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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 34, Num. 1 (2019), 1 – 17

doi:10.17398/2605-5686.34.1.1

Available online February 3, 2019

Ideal operators and relative Godun sets

S. Dutta∗, C.R. Jayanarayanan, Divya Khurana

Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, India

Department of Mathematics, Indian Institute of Technology Palakkad, India
crjayan@iitpkd.ac.in

Department of Mathematics, Weizmann Institute of Science, Israel

divya.khurana@weizmann.ac.il , divyakhurana11@gmail.com

Received August 13, 2017 Presented by David Yost
Accepted July 2, 2018

Abstract: In this paper we study ideals in Banach spaces through ideal operators. We provide
characterisation of recently introduced notion of almost isometric ideal which is a version of Principle

of Local Reflexivity for a subspace of a Banach space. Studying ideals through ideal operators give

us better insight in to the properties of these subspaces vis-a-vis properties of the space itself. We
provide a few applications of our characterisation theorem.

Key words: Ideals, almost isometric ideals, strict ideals, maximal ideal operator, Godun sets, VN-

subspaces.

AMS Subject Class. (2010): 46E30, 46B20.

1. Introduction and preliminaries

Principle of local reflexivity (henceforth PLR) states that finite dimen-
sional subspaces of X∗∗, the bidual of a Banach space X, are almost isometric
to finite dimensional subspaces of X. PLR also provides control over actions
of a fixed finite dimensional subspace F ⊆ X∗ on a finite dimensional subspace
E ⊆ X∗∗ and its almost isometric copy in X. It is immediately realised that
PLR is a consequence of finite representability of X∗∗ in X and X∗ is norming
for X∗∗, which in turn is a consequence of existence of norm one projection in
X∗∗∗ with range X∗ and kernel X⊥. Since finite representability is a notion
which can be defined for arbitrary pair of Banach spaces, the notion of ideals
were introduced and studied extensively. A closed subspace Y of a Banach
space X is said to be an ideal in X if there exists a norm one projection
P : X∗ −→ X∗ with ker(P) = Y ⊥. The following characterisation of ideals,
though it was widely known, was stated explicitly in [12].

∗Professor Dutta passed away while the paper was being finalized.

ISSN: 0213-8743 (print), 2605-5686 (online)

https://doi.org/10.17398/2605-5686.34.1.1
mailto:crjayan@iitpkd.ac.in
mailto:divya.khurana@weizmann.ac.il
mailto:divyakhurana11@gmail.com
https://www.eweb.unex.es/eweb/extracta/
https://creativecommons.org/licenses/by-nc/3.0/


2 s. dutta, c.r. jayanarayanan, d. khurana

Theorem 1.1. Let Y be a subspace of a Banach space X. Then Y is
an ideal in X if and only if there exists a Hahn–Banach extension operator
φ : Y ∗ −→ X∗ such that for every � > 0 and every finite dimensional subspaces
E ⊂ X and F ⊂ Y ∗ there exists T : E −→ Y such that

(a) Te = e for all e ∈ Y ∩E,
(b) ||Te|| ≤ (1 + �)||e|| for all e ∈ E,
(c) φf∗(e) = f∗(Te) for all e ∈ E, f∗ ∈ F .

Clearly the notion of ideal imitates PLR. However it lacks two important
aspects of X in X∗∗ situation; namely, we do not get almost isometry from
the finite dimensional subspace E ⊆ X to Y and Range(P) is not norming
for X, where by norming we mean 1-norming. So following two possible
strengthening of notion of ideals were considered.

Definition 1.2. [9] A subspace Y of a Banach space X is said to be a
strict ideal in X if Y is an ideal in X and Range(P) is norming for X where
P : X∗ −→ X∗ is a norm one projection with ker(P) = Y ⊥.

Definition 1.3. [2] A subspace Y of a Banach space X is said to be an
almost isometric ideal (henceforth ai-ideal) in X if for every � > 0 and every
finite-dimensional subspace E ⊂ X there exists T : E −→ Y which satisfies
condition (a) in Theorem 1.1 and (1 − �)||e|| ≤ ||Te|| ≤ (1 + �)||e|| for all
e ∈ E.

On the other hand there is a notion of u-ideal, which is a generalisation of
M-ideal and is also a strengthening of PLR.

Definition 1.4. [9] A subspace Y of a Banach space X is said to be
a u-ideal in X if there exists a norm one projection P : X∗ → X∗ with
ker(P) = Y ⊥ and ‖IX∗ − 2P‖ = 1, where IX∗ denotes the identity operator
on X∗.

It is straightforward to see that strict ideals are ai-ideals and ai-ideals are
of course ideals. The inclusions are strict as shown by examples in [2].

If we view above notion of ideals and its subsequent strengthening as gen-
eralisations of X in X∗∗ situation, many isometric properties of X are carried
to Y and much of the studies in this area are devoted to that.

However PLR can be viewed simply as: identity operator on X∗∗ is an
extension of identity operator on X and X∗ is norming for X. On the same



ideal operators and relative godun sets 3

vein, the property of Y being an ideal in X may be viewed as there exists
T : X → Y ∗∗ such that ‖T‖ ≤ 1 and T|Y = IY . To see this consider the
following.

Suppose Y ⊆ X and T : X → Y ∗∗ is such that T|Y = IY and ‖T‖ ≤ 1.
We will refer T as an ideal operator. Given T we define P : X∗ → X∗
as P(x∗) = T∗|Y ∗ (x∗|Y ) and φ : Y ∗ → X∗ by φ(y∗) = T∗|Y ∗ (y∗). Then
P is a norm one projection on X∗ with ker(P) = Y ⊥ and φ is a Hahn–
Banach extension operator. In the sequel, we will refer these P and φ as
ideal projection and Hahn–Banach extension operator corresponding to T.
We also note that this definition is reversible, in the sense that given an ideal
projection P , we may define T as above. Same is true for φ.

With this view point, in Section 2 of this paper we provide characteri-
sations of ideal, strict ideal and ai-ideal in terms of the properties of ideal
operator T. In the case of ideal and strict ideal (see Proposition 2.1) the
results are essentially known.

Proposition 2.1 has some immediate corollaries. It is straightforward to see
that if Y is 1-complemented in X then Y is an ideal in X (if P : X → X is a
projection with ‖P‖ = 1 then P∗ is an ideal projection). For a space Y which
is 1-complemented in its bidual Y ∗∗ we show Y is an ideal in any superspace X
if and only if Y is 1-complemented in X. In particular L1(µ) or any reflexive
space is an ideal in a superspace if and only if it is 1-complemented. It also
follows that any ideal in L1(µ) is 1-complemented.

Coming to ai-ideals we show that reflexive spaces with a smooth norm can-
not have any proper ai-ideal. Situation becomes more interesting for the space
C[0, 1], the space of all real-valued continuous functions on [0,1] equipped
with the supremum norm. It is known that C[0, 1] is universal for the class
of separable Banach spaces. We show that any ai-ideal of C[0, 1] inherits the
universality property from C[0, 1]. Also, any separable ai-ideal in L1(µ), µ
non-atomic, is isometric to L1[0, 1].

Section 3 of this paper is mainly devoted to the study of properties of ideal
operator T. There is a need to exercise some caution while dealing with this
operator. It may well be the case that for some ideal operator, Y has ‘nice’
property in X but there are other ideal operators for which such properties
fail. For example, let Y be a u-ideal in Y ∗∗. Then Y is always a strict ideal
in Y ∗∗ under canonical projection π determined by Y ∗∗∗ = Y ∗ ⊕ Y ⊥. But
canonical projection may not satisfy u-ideal condition, namely, ‖I − 2π‖ = 1.
So we introduce the notion of a maximal ideal operator (vis-a-vis, maximal
ideal projection) and discuss properties of maximal ideal operator. However,



4 s. dutta, c.r. jayanarayanan, d. khurana

there are situations where there is only one possible ideal operator. Here we
introduce relative Godun set G(Y,X) of X with respect to Y . We recall that
Y ⊆ X is said to have unique ideal property (henceforth UIP) in X if there is
only one possible norm one projection P on X∗ with ker(P) = Y ⊥. Similarly
Y ⊆ X is said to have unique extension property (henceforth UEP) in X if
there is only one possible T : X → Y ∗∗ such that ‖T‖ ≤ 1 and T|Y = IY .
From the above relation of P and T it is clear that Y has UIP in X if and
only if it has UEP in X (and in this case φ is also unique).

If Y has UEP in Y ∗∗ then we just say Y has UEP.
UEP also provides a sufficient condition for an ai-ideal in a dual space to

be a local dual for predual. See [6] and references there in for recent results
on local duals. In particular a pertinent question in this area is if a separable
Banach space with non separable dual always has a separable local dual.

We now provide some brief preliminaries needed throughout this paper.
To show that finite representability of X in a subspace Y with condition (a)

of Theorem 1.1 is enough to characterise ideals through a global property we
use following two lemmas from [12].

Lemma 1.5. Let E be a finite dimensional Banach space and T : E −→
Y ∗∗ be a linear map for any Banach space Y . Then there exists a net (Tα),
Tα : E −→ Y such that

(a) ||Tα|| −→ ||T ||,
(b) Tαe −→ Te for all e ∈ T−1(Y ),
(c) T∗αy

∗ −→ T∗y∗ for all y∗ ∈ Y ∗.

Next result from [12] shows that if we are given with T and (Tα) satisfying
the conditions of Lemma 1.5, then we can modify (Tα) so that it satisfies the
conditions of the following lemma.

Lemma 1.6. Let E be a finite dimensional Banach space and T : E −→
Y ∗∗ be a linear map for any Banach space Y . Let F ⊆ Y ∗ be a finite
dimensional Banach space, then there exists a net (Tα), Tα : E −→ Y such
that

(a) ||Tα|| −→ ||T||,
(b) Tαe = Te for all e ∈ T−1(Y ),
(c) T∗αy

∗ = T∗y∗ for all y∗ ∈ F,
(d) T∗αy

∗ −→ T∗y∗ for all y∗ ∈ Y ∗.



ideal operators and relative godun sets 5

In Section 3 we will refer to VN-subspaces (very non-constraint subspaces)
and their characterisation done in [3]. We recall definitions of VN-subspaces
of a Banach space and nicely smooth spaces from [3] and [7] respectively.

Definition 1.7. Let Y be a subspace of a Banach space X.

(a) The ortho-complement O(Y,X) of Y in X is defined as

O(Y,X) = {x ∈ X : ‖x−y‖≥‖y‖ for all y ∈ Y}.

We denote O(X,X∗∗) by O(X).

(b) Y is said to be a VN-subspace of X if O(Y,X) = {0}.
(c) X is said to be nicely smooth if it is a VN-subspace of its bidual.

In this article, we consider only Banach spaces over the real field R and
all subspaces we consider are assumed to be closed.

2. Generalisation of PLR

We start with the following proposition which is essentially known. How-
ever it provides a way to look ideals and strict ideals through some global
property. Certain known properties of ideals and strict ideals follow trivially
if we take this global view point.

Proposition 2.1. Let X be a Banach space and Y be a subspace of X.
Then

(a) Y is an ideal in X if and only if there exists T : X −→ Y ∗∗ such that
||T|| ≤ 1 and T|Y = IY .

(b) Y is a strict ideal in X if and only if there exists an isometry
T : X −→ Y ∗∗ such that T|Y = IY .

We now note some immediate corollaries.

Corollary 2.2. Let Y be a subspace of a Banach space X and Y be
1-complemented in its bidual. Then Y is an ideal in X if and only if Y is
1-complemented in X.

Proof. If Y is 1-complemented in X then trivially Y is an ideal in X.
For the converse consider the map T : X −→ Y ∗∗ from Proposition 2.1 (a).

Let P : Y ∗∗ −→ Y ∗∗ be a norm one projection with Range(P) = Y . If we take
Q = PT : X −→ X then ||Q|| = ||PT|| ≤ 1, Q2 = Q and Range(Q) = Y .



6 s. dutta, c.r. jayanarayanan, d. khurana

Corollary 2.3. (a) If Y is isometric to any dual space then Y ⊆ X is
an ideal if and only if Y is 1-complemented in X. In particular no reflexive
space can simultaneously be a VN-subspace and an ideal in any superspace.

(b) L1(µ) ⊆ X is an ideal in X if and only if L1(µ) is 1-complemented in
X.

(c) Y is an ideal in L1(µ) if and only if Y is 1-complemented in L1(µ).

(d) No infinite dimensional reflexive space can be an ideal in a space with
Dunford Pettis property (see [4, Definition 1.10]).

Proof. Proofs of (a) and (b) follow from Corollary 2.2.

(c) Let Y be an ideal in L1(µ). Then Y
∗ is isometric to a 1-complemented

subspace of L∞(µ). Thus Y is isometric to L1(ν) for some positive measure
ν. Hence Y is 1-complemented in its bidual. Then, by Corollary 2.2, Y is
1-complemented in L1(µ).

(d) If Y is an infinite dimensional reflexive space and Y ⊆ X is an ideal then by
Corollary 2.2 Y is 1-complemented in X. However complemented subspaces
of a space with Dunford Pettis property have Dunford Pettis property and a
reflexive space with Dunford Pettis property is finite dimensional.

The notion of ai-ideals is strictly in between the notions of ideals and strict
ideals. In the next theorem we characterise ai-ideals in terms of the operator
T defined in Theorem 2.1.

Definition 2.4. Let X and Z be Banach spaces. For � > 0, an operator
T : X → Z is said to be an �-isometry if ‖T‖‖T−1‖≤ 1 + �.

Theorem 2.5. Let Y be a subspace of X. Then Y is an ai-ideal in X if
and only if following condition is satisfied.

Given a finite dimensional subspace E of X and � > 0, there exists a
bounded linear map TE : X → Y ∗∗ such that TE|Y = IY and TE is an
�-isometry on E ∩ (TE)−1(Y ).

Proof. Let E be a finite dimensional subspace of X. Without loss of
generality, (by possibly adding an element from Y ) we assume E ∩Y 6= {0}.
Now consider a net (Eα) of finite dimensional subspaces of X such that Eα ⊇
E and the �Eα-isometry T̃

�Eα
Eα

: Eα → Y with T̃
�Eα
Eα

is identity on E ∩Y . Let
TE be a weak∗ limit point of this net in the sense that for all x ∈ X and
x∗ ∈ X∗, x∗(TEx) = limα x∗(T̃

�Eα
Eα

x). Let Tα = T̃
�Eα
Eα
|E and TE = TE|E.



ideal operators and relative godun sets 7

It is straightforward to verify that

(a) ||Tα|| −→ ||TE||,

(b) Tαy −→ TEy for all y ∈ T−1E (Y ),

(c) T∗αy
∗ −→ T∗Ey

∗ for all y∗ ∈ Y ∗.

Thus Tα and the operator TE satisfy conditions of Lemma 1.5.

Now applying a perturbation argument as in Lemma 1.6, given any � > 0
we can find Sα satisfying conditions of Lemma 1.6 and ||Sα − Tα|| < � for
large dim(Eα). Hence Sα is (�Eα + �)-isometry and Sα −→ TE in the weak∗
topology. Thus TE is an �-isometry on E ∩T−1(Y ).

Conversely, let E be a finite dimensional subspace of X and there exists an
operator TE satisfying the condition of the theorem. Consider TE|E : E →
Y ∗∗ and apply PLR to get the desired operator satisfying the definition of
ai-ideal.

Remark 2.6. Following example of ai-ideal in c0 was considered in [2].

Let Y = {(an) ∈ c0 : a1 = 0}. Then T : c0 −→ Y ∗∗ in this case is given by
T(a) = (0,a2,a3, . . .). Hence T can not be extended as an isometry beyond
T−1(Y ).

Corollary 2.7. Let Y ⊆ X be an ai-ideal and Y be reflexive. If Y has
UEP in X, then Y is isometric to X. In particular, in the following cases Y
is isometric to X.

(a) The norm of X is smooth on Y .

(b) Y is a u-ideal in X [2, Theorem 2.3].

Proof. Given x ∈ X, consider E = span{x} and map TE : X −→ Y ∗∗
as in Theorem 2.5. Since Y is reflexive and TE|Y = I|Y we get TE is onto.
Thus (TE)−1(Y ) = X. Since Y has UEP in X, there exists a unique T such
that TE = T for all E. It follows that the ideal operator T is one-one as well.
Hence T is an isometry.

Let µ be a non-atomic σ-finite measure. It is proved in [2] that any copy of
`1 in L1(µ) can not be an ai-ideal. However L1(µ) contains a 1-complemented
copy of `1. Hence a copy of `1 can be an ideal in L1(µ). It follows from Corol-
lary 2.7 that any copy of `p in Lp(µ) can not be an ai-ideal for 1 < p < ∞.



8 s. dutta, c.r. jayanarayanan, d. khurana

We will now prove that ai-ideals of C[0, 1] are universal for separable Ba-
nach spaces.

Proposition 2.8. Let Y be an ai-ideal in C[0, 1]. Then Y is universal
for separable Banach spaces.

Proof. Let Y be an ai-ideal in C[0, 1]. Then Y is an L1-predual space and
it follows from [2, Proposition 3.8] that Y inherits Daugavet property from
C[0, 1]. Also from [13, Theorem 2.6] it follows that Y contains a copy of `1.
Thus Y is an L1-predual with non separable dual and hence Y is also universal
for separable Banach spaces (see [10, Theorem 2.3]).

We now show that for a non-atomic measure µ, any separable ai-ideal in
L1(µ) is isometric to L1[0, 1].

Proposition 2.9. Let Y ⊆ L1(µ) be a separable ai-ideal where µ is a
non-atomic probability measure. Then Y is isometric to L1[0, 1].

Proof. Let Y ⊆ L1(µ) be a separable ai-ideal. Then by [2, Proposition
3.8], it follows that Y inherits Daugavet property from L1(µ) and thus Y

∗

is non separable. Since Y is an ideal in L1(µ) we have Y
∗ is isometric to a

1-complemented subspace of L∞(µ) and thus Y is isometric to L1(ν) space for
some measure ν. But Y has Daugavet property so ν can not have atoms. Now
since ν is non atomic and Y is separable we can conclude that Y is isometric
to L1[0, 1].

The property of being an ai-ideal is inherited from bidual.

Proposition 2.10. Suppose Y ⊆ X and Y ⊥⊥ is an ai-ideal in X∗∗. Then
Y is an ai-ideal in X∗∗ and hence in particular in X.

Proof. We note that the property of being ai-ideal is transitive. Since Y
is always an ai-ideal in Y ⊥⊥ and Y ⊥⊥ is an ai-ideal in X∗∗, Y is ai-ideal in
X∗∗ as well.

We end this section stating a result which connects ai-ideals in a dual
space to local dual of preduals.

Definition 2.11. [6] A closed subspace Z of X∗ is said to be a local dual
space of a Banach space X if for every � > 0 and every pair of finite dimensional
subspaces F of X∗ and G of X, there exists an �-isometry L : F → Z satisfying
the following conditions.



ideal operators and relative godun sets 9

(a) L(f)|G = f|G for all f ∈ F,
(b) L(f) = f for f ∈ F ∩Z.

The following result is simple to observe from above definition.

Proposition 2.12. Let X be a Banach space and Z be a subspace of X∗.
If Z is a local dual of X, then Z is an ai-ideal in X∗.

Remark 2.13. Let Y be an ai-ideal in a Banach space X. Since X is an
ai-ideal in X∗∗, Y is an ai-ideal in X∗∗. But Y cannot be a local dual space
of X∗ as it is not norming for X∗. So the converse of Proposition 2.12 need
not be true.

Theorem 2.14. Let X be a Banach space. Let Z be an ai-ideal in X∗

with UEP and Z be norming for X. Then Z is a local dual space of X.

Proof. Let F and G be finite dimensional subspaces of X∗ and X respec-
tively. Also, let � > 0. Now let Ĝ = span{ĝ|Z : g ∈ G}, where ĝ is the
canonical image of g in X∗∗. Then, by [2, Theorem 1.4], there exists a Hahn–
Banach extension operator ϕ : Z∗ → X∗∗ and an �-isometry L : F → Z such
that Lf = f for all f ∈ F ∩ Z and ϕ(ĝ|Z)(f) = (ĝ|Z)(Lf) = (Lf)(g) for all
g ∈ G and f ∈ F. Now to prove Z is a local dual space of X, it is enough to
prove that L(f)(g) = f(g) for all f ∈ F and g ∈ G. Now let g ∈ G. Since Z is
norming for X, ĝ is a Hahn–Banach extension of ĝ|Z. Further, by UEP, ĝ is the
only Hahn–Banach extension of ĝ|Z. Therefore (Lf)(g) = ϕ(ĝ|Z)(f) = f(g)
for all f ∈ F and g ∈ G. Hence Z is a local dual space of X.

3. Properties of ideal operators

In this section we explore conditions for the ideal operator T to be unique
or one-one. Any nicely smooth space has UEP (see [3, 7]). However any Ba-
nach space X is a strict ideal in X∗∗. So for an ideal Y in X, to get uniqueness
we mostly have to consider Y to be a strict ideal in X. As mentioned in the
introduction, while considering UEP one needs to exercise some caution here.
For an ideal Y in X we will first make sense of a maximal ideal projection
through the use of Godun set of X with respect to Y .

We formulate the following lemma for which the equivalence of first three
parts is established in [9, Lemma 2.2] and the proof for the fourth part goes
verbatim as in the proof of (2) ⇒ (4) and (4) ⇒ (3) in [9, Proposition 2.3].



10 s. dutta, c.r. jayanarayanan, d. khurana

Lemma 3.1. Let Y be an ideal in X and T be the corresponding ideal
operator. For λ,a ∈ R, the following assertions are equivalent.

(a) ‖I −λP‖≤ a.
(b) For any � > 0, x ∈ X and convex subset A of Y such that Tx is in the

weak∗ closure of A there exists y ∈ A such that ‖x−λy‖ < a‖x‖ + �.
(c) For any x ∈ X there exists a net (yα) ⊆ Y such that (yα) converges to

Tx in the weak∗ topology and lim supα‖x−λyα‖≤ a‖x‖.

Moreover, if Y is a strict ideal in X and T is the corresponding strict ideal
operator then above assertions are also equivalent to the following.

(d) For � > 0 and any sequence (yn) in BY with (yn) converges in the weak
∗

topology to Tx for some x ∈ BX, there exist n and u ∈ co{yk}nk=1,
t ∈ co{yk}∞k=n+1 such that ‖t−λu‖ < a + �.

Let π be the canonical projection of X∗∗∗ onto X∗. The Godun set G(X)
is defined to be G(X) = {λ : ‖I −λπ‖ = 1} (see [9]).

For ideal Y ⊆ X and ideal projection P we define Godun set of X with
respect to Y and P as GP (Y,X) = {λ : ‖I −λP‖ = 1}. Then it follows that
0 ∈ GP (Y,X) and GP (Y,X) is a closed convex subset of [0, 2] and thus itself
an interval.

Our next result is an analogue of [9, Lemma 2.5] and has interesting con-
sequences.

Lemma 3.2. Let Y be a strict ideal in X and P be the corresponding
strict ideal projection.

(a) If Z is a subspace of X such that Y ⊆ Z ⊆ X then there exists an ideal
projection Q on Z∗ with ker(Q) = Y ⊥ such that GP (Y,X) ⊆ GQ(Y,Z).

(b) If Z is a closed subspace of Y then Y/Z is an ideal in X/Z and there
exists an ideal projection P̃ on (X/Z)∗ such that GP (Y,X) ⊆
G
P̃

(Y/Z,X/Z).

Proof. (a) Let T be the corresponding strict ideal operator from X to
Y ∗∗. Consider TZ = T|Z : Z → Y ∗∗. We define Q : Z∗ → Z∗ as Q(z∗) =
(TZ)

∗(z∗|Y ). It is straightforward to check that ker(Q) = Y ⊥ ⊆ Z∗. Thus Q
is an ideal projection on Z∗. The proof now follows from equivalence of (d)
and (a) in Lemma 3.1.

(b) We again let T be the corresponding strict ideal operator from X to Y ∗∗.
Let q : X → X/Z be the quotient map. We define T̃ : X/Z → (Y/Z)∗∗ by



ideal operators and relative godun sets 11

T̃(x) = q∗∗(Tx), where x is the equivalence class containing x. Let P̃ be the
ideal projection corresponding to T̃. Again a straightforward application of
equivalence of (d) and (a) in Lemma 3.1 gives the desired conclusion.

For an ideal Y in X, we define Godun set of X with respect to Y as
G(Y,X) = ∪{GP (Y,X) : P is an ideal projection}. We now verify that
G(Y,X) is GP (Y,X) for some ideal projection P . In the sequel we will re-
fer such projection as maximal ideal projection and the corresponding T as
maximal ideal operator.

Theorem 3.3. Let Y be an ideal in X. Then there exists an ideal pro-
jection P such that G(Y,X) = GP (Y,X).

Proof. Suppose for all ideal projection P , GP (Y,X) = {0}. Then G(Y,X)
= {0} and we choose any P as maximal ideal projection.

Suppose there exists an ideal projection P and λ ∈ GP (Y,X) with λ 6= 0.
Then we claim that [0, 1] ⊆ GP (Y,X). To see this, suppose on the contrary
that GP (Y,X) ⊆ [0,γ] for some 0 < γ < 1. We choose µ ∈ (0,γ). It is
straightforward to verify that γ+µ−γµ ∈ GP (Y,X) as well. Thus γ+µ−γµ ≤
γ. Hence µ(1 −γ) ≤ 0 which is a contradiction.

Now let us consider λ = sup{µ : µ ∈ G(Y,X)}.
If λ 6= 0, then by above argument either λ = 1 or 1 < λ ≤ 2. In the case

λ = 1, there exists an ideal projection P such that G(Y,X) = GP (Y,X) =
[0, 1].

If λ > 1, then choose a sequence (λn) in G(Y,X) such that λn > 1 and
λn converges to λ. Let Pn be an ideal projection corresponding to Y with
‖I −λnPn‖ = 1 for all n. Since B(X∗) is isometric to the dual of projective
tensor product of X∗ and X, there exists a bounded linear map P : X∗ → X∗
and a subsequence (denoted again by (Pn)) of (Pn) such that for every x

∗ ∈
X∗, Pn(x

∗) converges to P(x∗) in the weak∗ topology. Since for every x∗ ∈ X∗
and n ∈ N, Pn(x∗) is a Hahn-Banach extension of x∗|Y , we can see that
P(x∗) is also a Hahn-Banach extension of x∗|Y . Thus ker(P) = Y ⊥. For any
x∗ ∈ X∗, since x∗−P(x∗) ∈ Y ⊥ = ker(P), we can see that P(P(x∗)) = P(x∗).
Hence P is an ideal projection corresponding to Y . Since, for every x∗ ∈ X∗,
x∗ − λnPn(x∗) converges to x∗ − λP(x∗) in the weak∗ topology, we can see
that ‖(I − λP)(x∗)‖ ≤ lim infn‖(I − λnPn)(x∗)‖ ≤ ‖x∗‖ for every x∗ ∈ X∗.
Thus ‖I −λP‖ = 1. Hence G(Y,X) = GP (Y,X) = [0,λ].

Remark 3.4. We note that G(Y,X) = {0} is possible. If Y = `1 and



12 s. dutta, c.r. jayanarayanan, d. khurana

X = Y ∗∗, then following the same argument used in [9, Proposition 2.6] it
follows that G(Y,X) = {0}.

We now show that if Y is nicely smooth and Y embeds in a superspace X
as a strict ideal then strict ideal operator T is unique.

Proposition 3.5. Let Y be a nicely smooth Banach space and Y be a
strict ideal in a superspace X. Then the strict ideal operator is unique.

Proof. Let T1 and T2 be two strict ideal operator. Then for any x ∈
X, ‖T1x−y‖ = ‖T2x−y‖ for all y ∈ Y . Hence T1x−T2x ∈ O(Y ) (see [3]).
But since Y is nicely smooth, O(Y ) = {0} and hence T1x = T2x.

We will now give a sufficient condition for a strict ideal Y in X to be a
VN-subspace of X. We will first provide an analogue of [9, Proposition 2.7].

Proposition 3.6. Let Y be a strict ideal in X and P a strict ideal pro-
jection for Y in X∗. If 1 < λ ≤ 2 and ‖I −λP‖ = a < λ then for any proper
subespace M ⊆ X∗, M norming for Y , we have M is weak∗ dense in X∗.

Proof. Let rY (M) be the greatest constant r such that supx∗∈SM |x
∗(y)| ≥

r‖y‖ for all y ∈ Y .
We will first show that for any weak∗ closed subspace M ⊆ X∗, rY (M) ≤

λ−1a. Without loss of generality let M = ker(x) for some x ∈ SX.
Consider the isometry T : X −→ Y ∗∗ corresponding to P . Then by

Lemma 3.1 there exists a net {yα} ⊆ Y such that yα −→ Tx in the weak∗
topology and lim sup‖x−λyα‖≤ a‖x‖.

Now since T is an isometry we have ‖yα‖ −→ 1. For any x∗ ∈ SM ,
λ|x∗(yα)| = |x∗(x−λyα)| ≤ ‖x−λyα‖.

Since supx∗∈SM |x
∗(yα)| ≥ rY (M)‖yα‖ and ‖yα‖ −→ 1 it follows that

rY (M) ≤ λ−1a.
If there exists 1 < λ ≤ 2 and ‖I − λP‖ = a < λ then it follows that for

any weak∗ closed proper subspace M of X∗ which is norming for Y we have
rY (M) < 1. This contradicts M is norming for Y and hence we have the
result.

Theorem 3.7. Let Y ⊆ X be a strict ideal such that ‖I − λP‖ < λ for
some 1 < λ ≤ 2 where P is a strict ideal projection. Then Y is a VN-subspace
of X. In particular a strict u-ideal is always a VN-subspace.



ideal operators and relative godun sets 13

Proof. Let Y ⊆ X be a strict ideal such that ‖I −λP‖ < λ for some 1 <
λ ≤ 2 where P is a strict ideal projection. Then it follows from Proposition 3.6
that any norming subspace for Y separates points in X. Hence Y is a VN-
subspace of X (see [3]).

Corollary 3.8. Let X be a Banach space. Then the following assertions
are equivalent.

(a) X∗ is separable.

(b) There exists a renorming of X such that X is nicely smooth, that is X∗

has no proper norming subspace.

(c) There exists a renorming of X such that every subspace and quotient of
X in the new norm are nicely smooth.

Proof. (a) ⇒ (b) From [9, Theorem 2.9] it follows that given 1 < λ < 2
there exists a renorming of X such that λ ∈ G(X). Conclusion follows from
Theorem 3.7.

(b) ⇒ (c) Follows from Lemma 3.2 and Theorem 3.7.
(c) ⇒ (b) ⇒ (a) Is trivial.

Corollary 3.9. Let Y be a strict ideal in X with strict ideal projection
P and O(Y,X) 6= {0}. Then either GP (Y,X) = {0} or GP (Y,X) = [0, 1] and
the later happens only if P is bicontractive.

Proof. Let 0 6= x ∈ O(Y,X) and M = ker(x). Then M is norming for Y .
Now if we assume that ‖I−λP‖ < λ then rY (M) ≤‖I−λP‖λ−1 < 1. But M
is norming for Y so it follows that ‖I −λP‖≥ λ and thus GP (Y,X) ⊆ [0, 1].

If 1 /∈ GP (Y,X) that is ‖I − P‖ > 1 then GP (Y,X) = {0}. Hence the
conclusion follows.

We now provide a sufficient condition for x ∈ X to be in O(Y,X).

Proposition 3.10. Let Y ⊆ X be an ideal and T be the corresponding
ideal operator. If Tx = 0, then x ∈ O(Y,X). Consequently, if Y is also a
VN-subspace of X, then any ideal operator T is one-one.

Proof. Let Tx = 0. Then Px∗(x) = 0 for all x∗ ∈ X∗ where P is the ideal
projection corresponding to T . Thus Range(P) ⊆ ker(x). But Range(P) is
norming for Y , hence ker(x) is norming for Y and x ∈ O(Y,X).



14 s. dutta, c.r. jayanarayanan, d. khurana

We now present an extension of [9, Theorem 7.4]. Towards this, for an ideal
Y in X with associated ideal operator T, we define

Ba(Y,X) = {x ∈ X : there exists {yn}⊆ Y such that yn −→ Tx
in the weak∗ topology}

and

kYu (x) = inf
{
a : Tx =

∑
yn in weak

∗ topology and for any n,

‖
∑n

k=1 θkyk‖≤ a,θk = ±1
}
.

It follows from [11] that if Y does not contain a copy of `1 then Ba(Y,X) =
X. As considered in [9], we will say pair (Y,X) has property u if kYu (x) < ∞
for all x ∈ X. In this case by closed graph theorem there exists a constant C
such that kYu (x) ≤ C||Tx|| for all x ∈ Ba(Y,X). We denote least constant C
by kYu (X).

We will need the following lemma.

Lemma 3.11. Let Y be a strict ideal in X such that Y does not contain
a copy of `1 and T be a strict ideal operator. Then Y is a u-ideal in X if and
only if kYu (X) = 1.

Proof. If Y is a u-ideal in X then the result follows from [9, Lemma 3.4].
Conversely, by following the similar arguments as in [9, Lemma 5.3] that

in this case ‖I − 2P‖≤ kYu (X) where P is the ideal projection corresponding
to the ideal operator T. Thus ‖I−2P‖ = 1 and Y is a strict u-ideal in X.

Theorem 3.12. Let Y be a Banach space not containing `1. Then the
following assertions are equivalent.

(a) Y is a u-ideal in Y ∗∗.

(b) Whenever Y is a strict ideal in X, Y is a strict u-ideal in X.

(c) Whenever Y is a strict ideal in X, kYu (X) < 2.

Proof. (a) ⇒ (b) Since Y is a strict ideal in X, the ideal operator T is an
extension of identity operator on Y and we have the results by [9, Proposition
3.6].
(b) ⇒ (c) Follows from Lemma 3.11.
(c) ⇒ (a) Follows from [9, Theorem 7.4] by taking X = Y ∗∗.



ideal operators and relative godun sets 15

We will now give examples where the ideal operator T is unique/ one-one.
We first see how does the ideal operator corresponding to an M-ideal in

C(K) behave. It is well-known that M-ideals in C(K) are precisely of the
form JD = {f ∈ C(K) : f|D = 0} for some closed subset D of K.

Proposition 3.13. Let D be a closed subset of a compact Hausdorff
space K. Then the following are equivalent.

(a) JD is a strict ideal in C(K).

(b) JD is a VN-subspace of C(K).

(c) K \D = K.

Proof. (a) ⇐⇒ (b) Observe that J∗D is norming if and only if K \D = K.
(b) ⇐⇒ (c) Follows from standard arguments.

Example 3.14. Since the ideal projection corresponding to an M-ideal
is unique, the ideal operator T corresponding to JD is unique. In addition,
if K \D = K, then, by Proposition 3.13, the unique ideal operator T corre-
sponding to JD is an isometry.

We next note that the ideal operators corresponding to C(K,X) and
C(K,X∗) are isometries.

For a compact Hausdorff space K and for any Banach space X, let
WC(K,X) denote the space of X-valued functions on K that are contin-
uous when X has the weak topology, equipped with the supremum norm.
Also, W∗C(K,X∗) denotes the space of X∗-valued functions on K that are
continuous when X∗ has the weak∗ topology, equipped with the supremum
norm.

Proposition 3.15. Let X be a Banach space. Then C(K,X) is a strict
ideal in WC(K,X). Moreover, C(K,X∗) is a strict ideal in W∗C(K,X∗).

Proof. The former conclusion follows from the fact that there exists an
isometry from WC(K,X) to C(K,X)∗∗ whose restriction to C(K,X) is the
canonical embedding.

To prove the later conclusion recall that C(K,X∗) = K(X,C(K)), the
space of compact operators from X to C(K) and W∗C(K,X∗) = L(X,C(K)),
the space of bounded linear operators from X to C(K). It follows from [8,
Lemma 2] that if Y is a Banach space having metric approximation property
(in short MAP), then there exists an isometry from L(X,Y ) to K(X,Y )∗∗



16 s. dutta, c.r. jayanarayanan, d. khurana

whose restriction to K(X,Y ) is the canonical embedding. Since C(K) has
MAP, it follows that C(K,X∗) is a strict ideal in W∗C(K,X∗).

We know that C(K,X) ⊆ Ba(K,X) ⊆ C(K,X)∗∗, where Ba(K,X) de-
notes the class of Baire-1 functions from K to X. Since C(K,X) is a strict
ideal in C(K,X)∗∗, it follows that C(K,X) is also a strict ideal in Ba(K,X).
So the corresponding ideal operator is an isometry.

If X has MAP, then, by [8, Lemma 2], K(X) is a strict ideal in L(X).
Now it follows from [5] that a reflexive space with compact approximation
property has MAP. Hence if X is a reflexive space with compact approximation
property such that either weak∗ denting points of BX∗ separates points of X

∗∗

or denting points of BX separates points of X
∗, then K(X) is a strict ideal in

L(X) and is also a VN-subspace of L(X) (see [3]). So the ideal operator T is
an isometry.

Acknowledgements

The second author is grateful to the NBHM, India for its financial
support (No.2/40(2)/2014/R&D-II/6252). He also likes to thank Prof.
Pradipta Bandyopadhyay for the discussion regarding local dual spaces.

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	Introduction and preliminaries
	Generalisation of PLR
	Properties of ideal operators