L- and M-structure in lush spaces
Let X be a Banach space which is lush. It is shown that if a subspace of X is either an L-summand or an M-ideal then it is also lush.
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irk-123456789-1066652016-10-02T03:02:32Z L- and M-structure in lush spaces Pipping, E. Let X be a Banach space which is lush. It is shown that if a subspace of X is either an L-summand or an M-ideal then it is also lush. 2011 Article L- and M-structure in lush spaces / E. Pipping // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 1. — С. 87-95. — Бібліогр.: 17 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106665 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Let X be a Banach space which is lush. It is shown that if a subspace of X is either an L-summand or an M-ideal then it is also lush. |
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L- and M-structure in lush spaces |
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L- and M-structure in lush spaces |
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l- and m-structure in lush spaces |
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L- and M-structure in lush spaces / E. Pipping // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 1. — С. 87-95. — Бібліогр.: 17 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT pippinge landmstructureinlushspaces |
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2025-07-07T18:50:05Z |
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1837015191414374400 |
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Journal of Mathematical Physics, Analysis, Geometry
2011, vol. 7, No. 1, pp. 87�95
L- and M-structure in lush spaces
Elias Pipping
Fachbereich Mathematik und Informatik, Freie Universität Berlin
10115, Berlin, Germany
E-mail:pipping@math.fu-berlin.de
Received September 9, 2010
Let X be a Banach space which is lush. It is shown that if a subspace of
X is either an L-summand or an M-ideal then it is also lush.
Key words: Lushness, M-summand, M-ideal, L-summand.
Mathematics Subject Classi�cation 2000: 46B20, 46B04.
Introduction
Toeplitz de�ned [1] the numerical range of a square matrix A over the �eld F
(either R or C), i.e. A ∈ Fn×n for some n ≥ 0, to be the set
W (A) = {〈Ax, x〉 : ‖x‖ = 1, x ∈ Fn},
which easily extends to operators on Hilbert spaces. In the 1960s, Lumer [2]
and Bauer [3] independently extended this notion to arbitrary Banach spaces.
For a Banach space X whose unit sphere we denote by SX and an operator
T ∈ B(X) = {T : X → X : T linear, continuous}, we thus call
V (T ) = {x∗(Tx) : x∗(x) = 1, x∗ ∈ SX∗ , x ∈ SX} and v(T ) = sup{|λ| : λ ∈ V (T )}
the numerical range and radius of T , respectively. By construction, we have
v(T ) ≤ ‖T‖ for all T ∈ B(X). The greatest number m ≥ 0 that satis�es
m‖T‖ ≤ v(T ) for every T ∈ B(X)
is called the numerical index of X and denoted by n(X). A summary of what
is and what is not known about the numerical index can be found in [4] and [5].
In the special case n(X) = 1 the operator norm and the numerical radius coincide
on B(X).
c© Elias Pipping, 2011
Elias Pipping
Several attempts have been made to characterize the spaces with numerical
index one among all Banach spaces geometrically, one of them in [6]. We denote
by
S(BX , x∗, α) := {x ∈ BX : Rex∗(x) > 1− α}
for any x∗ ∈ SX∗ and α > 0 an open slice of the unit ball. Setting T :=
{ω ∈ F : |ω| = 1} and writing co(F ) for the convex hull of a subset F ⊆ X al-
lows us to write the absolutely convex hull of F as co(TF ).
De�nition. Let X be a Banach space. If for every two points u, v ∈ SX and
ε > 0 there is a functional x∗ ∈ SX∗ that satis�es
u ∈ S(BX , x∗, ε) and dist(v, co(TS (BX , x ∗, ε))) < ε,
the space X is said to be lush.
Unfortunately, whilst lush spaces do have numerical index one, spaces with
numerical index one need not be lush [7, Rem. 4.2]. Lushness has proved invalu-
able in constructing a Banach space whose dual has strictly smaller numerical
index � answering a question that up until then had been open for decades.
Consequently, the property deserves attention.
Let us recall some results about sums of Banach spaces.
Proposition (M. Martín and P. Payá [8, Prop. 1]). Let (Xn)n∈N be a sequence
of Banach spaces. Then
n
(
c0((Xn)n∈N)
)
= n
(
`1((Xn)n∈N)
)
= n
(
`∞((Xn)n∈N)
)
= inf
n∈N
n(Xn).
In particular, the following statements are equivalent:
(i) every Xn has numerical index one,
(ii) the space c0
(
(Xn)n∈N
)
has numerical index one,
(iii) the space `1
(
(Xn)n∈N
)
has numerical index one, and
(iv) the space `∞
(
(Xn)n∈N
)
has numerical index one.
A notion that has been introduced in [9] is that of a CL space. Originally
de�ned for real spaces, it has proven inappropriate for complex spaces. Thus we
will deal with a weakening introduced in [10] that had previously been used in
[11] but remained unnamed.
De�nition. Let X be a Banach space. If for every convex subset F ⊆ SX
that is maximal in SX with respect to convexity, co(TF ) = BX holds, then X is
called an almost-CL space.
88 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
L- and M-structure in lush spaces
Almost-CL spaces are easily seen to be lush spaces but the converse does not
hold [6, Ex. 3.4(c)]. With regard to sums, the following result has been obtained.
Proposition (M. Martín and P. Payá [12, Prop. 8 & 9]). Let (Xn)n∈N be a
sequence of Banach spaces. Then the following are equivalent:
(i) every Xn is an almost-CL space,
(ii) the space c0
(
(Xn)n∈N
)
is almost-CL, and
(iii) the space `1
(
(Xn)n∈N
)
is almost-CL.
For the recently introduced lushness property, however, only part of the cor-
responding equivalence has been shown.
Proposition (Boyko et al. [13, Prop. 5.3]). Let (Xn)n∈N be a sequence of
Banach spaces. If every Xn is lush, then so are the spaces
c0((Xn)n∈N), `1((Xn)n∈N), and `∞((Xn)n∈N).
We seek to improve this result, bringing it up to par with what has been
proved for almost-CL spaces and spaces with numerical index one.
Inheritance of Lushness
To this end we will show that if X and Y are arbitrary Banach spaces and one
of the two spaces X ⊕1 Y or X ⊕∞ Y is lush, then X and Y are lush themselves.
Such a relation between the spaces X, Y , and their sum can also be expressed
in terms of projections.
De�nition. Let Z be a Banach space and P : Z → Z a linear projection that
satis�es ‖z‖ = max{‖Pz‖, ‖z − Pz‖} for every z ∈ Z. Then P and ranP are
called an M-projection and an M-summand, respectively.
De�nition. Let Z be a Banach space and P : Z → Z a linear projection that
satis�es ‖z‖ = ‖Pz‖ + ‖z − Pz‖ for every z ∈ Z. Then P and ran P are called
an L-projection and an L-summand, respectively.
Basic results of L- and M-structure theory that will be used from here on can
be found in [14, Sec. I.1]. If a subspace X ⊆ Z is an M-summand, its annihilator
X⊥ is an L-summand in Z∗. However, an L-summand of Z∗ need not be the
annihilator of any space X ⊆ Z, nor must subspaces X ⊆ Z for which X⊥ is
an L-summand in Z∗ be M-summands. Subspaces X ⊆ Z for which X⊥ is an
L-summand in Z∗ are referred to as M-ideals.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 89
Elias Pipping
M-summands
We can now proceed to show that M-summands inherit lushness.
Proposition 1. Let X be an M-summand in a lush space Z. Then X is lush.
P r o o f. Let u, v ∈ SX and ε ∈ (0, 1) be arbitrary. Since X is an M-
summand there is an M-projection P : Z → Z with ran(P ) = X. Because Z is
lush, there is a functional z∗ ∈ SZ∗ satisfying u ∈ S(BZ , z∗, ε/2) and
dist(v, co(TS (Bz , z ∗, ε/2 ))) < ε/2 .
Hence there are points z1, . . . , zn ∈ S(BZ , z∗, ε/2) and corresponding θ1, . . . , θn
∈ F that satisfy
∑n
k=1 |θk| ≤ 1 such that ‖∑n
k=1 θkzk − v‖ < ε/2 holds.
The projection P allows us to split these points up into
xk := Pzk and yk := Pxk − xk,
of which the xk appear to approximate v mostly by themselves:
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkzk − v
∣∣∣∣∣
∣∣∣∣∣ = max
{∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkyk
∣∣∣∣∣
∣∣∣∣∣ ,
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkxk − v
∣∣∣∣∣
∣∣∣∣∣
}
.
By Re z∗(x) > 1− ε/2 and ‖z∗‖ = 1 we clearly have Re z∗(yk) ≤ ε/2‖xk‖ ≤ ε/2
for every k and thus
Re z∗(xk) = Re z∗(zk)− Re z∗(yk) > 1− ε,
leaving us with xk ∈ S(BX , z∗, ε), and therefore
dist(v, co(TS (BX , z ∗, ε))) < ε.
By restricting z∗ to X and normalizing the restriction, we obtain the desired
functional.
M-ideals
The celebrated principle of local re�exivity due to Lindenstrauss and Rosenthal
[15] can be used to extend Proposition 1 to M-ideals. More precisely, we require
a re�ned statement.
Theorem (Johnson et al. [16, Sec. 3]). Let X be a Banach space, E ⊆ X∗∗
and F ⊆ X∗ �nite dimensional and ε > 0 arbitrary. Then there is an operator
T : E → X with ||T ||||T−1|| ≤ 1+ ε that satis�es (T ◦ iX)(x) = x for every x ∈ X
with iX(x) ∈ E and x∗∗(x∗) = x∗(Tx∗∗) for every x∗ ∈ F , x∗∗ ∈ E.
90 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
L- and M-structure in lush spaces
An elementary proof is given in [17, Th. 2].
Remark 1. We shall only be concerned with the case X 6= { 0} in the above
theorem. Without loss of generality, we can then assume E ∩ iX(X) 6= { 0}.
Consequently, the ε-isometry T can be chosen to satisfy
1− ε ≤ ||Tz∗∗|| ≤ 1 + ε for every z∗∗ ∈ SE .
With that in mind extending Proposition 1 to M-ideals is straightforward.
Theorem 2. Let X be an M-ideal in a lush space Z. Then X is lush as well.
P r o o f. Let the points u, v ∈ SX be arbitrary and ε > 0. The lushness
of Z now guarantees that there is a functional z∗ ∈ SZ∗ with u ∈ S(BZ , z∗, ε/2)
as well as an absolutely convex combination of points z1, . . . , zn ∈ S(BZ , z∗, ε/2)
and corresponding scalars θ1, . . . , θn ∈ F such that ‖∑n
k=1 θkzk − v‖ < ε/2 and∑n
k=1 |θk| ≤ 1. We observe Z∗∗ = X⊥⊥ ⊕∞ M for some subspace M ⊆ Z∗∗.
For k ∈ { 1, . . . , n} we can now �nd a decomposition iZ(zk) = x∗∗k + y∗∗k with
x∗∗k ∈ X⊥⊥ and y∗∗k ∈ M . By
Re
(
iZ∗(z∗)
)(
iZ(u)
)
= Re z∗(u) > 1− ε/2,
we clearly have
|y∗∗(z∗)| ≤ ε/2 for every y∗∗ ∈ SM .
The functionals x∗∗k satisfy
Re x∗∗k (z∗) = Re z∗(zk)− Re y∗∗k (z∗) > 1− ε
and in particular
1− ε ≤ ‖x∗∗k ‖ ≤ ‖zk‖ = 1.
We also remark
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkzk − v
∣∣∣∣∣
∣∣∣∣∣ = max
{∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θky
∗∗
k
∣∣∣∣∣
∣∣∣∣∣ ,
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkx
∗∗
k − iZ(v)
∣∣∣∣∣
∣∣∣∣∣
}
.
Since X⊥⊥ and X∗∗ can be identi�ed, we have shown that the functionals x∗∗k
meet the requirements of lushness for iX(u) and iX(v) in X∗∗.
In applying the principle of local re�exivity to the �nite dimensional subspace
E := lin {x∗∗1 , . . . , x∗∗n , iZ(v)} ⊆ X∗∗, we obtain an operator T : E → X that
satis�es
• (T ◦ iX)x = x for every x ∈ X with iX(x) ∈ E,
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 91
Elias Pipping
• z∗(Tz∗∗) = z∗∗(z∗) for z∗∗ ∈ E and
• 1− ε/2 ≤ ‖Tz∗∗‖ ≤ 1 + ε/2 for z∗∗ ∈ SE (as per Remark 1).
We can now project x∗∗k onto X with any relevant structure preserved. For xk :=
Tx∗∗k ∈ X we observe
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkxk − v
∣∣∣∣∣
∣∣∣∣∣ =
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkTx∗∗k − (T ◦ iZ)v
∣∣∣∣∣
∣∣∣∣∣ ≤ (1 + ε/2)
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkx
∗∗
k − iZ(v)
∣∣∣∣∣
∣∣∣∣∣ < ε
and Re z∗(xk) = Rex∗∗k (z∗) > 1 − ε. What remains to be done is normalizing.
We thus continue to set x̃k := xk/||xk|| and obtain
‖xk − x̃k‖ = |‖xk‖ − 1|
≤ |‖xk‖ − ‖x∗∗k ‖|+ |‖x∗∗k ‖ − 1|
≤ |‖Tx∗∗k ‖ − ‖x∗∗k ‖|+ ε/2
= ε‖x∗∗k ‖/2 + ε/2
≤ ε,
and therefore
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkx̃k − v
∣∣∣∣∣
∣∣∣∣∣ ≤
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θk(xk − x̃k)
∣∣∣∣∣
∣∣∣∣∣+
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkxk − v
∣∣∣∣∣
∣∣∣∣∣ ≤ max
k≤n
‖xk − x̃k‖+ε ≤ 2ε
as well as
Re z∗(x̃k) ≥ Re z∗(xk)− ‖xk − x̃k‖ > 1− 2ε.
L-summands
Lushness is also inherited by L-summands. To see this we replace the comple-
mentary parts yk of zk with elements ξk ∈ X on which the functional z∗ nearly
attains its norm, such that the θkξk nearly add up to zero.
Theorem 3. Let X be an L-summand of a lush space Z. Then X is lush.
P r o o f. Let u, v ∈ SX and ε > 0 be arbitrary. Since Z is lush, for any η > 0
there is a functional z∗ ∈ SZ∗ as well as z1, . . . , zn ∈ S(BZ , z∗, η) and θ1, . . . , θn
∈ F with
∑n
k=1 |θk| ≤ 1 satisfying u ∈ S(BZ , z∗, η) and ‖∑n
k=1 θkzk − v‖ < η.
Let P be the L-projection onto X. We set xk := Pzk, yk := zk − xk and note
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkzk − v
∣∣∣∣∣
∣∣∣∣∣ =
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkxk − v
∣∣∣∣∣
∣∣∣∣∣ +
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkyk
∣∣∣∣∣
∣∣∣∣∣ .
92 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1
L- and M-structure in lush spaces
In particular, this gives ‖∑n
k=1 θkxk − v‖ < η and ‖∑n
k=1 θkyk‖ < η. Replacing
yk with ξk := ‖yk‖/‖u‖u by setting x̃k := xk + ξk yields ‖x̃k‖ ≤ ‖zk‖ ≤ 1 and
Re z∗(x̃k) = Re z∗(zk − yk + ξk)
> (1− η)− ‖yk‖+ (1− η)‖yk‖
= 1− η − η‖yk‖
≥ 1− 2η.
We observe
Re z∗(yk) = Re z∗(zk)− Re z∗(xk) ≥ (1− η)− ‖xk‖ ≥ ‖yk‖ − η, (1)
which we will utilize to prove
(
Im z∗(yk)
)2 ≤ 2‖yk‖η. (2)
Since (2) trivially holds if ‖yk‖ ≤ η is satis�ed, we shall assume ‖yk‖ > η, leaving
us with
(
Im z∗(yk)
)2 ≤ (Re z∗(yk))2 + (Im z∗(yk))2 − (‖yk‖ − η)2
= |z∗(yk)|2 − ‖yk‖2 + 2‖yk‖η − η2
≤ 2‖yk‖η − η2
< 2‖yk‖η.
We therefore have
∣∣∣∣∣
n∑
k=1
θk Re z∗(yk)
∣∣∣∣∣ =
∣∣∣∣∣
n∑
k=1
θkz
∗(yk)− i
n∑
k=1
θk Im z∗(yk)
∣∣∣∣∣
≤
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkyk
∣∣∣∣∣
∣∣∣∣∣ + max
k≤n
|Im z∗(yk)|
≤ η + max
k≤n
√
2‖yk‖η
≤ η + 2
√
η.
Applying (1) to δk := ‖yk‖ − Re z∗(yk) yields |δk| ≤ η; we conclude
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkξk
∣∣∣∣∣
∣∣∣∣∣ ≤
∣∣∣∣∣
n∑
k=1
θk Re z∗(yk)
∣∣∣∣∣ +
∣∣∣∣∣
n∑
k=1
θkδk
∣∣∣∣∣
≤ 2η + 2
√
η
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 1 93
Elias Pipping
and thus
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkx̃k − v
∣∣∣∣∣
∣∣∣∣∣ =
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θk (xk + ξk)− v
∣∣∣∣∣
∣∣∣∣∣
≤
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkxk − v
∣∣∣∣∣
∣∣∣∣∣ +
∣∣∣∣∣
∣∣∣∣∣
n∑
k=1
θkξk
∣∣∣∣∣
∣∣∣∣∣
≤ 3η + 2
√
η.
Going back and choosing η such that 3η +2
√
η < ε and 2η < ε are satis�ed yields
Re z∗(x̃k) > 1− ε for every k ∈ { 1, . . . , n}
and
dist(v, co(TS (BX , z ∗, ε))) < ε
as desired.
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