triple
product
{., ., .}:E⊗E⊗E→E
(x,y,z)→{x,y,z}
which is bilinear and symmetric in the outer variables and conjugate linear in the
middle one and satisfies:
(a)(Jordan Identity)
L(x,y){a,b,c}={L(x,y)a,b,c}−{a,L(y,x)b,c}+{a,b,L(x,y)c},
for all x,y,a,b,c∈E,where L(x,y):E→Eis the linear mapping given
by L(x,y)z={x,y,z};
(b)The map L(x,x)is an hermitian operator with non-negative spectrum for all
x∈E;
(c){x,x,x}=x3for all x∈E.
Every C∗-algebra is a JB∗-triple with respect to
{x,y,z}2−1=(xy∗z+zy∗x),
every JB∗-algebra is a JB∗-triple with triple product
{a,b,c}=(a◦b∗)◦c+(c◦b∗)◦a−(a◦c)◦b∗,
and the Banach space B(H,K)of all bounded linear operators between two
complex Hilbert spaces H,Kis also an example of a JB∗-triple with respect to
{R,S,T}=2−1(RS∗T+TS
∗R).
A subspace Iof a JB*-triple Eis called a triple ideal if {E,E,I}+{E,I,E}⊂
I.IfIsatisfies that {I,E,I}⊂I,thenIcalled an inner ideal of E.
A JBW*-triple is a JB*-triple which is also a dual Banach space. The bidual
of a JB*-triple Eis a JBW*-triple with a triple product extending the one on E
(compare [14]).
Every JB*-triple is a Jordan triple. Therefore, given a tripotent e(i.e. {e,e,e}=
e)in a JB*-triple E,there exists a decomposition of Ein terms of the eigenvalues
of L(e,e):
E=E0(e)⊕E1(e)⊕E2(e)
where Ek(e)={x∈E:L(e,e)x=k
2x}for k=0,1,2.Ek(e)is called the
Peirce k-space of e.The following rules are satisfied:
1. {Ei(e), Ej(e), Ek(e)}⊆Ei−j+k(e), where i,j,k=0,1,2andEl(e)=0for
l= 0,1,2.
2. {E0(e), E2(e), E}={E2(e), E0(e), E}=0.

886 A. M. Peralta, I. Villanueva
The projection Pk(e)of Eonto Ek(e)is called the Peirce k-projection of e.
These projections are given by
P2(e)=Q(e)2;
P1(e)=2(L(e,e)−Q(e)2);
P0(e)=IdE−2L(e,e)+Q(e)2.
where Q(e)is the conjugate linear operator on Edefined by
Q(e)(x):= {e,x,e}.
Let Ebe a JB*-triple. A tripotent e∈Eis called minimal if E2(e)coincides
with Ce. Two tripotents e,f∈Eare called orthogonal whenever L(e,f)=0
(equivalently, L(f,e)=0).
2 Complemented copies of 2in the projective tensor product
of Banach spaces
As we said in the introduction, in this section we extend some of the results in [9]
to a wider class of Banach spaces, which in particular includes C∗-algebras and
non reflexive JB∗-triples.
We start by recalling some known notions. Given Xand 1 ≤p≤∞,wesaythat
a (finite or infinite) sequence (xn)n⊂Xis strongly p-summable if (xn)n∈p.
In that case we define its strong p-summing norm by
(xn)np=
nxnp1
p
.
Analogously we say that (xn)nis weakly p-summable if, for every x∗∈X∗,
(x∗,xn)n∈p. In that case we define the weak p-summing norm of (xn)by
(xn)nω
p=sup 


nx∗,xnp1
p
:x∗∈BX∗


.
We recall that, if 1 ≤p<+∞, an operator T:X−→ Yis called p-summing
it takes weakly p-summable sequences to strong p-summable sequences, that is,
if there exists a constant K>0 such that, for every p-weak summing sequence
(xn)n⊂X

nT(xn)p1
p
≤K(xn)ω
p(1)
In that case, we define the p-summing norm of Tby πp(T)=min{K:
Kverifies (1)}.
The following result is proved in [9].

DP1 on tensor products 887
Lemma 2.1 Let (bn)⊂C(K2)be a bounded sequence (for instance,
bn≤C for every n ∈N)and let (en)⊂C(K1)be a sequence such that
(en)nω
1≤1. Then, the sequence (en⊗bn)⊂C(K1)ˆ
⊗πC(K2)verifies that
(en⊗bn)nw
2≤√2C.
We state now our first result, which gives sufficient conditions for the projective
tensor product to have a complemented copy of 2.
Theorem 2.2 Let E,F be Banach spaces such that E contains c0and F contains
aC(K)space G containing 1.ThenEˆ
⊗πF contains a complemented copy of 2.
Proof The proof is a generalization of [9, Theorem 1]. Let (xn)n⊂Ebe a sequence
equivalent to the canonical basis of c0,andlet(x∗
n)n⊂E∗be a sequence bior-
thogonal to (xn)n.SinceGcontains a copy of 1, [13, Corollary 4.16] allows us
to consider a surjective operator q:G−→ 2. By Grothendieck’s theorem q
is 2-summing, hence, according to [13, Theorem 4.15], qcan be extended to a
quotient ˜q:F−→ 2.Let(bn)n⊂G⊂Fbe a bounded sequence such that
˜q(bn)=q(bn)=en,where (en)is the canonical basis in 2. According to Lemma
2.1 the sequence (xn⊗bn)nis 2-weak summing when considered in c0ˆ
⊗πG.
Therefore, since the natural operator c0ˆ
⊗πG−→ Eˆ
⊗πFis continuous, (xn⊗bn)
is also 2-weak summing when considered in Eˆ
⊗πF. Therefore (see for example
[13, Proposition 2.2]) we can define an operator
θ:2−→ Eˆ
⊗πF
by
θ(en)=xn⊗bn.
Defined this way θis bounded by KGtimes a bound for the sequence (bn)n,where
KGis Grothendieck’s constant.
We also define an operator
ϕ:Eˆ
⊗πF−→ 2
by
ϕ(a⊗b)=(x∗
n(a)q(b)n)n.
It is easy to see that ϕis well defined, continuous, and that ϕ◦θis the identity map
on 2, which finishes the proof. 
Remark 2.3 Let us recall that, given λ>1, a Banach space Xis said to be an
L∞,λ space if every finite dimensional subspace Y⊂Xis contained in a finite
dimensional space Z⊂Xfor which there exists an isomorphism v:Z−→ dim Z
∞
such that vv−1<λ.Wesaythat Xis an L∞space if it is an L∞,λ space
for some λ>1. L∞(µ) and C(K)spaces are the basic examples of L∞-spaces.
Clearly, our previous Theorem remains true if Gis any L∞space.

888 A. M. Peralta, I. Villanueva
It has been recently shown in [3] that the projective tensor product of two JB∗-
triples fails the DPP if and only if at least one of them contains 1. As an application
of our previous result we prove that in that case, actually there is a complemented
copy of 2in the projective tensor product. We recall first some known results.
Let xbe an element in a complex JB∗-triple E, and denote by E(x)the JB∗-
subtriple of Egenerated by x. It is known that there exists a locally compact subset
Sxof (0,+∞)such that Sx∪{0}is compact and E(x)is JB∗-triple isomorphic to
the C∗-algebra C0(Sx),viaatripleisomorphism, which satisfies (x)(t)=t
(t∈Sx) (cf. [21, 4.8], [22, 1.15] and [16]). The subset Sxis called the triple
spectrum of x.
It is well known that for every infinite dimensional C*-algebra Athere exists
an infinite sequence (xn)in A+satisfying xnxm=xmxn=0andxn=1,for all
n= m(compare [20, 4.6.13]). Since the subtriple generated by a single element
xin a complex JB*-triple Eis isomorphic as JB*-triple (and hence isometric as
Banach space) to a C*-algebra, we can conclude, from [11, Theorem 6] and [6,
Proposition 4.5], that for every non-reflexive complex JB*-triple, E,there exists an
infinite sequence (xn)in Esatisfying L(xn,xm)=L(xm,xn)=0andxn=1,
for all n= m. It is well known that (xn)is equivalent to the basis of c0(compare,
for example, [19, §4]). Therefore, we have:
Corollary 2.4 Let E,FbeJB
∗-triples such that E is not reflexive and F contains
1.ThenEˆ
⊗πF contains a complemented copy of 2.
The corresponding result in the setting of C*-algebras follows immediately
from the above Corollary.
Corollary 2.5 Let A,B be two infinite dimensional C∗-algebras such that B con-
tains 1.Then Aˆ
⊗πB contains a complemented copy of 2.
There is also a local version of Theorem 2.2.
Theorem 2.6 Let E,F be Banach spaces each one of them containing an iso-
morphic copy of L∞spaces G,H. Then E ˆ
⊗πF contains uniformly complemented
copies of the n
2’s.
Proof Let us fix n∈N. There exists an n-dimensional space X⊂Gsuch that
d(X;n
∞)≤C,whereCis a constant independent of n.InXwe consider the
canonical basis (xn
m)n
m=1and we consider a sequence (xn∗
m)n
m=1⊂E∗biorthogo-
nal to it.
We fix an >0 and Dvoretzky’s Theorem, assures the existence of a number
N∈Nsuch that n
2is (1+)-contained in N
1. Therefore, there exists a quotient
qn:N
∞−→ n
2and a constant Ksuch that, for every 1 ≤n≤mthere exists
bm
n∈N
∞with bm
n≤Kand qn(bm
n)=em,whereKis a constant independent
of nand 1 ≤n≤m.WeconsideraspaceY⊂Hverifying d(Y,
N
∞)≤C,where
Cis again independent of N. We consider the quotient ˆqn:Y−→ n
2induced by
qn. The 2-summing norm of ˆqnis bounded by qnKGC(1+) (independently of
n) and hence applying again [13, Theorem 4.15] ˆqncan be extended to a quotient
˜qn:F−→ n
2with no greater norm. Now the proof proceeds as in Theorem 2.2.


DP1 on tensor products 889
Corollary 2.7 Let E,F be non reflexive JB∗-triples. Then E ˆ
⊗πF contains uni-
formly complemented copies of the n
2’s. In particular, the result remains true
whenever E and F are infinite dimensional C∗-algebras.
We will substantially improve this last corollary in our next result.
In [3, Remark 2.7], the authors prove the existence of JB∗-triples E,Fsuch
that Eˆ
⊗πFhas DPP, (Eˆ
⊗πF)∗is a Schur space and (Eˆ
⊗πF)∗∗ does not have
DPP.
Until quite recently, the (essentially) unique example of a space Xwith the DPP,
such that X∗does not have DPP was Stegall’s example c0(n
2);(c0(n
2))∗=1(n
2)
is a Schur space, but (c0(n
2))∗∗ =∞(n
2)contains a complemented copy of 2,
and therefore does not have DPP [25].
Recently it was known that the same situation happens with c0ˆ
⊗πc0:thatis,
(c0ˆ
⊗πc0)∗is a Schur space but (c0ˆ
⊗πc0)∗∗ does not have DPP ([17,8]). Very
recently, it was proved in [9] that c0ˆ
⊗πc0contains a complemented copy of c0(n
2).
We prove next that the above mentioned projective tensor product of JB∗-tri-
ples appearing in [3] contains a complemented copy of c0ˆ
⊗πc0, and hence it also
contains a complemented copy of c0(n
2). Therefore, Stegall’s example remains
essentially unique.
Other interesting examples of JB*-triples are constituted by the so-called Car-
tan factors, Cα(α =1,...,6), defined as follows: let Hand Kbe complex
Hilbert spaces and let j:H→Hbe a conjugation (conjugate linear isom-
etry of period-2) on H,C1=L(H,K),C2={x∈L(H):jx∗j=−x},
C3={x∈L(H):jx∗j=x},C4is a complex spin factor (that is, a renormed
Hilbert space, see for example [3, Page 9]), C5=M1,2(O),andC6=H3(O),
the hermitian 3 ×3 matrices with entries in the eight-dimensional Cayley division
algebra O. The elementary JB*-triples, Kα(α =1,...,6), introduced in [5, page
330], can be described in the following way: K1=K(H,K),Kα=Cα∩K(H),
for α=2,3, and Kα=Cα,for α=4,5,6.
The following result, which is needed to prove Corollary 2.9, is probably inter-
esting by itself.
Theorem 2.8 Let E be a non reflexive JB*-triple not containing 1. Then E con-
tains a complemented copy of c0.
Proof Let Adenote the weak*-closed triple ideal generated by all minimal tripo-
tents of E∗∗,andletK0(E)=E∩A.ItisknownthatK0(E)is an inner ideal of E
(compare [6, Corollary 3.5]). It is also known that K0(E)is a c0-sum of a family,
{Ki}i∈, of elementary JB*-triples [6, Lemma 3.3].
It is known that a JB*-triple satisfies the RNP if and only if it is reflexive (com-
pare [11, Theorem 6]). Now it follows from [5, Theorem 3.4] and [6, Proposition
4.5] that the triple spectrum of every element x∈Eis countable and there exists
at least one xin Ewhose triple spectrum, Sx, must be infinite. Arguing as in the
proof of [5, Theorem 3.4, (iv) ⇒(v)] (see also the proof of [5, Proof of Lemma
3.2], we deduce that, for each n∈N,Econtains a set {e1,...,en}of pairwise
orthogonal minimal tripotents of E∗∗. In particular, we have K0(E)={0}. From
[2, Lemma 3.7 and Theorem 3.8] we deduce that there exists a sequence (en)⊂E
of pairwise orthogonal tripotents of E∗∗.
Therefore, there exists an inner ideal Iof Econtaining a sequence (en)of
pairwise orthogonal tripotents of E∗∗.Theseriesnenis weak*-convergent to a

890 A. M. Peralta, I. Villanueva
tripotent u∈I∗∗ (compare [19]). Since for each x∈I,theseriesn∈NP2(en)(x)
converges in the weak*-topology of I∗∗ to P2(u)(x)we deduce that the map-
ping P:I→Lin{en}=c0,P(x):= (P2(en)(x))n∈Nis a bounded linear
projection of Ionto c0.SinceIis an inner ideal of E, then for each x∈Eand
n∈N,we have P2(en)(x)={en,{en,x,en},en}∈Iand hence (P2(en)(x))n=
(P2(en)P2(en)(x))n∈c0.Therefore, P:E→c0is a well-defined contractive
projection of Eonto c0.
The case E=∞tells us that we can not just remove the condition that Edoes
not contain 1in the above theorem.
The following announced corollary follows now immediately.
Corollary 2.9 Let E and F be two infinite dimensional JB*-triples not containing
1and satisfying the DPP. Then E ˆ
⊗πF contains a complemented subspace iso-
morphic to c0ˆ
⊗πc0. As a consequence, E ˆ
⊗πF contains a complemented subspace
isomorphic to c0(n
2).
Proof We just need to observe that infinite dimensional reflexive Banach spaces
never have DPP. The last statement follows now from [9], where it is proved that
c0ˆ
⊗πc0contains a complemented copy of c0(n
2).
3 The alternative Dunford-Pettis property on tensor products
In this section we find sufficient conditions to assure that the projective tensor prod-
uct of two Banach spaces does not have DP1. These conditions will completely
determine those C∗-algebras whose projective tensor product satisfies DP1. They
also solve the same problem for many, but not all, cases of JB∗-triples.
We develop first some necessary tools.
Let X1,X2,Y1,Y2be real Banach spaces. Every bounded linear operator T
from X1⊕∞X2to Y1⊕1Y2,can be written as a matrix
T=T1T2
T3T4,
where T1:X1→Y1,T2:X2→Y1,T3:X1→Y2,and T4:X2→Y2are
bounded linear operators.
Lemma 3.1 Let X1,X2,Y1,Y2be Banach spaces. The law
T=T1T2
T3T4→ P(T)=T10
0T4,
defines a contractive linear projection on L(X1⊕∞X2,Y1⊕1Y2). Moreover,
P(T)=T1+T4.
Proof Suppose first that X1,X2,Y1and Y2are real Banach spaces. Let T∈
L(X1⊕∞X2,Y1⊕1Y2).
T= sup
xi≤1T(x1,x2)= sup
xi∈BXi
sup
ϕi∈BY∗
i|(ϕ1,ϕ
2)T(x1,x2)|

DP1 on tensor products 891
=sup
xi∈BXi
sup
ϕi∈BY∗
i|ϕ1T1(x1)+ϕ1T2(x2)+ϕ2T3(x1)+ϕ2T4(x2)|
=sup
xi∈BXi
sup
ϕi∈BY∗
i|ϕ1T1(x1)+ϕ1T2(x2)|+|ϕ2T3(x1)+ϕ2T4(x2)|
=sup
xi∈BXi,ϕi∈BY∗
i
max{|ϕ1T1(x1)+ϕ1T2(x2)|+|ϕ2T3(x1)+ϕ2T4(x2)|,
|ϕ1T1(x1)−ϕ1T2(x2)|+|ϕ2T3(x1)−ϕ2T4(x2)|}
sup
xi∈BXi,ϕi∈BY∗
i
max{|ϕ1T1(x1)+ϕ1T2(x2)+ϕ2T3(x1)+ϕ2T4(x2)|,
|ϕ1T1(x1)+ϕ1T2(x2)−ϕ2T3(x1)−ϕ2T4(x2)|,|ϕ1T1(x1)−ϕ1T2(x2)
+ϕ2T3(x1)−ϕ2T4(x2)|,|ϕ1T1(x1)−ϕ1T2(x2)−ϕ2T3(x1)+ϕ2T4(x2)|}
=sup
xi∈BXi,ϕi∈BY∗
i
max{|ϕ1T1(x1)+ϕ2T4(x2)|+|ϕ1T2(x2)+ϕ2T3(x1)|,
|ϕ1T1(x1)−ϕ2T4(x2)|+|ϕ1T2(x2)−ϕ2T3(x1)|}
≥sup
xi∈BXi,ϕi∈BY∗
i
max{|ϕ1T1(x1)+ϕ2T4(x2)|,|ϕ1T1(x1)−ϕ2T4(x2)|}
=sup
xi∈BXi,ϕi∈BY∗
i|ϕ1T1(x1)|+|ϕ2T4(x2)|=T1+T4=P(T).
When X1,X2,Y1and Y2are complex Banach spaces we can apply the formula
max
θ∈[0,2π]{|eiθa+b|,|eiθa−b|} = |a|+|b|,
(a,b∈C,) instead of |a|+|b|=max{|a+b|,|a−b|},(a,b∈R,)togetthe
statement in the complex case. 
We need the following proposition, which might be useful also in different
settings.
Proposition 3.2 Let X1,X2,Y1and Y2be Banach spaces. Then X1ˆ
⊗πY1⊕∞
X2ˆ
⊗πY2is complemented in X1⊕∞X2ˆ
⊗πY1⊕∞Y2.

892 A. M. Peralta, I. Villanueva
Proof Let p1and p2denote the contractive projections of X1⊕∞X2and Y1⊕∞Y2
onto X1and Y1, respectively, and let q1=1−p1and q2=1−p2. By [12,
Proposition 3.9] we conclude that the mapping =p1⊗p2+q1⊗q2is a
bounded linear projection on X1⊕∞X2ˆ
⊗πY1⊕∞Y2,≤2and
(α)=p1⊗p2(α) +q1⊗q2(α)≥max{p1⊗p2(α),q1⊗q2(α)}.
We claim that, in fact
(α)=p1⊗p2(α) +q1⊗q2(α)=max{p1⊗p2(α),q1⊗q2(α)}.
Indeed, let us denote α1=p1⊗p2(α) and α2=q1⊗q2(α).
α1+α2= sup
T∈BL(X1⊕∞X2,Y∗
1⊕1Y∗
2)|T(α1)+T(α2)|
=sup
1≥T≥T1+T4|T1(α1)+T4(α2)|
≤sup
1≥T1+T4|T1(α1)|+|T4(α2)|
=max{α1,α2}.

Let Xbe a Banach space. We say that two elements x,y∈Xare M-orthogonal
if and only if λx+µy=max{λx,µy},forallλ, µ ∈K.
Corollary 3.3 Let X and Y be Banach spaces. Let x1,x2∈X and y1,y2∈Ybe
such that x1is M-orthogonal to x2and y1is M-orthogonal to y2. Then, if .π
denotes the projective tensor norm in X ˆ
⊗πY,we have x1⊗y1+x2⊗y2π=
max{x1⊗y1π,x2⊗y2π}.
Proof Let A,Bbe Banach spaces. Following the notation in [12], in this proof
the projective tensor norm of an element α∈Aˆ
⊗πBwill also be denoted by
π(α;A,B). From [12, 3.2.(5)], we conclude that
π(α;X,Y)=inf π(α;M,N):α∈M⊗N;M⊂Xand
N⊂Yfinite dimensional .(2)
Let M(respectively, N) denote the linear span of x1and x2(respectively, y1
and y2). Clearly, M=Kx1⊕∞Kx2and N=Ky1⊕∞Ky2.From (2)we have
x1⊗y1+x2⊗y2π=π(x1⊗y1+x2⊗y2;X,Y)
≤π(x1⊗y1+x2⊗y2;M,N).
Now Proposition 3.2 gives that
π(x1⊗y1+x2⊗y2;M,N)=max{π(x1⊗y1;M,N), π (x2⊗y2;M,N)}
=max{x1⊗y1π,x2⊗y2π},

DP1 on tensor products 893
showing that
x1⊗y1+x2⊗y2π≤max{x1⊗y1π,x2⊗y2π}.
To see the reverse inequality, suppose that max{x1⊗y1π,x2⊗y2π}=
x1⊗y1π.Let ϕbe the norm-one functional in M∗given by ϕ(λx1+µx2)=
λx1,andletφ∈X∗a Hahn-Banach extension of ϕ.Foreveryε>0 we can take
a norm-one functional ψ∈Y∗such that ψ(y1)>y1−ε.LetT∈L(X,Y∗)be
the norm-one linear operator defined by T(x)(y):= (φ ⊗ψ)(x⊗y)=φ(x)ψ(y).
The it follows that
x1⊗y1+x2⊗y2π≥|T(x1⊗y1+x2⊗y2)|=x1(y1−ε),
which gives the desired inequality. 
Remark 3.4 In C[0,1]we can always find a norm one operator q:C[0,1]−→2
and a sequence, (yn), in the unit sphere of C[0,1]such that q(yn)=1
√2en,where
(en)n⊂2is the canonical basis. One way to do this is the following: we consider
λto be the Lebesgue measure on [0,1].Thelaw
f,g=1
0fgdλ, f,g∈C[0,1],
defines a positive sesquilinear form on C[0,1]with prehilbertian norm denoted by
·
λ. Then the completion Hλof (C[0,1],·
λ)is a Hilbert space with respect
to the norm .λ. The natural inclusion q:(C[0,1],·
∞)−→ Hλis clearly a
norm one linear operator and defining yn(t):= sin(2πnt)(t∈[0,1],n∈N)we
get the desired statement.
We prove now the main result of the paper.
Theorem 3.5 Let E,F be two Banach spaces such that E contains an isometric
copy of c0and F contains and isometric copy of C[0,1].ThenEˆ
⊗πF does not
have DP1.
Proof We first observe that C[0,1]contains C0,1
4∪3
4,1=C0,1
4⊕∞
C3
4,1isometrically and 1-complemented, with the inclusion
i:C0,1
4∪3
4,1−→ C[0,1]
given by
i(f,g)(t)=



















f(t)if t∈0,1
4
0ift=1
2
g(t)if t∈3
4,1
affine 1
4,1
2and 1
2,3
4

894 A. M. Peralta, I. Villanueva
and the natural projection
π:C[0,1]−→C0,1
4∪3
4,1
given by the restriction, that is,
π( f)(t)=f(t).
We consider the function 11:[0,1
4]−→Kof constant value 1, and we define
y0=i(11)∈C[0,1]. Reasoning as in Remark 3.4 we can also consider a sequence
(zn)n⊂C3
4,1of norm one functions and a mapping
p:C3
4,1−→ 2
such that p(zn)=1
√2en.Foreveryn∈N,wedefineyn=i(zn).Thenyn=1
for every n∈N. It follows from the definitions that, for every n∈N,y0is
M-orthogonal to yn. We can also consider the operator q:C[0,1]−→2which
naturally extends p,thatis,q=p◦π2,whereπ2:C[0,1]−→C3
4,1is again
the projection induced by the restriction.
We consider now the sequence (en⊗yn)n≥1⊂Eˆ
⊗πF. Clearly
en⊗ynEˆ
⊗πF=enyn=1.
Considering c0as c0(N∪{0}), we consider the vector e0⊗y0∈Eˆ
⊗πF,which
also has norm 1.
An application of Corollary 3.3 shows that
en⊗yn+e0⊗y0Eˆ
⊗πF=1.
It follows from Grothendieck’s theorem that q:C[0,1]−→2is 2-summing.
Therefore, [13, Theorem 4.15] allows us to extend qto an operator ˜q:F−→ 2.
We consider now a sequence (x∗
n)∈E∗biorthogonal to the c0-basis (en)⊂Eand
we define the operator
θ:Eˆ
⊗πF−→ 2
by
θ(a⊗b)=(x∗
n(a)˜q(b)n)n≥1
It is not difficult to see that θis well defined and continuous (the easiest way is to
consider θas a bilinear operator on E×F).
We consider now the sequence (θ∗(en))n⊂(Eˆ
⊗πF)∗,where(en)n⊂2
denotes now the canonical basis of 2. Clearly (θ∗(en))nweakly converges to 0.
Finally, noticing that
θ(e0⊗y0)=0

DP1 on tensor products 895
(both because x∗
n(e0)=0foreveryn≥1 and because ˜q(y0)=q(y0)=
p(π2)(y0)=p(0)=0) we have
θ∗(en)(en⊗yn+e0⊗y0)θ∗(en)(en⊗yn)+θ∗(en)(e0⊗y0)
=θ(en⊗yn)n+θ(e0⊗y0)n=1
√2+0→ 0.

Next we specialize the last result for the case of JB∗-triples.
Corollary 3.6 Let E,FbeJB
∗-triples such that E is not reflexive and F contains
1.ThenEˆ
⊗πF does not have DP1.
Proof By the remarks preceding Corollary 2.4, we know that Econtains an iso-
metric copy of c0.
Moreover, it is known that a JB*-triple Fcontains a subspace isomorphic to 1
if and only there exists a norm-one element x∈Fsuch that the triple spectrum of
xis not countable [5, Theorem 3.4]. Therefore, we may assume that there exists a
norm-one element x∈Fsuch that Sxis not countable. We recall that 1 ∈Sxis a
locally compact subset of (0,1],Sx∪{0}is compact and the JB*-subtriple of F
generated by xis triple isomorphic (and hence isometric) to C0(Sx)(the Banach
space of all complex-valued continuous functions on Sxvanishing at 0).
It is clear that C0(Sx)is also a JB*-subtriple of C(Sx∪{0}).SinceSxis not count-
able, it follows, again from [5, Theorem 3.4], that C0(Sx)and hence C(Sx∪{0})
contains a subspace isomorphic to 1. From [23] we conclude that Sx∪{0}is
non scattered and thus there exists a continuous surjection σ:Sx∪{0}→[0,1]
(compare [24, 8.5.4]).
The mapping :C([0,1])→C(Sx∪{0})defined by ( f)(t):= f(σ (t))
is an isometric linear JB*-triple embedding of C([0,1])into C(Sx∪{0}). If we
denote σ(0)=a∈[0,1]and C0([0,1]\{a})the Banach space of all complex-val-
ued continuous functions on [0,1]\{a}vanishing at a, it follows that T|C0([0,1]\{a})
is an isometric linear triple embedding of C0([0,1]\{a})into C0(Sx∪{0}). Since
C0([0,a)) or C0((a,1])is triple isomorphic (an hence isometric) to C0((0,1])we
may always assume that there exists a isometric linear JB*-triple embedding of
C0((0,1])into C0(Sx∪{0})and hence into F. Therefore, we may assume, without
lost of generality, that C0((0,1])is a JB*-subtriple of F.
The result follows now from our previous theorem. 
We have already commented that every infinite dimensional C*-algebra is not
reflexive, in fact, both properties are equivalent. In [7] it is shown that the DPP and
the DP1 are equivalent in every C*-algebra. Combining the above comments with
Theorem 3.6 and [3, Corollary 2.5] we get the following.
Corollary 3.7 Let A and B be infinite dimensional C*-algebras. Then the follow-
ing are equivalent:
(a)Aˆ
⊗πB satisfies the DP1;
(b)A and B satisfy the DPP and do not contain 1;
(c)Aˆ
⊗πB satisfies the DPP;

896 A. M. Peralta, I. Villanueva
Corollary 3.8 Let K1and K2be infinite compact Hausdorff spaces. Then the
following are equivalent:
(a)C(K1)ˆ
⊗πC(K2)satisfies the DP1;
(b)C(K1)and C(K2)satisfy the DPP and do not contain 1;
(c)C(K1)ˆ
⊗πC(K2)satisfies the DPP;
Problem 3.9 In order to determine when the projective tensor product of two JB∗-
triples satisfies DP1, we need to know whether E ˆ
⊗πF has DP1, when only one of
them is reflexive. In particular, we do not know whether c0ˆ
⊗π2or C[0,1]ˆ
⊗π2
have DP1. In both cases we have DPP in one of the factors and the Kadec-Klee
Property in the other one.
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