Post on 01-Jun-2020
Introduction Closed ideals Example References
Algebraic properties of the Banach algebra ofcompact-by-approximable operators
Henrik Wirzenius
University of Helsinki(Joint with Hans-Olav Tylli)
BANACH ALGEBRAS AND APPLICATIONS 2019Winnipeg Canada
July 15, 2019
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
• Let X and Y be Banach spaces and consider the followingclasses of linear operators:
L(X ,Y ) = {bounded operators X → Y }K(X ,Y ) = {compact operators X → Y }A(X ,Y ) = F(X ,Y ) = {approximable operators X → Y }F(X ,Y ) = {bounded finite rank operators X → Y }
• If X = Y we write L(X ) = L(X ,Y ), K(X ) = K(X ,Y ) etc.
• The quotient AX := K(X )/A(X ) is a (well-defined) radicalBanach algebra with norm
||T +A(X )|| = dist(T ,A(X )) = infA∈A(X )
||T − A||
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
• Let X and Y be Banach spaces and consider the followingclasses of linear operators:
L(X ,Y ) = {bounded operators X → Y }K(X ,Y ) = {compact operators X → Y }A(X ,Y ) = F(X ,Y ) = {approximable operators X → Y }F(X ,Y ) = {bounded finite rank operators X → Y }
• If X = Y we write L(X ) = L(X ,Y ), K(X ) = K(X ,Y ) etc.
• The quotient AX := K(X )/A(X ) is a (well-defined) radicalBanach algebra with norm
||T +A(X )|| = dist(T ,A(X )) = infA∈A(X )
||T − A||
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
• Let X and Y be Banach spaces and consider the followingclasses of linear operators:
L(X ,Y ) = {bounded operators X → Y }K(X ,Y ) = {compact operators X → Y }A(X ,Y ) = F(X ,Y ) = {approximable operators X → Y }F(X ,Y ) = {bounded finite rank operators X → Y }
• If X = Y we write L(X ) = L(X ,Y ), K(X ) = K(X ,Y ) etc.
• The quotient AX := K(X )/A(X ) is a (well-defined) radicalBanach algebra with norm
||T +A(X )|| = dist(T ,A(X )) = infA∈A(X )
||T − A||
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
? A Banach space X has the AP (approximation property) if forevery ε > 0 and every compact K ⊂ X there is T ∈ F(X )such that
supx∈K||Tx − x || < ε.
? (Grothendieck) X has the AP if and only ifA(Y ,X ) = K(Y ,X ) for every Banach space Y .
• It is unknown whether A(X ) = K(X ) implies that X hasthe AP.• Hence, AX 6= {0} only within the class of Banach spaces
X failing the AP.
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
? A Banach space X has the AP (approximation property) if forevery ε > 0 and every compact K ⊂ X there is T ∈ F(X )such that
supx∈K||Tx − x || < ε.
? (Grothendieck) X has the AP if and only ifA(Y ,X ) = K(Y ,X ) for every Banach space Y .
• It is unknown whether A(X ) = K(X ) implies that X hasthe AP.• Hence, AX 6= {0} only within the class of Banach spacesX failing the AP.
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
? A Banach space X has the AP (approximation property) if forevery ε > 0 and every compact K ⊂ X there is T ∈ F(X )such that
supx∈K||Tx − x || < ε.
? (Grothendieck) X has the AP if and only ifA(Y ,X ) = K(Y ,X ) for every Banach space Y .
• It is unknown whether A(X ) = K(X ) implies that X hasthe AP.• Hence, AX 6= {0} only within the class of Banach spacesX failing the AP.
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
? Characterization for reflexive spaces [Joh90]:
Let X be reflexive. Then X has the AP if and only if everycompact operator factorable through X is approximable.
Y Z
X
T
A B
? In [Dal13] Dales highlighted a number of open questions onAX .
? [TW19]: the quotient algebra AX is infinite-dimensional for anumber of different classes of Banach spaces X failing the AP.Furthermore, there are examples of X where AX isnon-commutative and where AX is non-nilpotent.
? The algebraic properties of AX remains poorly understood.
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
? Characterization for reflexive spaces [Joh90]:
Let X be reflexive. Then X has the AP if and only if everycompact operator factorable through X is approximable.
Y Z
X
T
A B
? In [Dal13] Dales highlighted a number of open questions onAX .
? [TW19]: the quotient algebra AX is infinite-dimensional for anumber of different classes of Banach spaces X failing the AP.Furthermore, there are examples of X where AX isnon-commutative and where AX is non-nilpotent.
? The algebraic properties of AX remains poorly understood.
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
? Characterization for reflexive spaces [Joh90]:
Let X be reflexive. Then X has the AP if and only if everycompact operator factorable through X is approximable.
Y Z
X
T
A B
? In [Dal13] Dales highlighted a number of open questions onAX .
? [TW19]: the quotient algebra AX is infinite-dimensional for anumber of different classes of Banach spaces X failing the AP.Furthermore, there are examples of X where AX isnon-commutative and where AX is non-nilpotent.
? The algebraic properties of AX remains poorly understood.
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Introduction
? Characterization for reflexive spaces [Joh90]:
Let X be reflexive. Then X has the AP if and only if everycompact operator factorable through X is approximable.
Y Z
X
T
A B
? In [Dal13] Dales highlighted a number of open questions onAX .
? [TW19]: the quotient algebra AX is infinite-dimensional for anumber of different classes of Banach spaces X failing the AP.Furthermore, there are examples of X where AX isnon-commutative and where AX is non-nilpotent.
? The algebraic properties of AX remains poorly understood.
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Closed ideals of AX
? In this talk we will show that AZ contains non-trivial closedideals of AZ for certain direct sums Z = X ⊕ Y where X hastype 2 and Y cotype 2.
? Let Q : K(X )→ AX be the quotient mappingT 7→ T +A(X ). Then I ↔ QI defines a one-to-onecorrespondence between closed ideals of K(X ) and closedideals of AX .
(recall that every closed ideal of K(X ) contains A(X )).
? Thus J is a non-trivial closed ideal of AX if and only if Q−1Jis a closed ideal of K(X ) such that A(X ) ( Q−1J ( K(X )
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Closed ideals of AX
? In this talk we will show that AZ contains non-trivial closedideals of AZ for certain direct sums Z = X ⊕ Y where X hastype 2 and Y cotype 2.
? Let Q : K(X )→ AX be the quotient mappingT 7→ T +A(X ). Then I ↔ QI defines a one-to-onecorrespondence between closed ideals of K(X ) and closedideals of AX .
(recall that every closed ideal of K(X ) contains A(X )).
? Thus J is a non-trivial closed ideal of AX if and only if Q−1Jis a closed ideal of K(X ) such that A(X ) ( Q−1J ( K(X )
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Closed ideals of AX
? In this talk we will show that AZ contains non-trivial closedideals of AZ for certain direct sums Z = X ⊕ Y where X hastype 2 and Y cotype 2.
? Let Q : K(X )→ AX be the quotient mappingT 7→ T +A(X ). Then I ↔ QI defines a one-to-onecorrespondence between closed ideals of K(X ) and closedideals of AX .
(recall that every closed ideal of K(X ) contains A(X )).
? Thus J is a non-trivial closed ideal of AX if and only if Q−1Jis a closed ideal of K(X ) such that A(X ) ( Q−1J ( K(X )
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Operators on direct sums
? We will use the operator matrix notation
T =
(T11 T12
T21 T22
)for operators T ∈ L(X1 ⊕ X2) where each componentTij ∈ L(Xi ,Xj).
? Recall that T ∈ K(X1 ⊕ X2) if and only if eachTij ∈ K(Xi ,Xj) and, similarly, T ∈ A(X1 ⊕ X2) if and only ifeach Tij ∈ A(Xi ,Xj).
? We also use the notation
K(X1 ⊕ X2) =
(K(X1) K(X2,X1)K(X1,X2) K(X2)
)
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Operators on direct sums
? We will use the operator matrix notation
T =
(T11 T12
T21 T22
)for operators T ∈ L(X1 ⊕ X2) where each componentTij ∈ L(Xi ,Xj).
? Recall that T ∈ K(X1 ⊕ X2) if and only if eachTij ∈ K(Xi ,Xj) and, similarly, T ∈ A(X1 ⊕ X2) if and only ifeach Tij ∈ A(Xi ,Xj).
? We also use the notation
K(X1 ⊕ X2) =
(K(X1) K(X2,X1)K(X1,X2) K(X2)
)
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
? Suppose that for Banach spaces X and Y we have
K(X ,Y ) = A(X ,Y ) (1)
andAX 6= {0} and AY 6= {0}. (2)
Then, considering operators on X ⊕ Y , the sets
I =
(A(X ) K(Y ,X )A(X ,Y ) K(Y )
)and J =
(K(X ) K(Y ,X )A(X ,Y ) A(Y )
)define distinct closed ideals of
K(X ⊕ Y ) =
(K(X ) K(Y ,X )A(X ,Y ) K(Y )
)such that
A(X ⊕ Y ) ( I,J ( K(X ⊕ Y ).
? Examples where condition (1) holds:
• X has type 2 and Y cotype 2. We will show this below.• X = P is a Pisier space and Y = P∗ [Joh90]. It is
however not known whether AP 6= {0}.Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
? Suppose that for Banach spaces X and Y we have
K(X ,Y ) = A(X ,Y ) (1)
andAX 6= {0} and AY 6= {0}. (2)
Then, considering operators on X ⊕ Y , the sets
I =
(A(X ) K(Y ,X )A(X ,Y ) K(Y )
)and J =
(K(X ) K(Y ,X )A(X ,Y ) A(Y )
)define distinct closed ideals of
K(X ⊕ Y ) =
(K(X ) K(Y ,X )A(X ,Y ) K(Y )
)such that
A(X ⊕ Y ) ( I,J ( K(X ⊕ Y ).
? Examples where condition (1) holds:
• X has type 2 and Y cotype 2. We will show this below.• X = P is a Pisier space and Y = P∗ [Joh90]. It is
however not known whether AP 6= {0}.Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
? Suppose that for Banach spaces X and Y we have
K(X ,Y ) = A(X ,Y ) (1)
andAX 6= {0} and AY 6= {0}. (2)
Then, considering operators on X ⊕ Y , the sets
I =
(A(X ) K(Y ,X )A(X ,Y ) K(Y )
)and J =
(K(X ) K(Y ,X )A(X ,Y ) A(Y )
)define distinct closed ideals of
K(X ⊕ Y ) =
(K(X ) K(Y ,X )A(X ,Y ) K(Y )
)such that
A(X ⊕ Y ) ( I,J ( K(X ⊕ Y ).
? Examples where condition (1) holds:
• X has type 2 and Y cotype 2. We will show this below.• X = P is a Pisier space and Y = P∗ [Joh90]. It is
however not known whether AP 6= {0}.Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Type and cotype
Let rk : [0, 1]→ {−1, 1}, k ∈ N be the Rademacher functionsdefined by
rk(t) = sign(sin 2kπt)
Definition
A Banach space X has type 2 if there is a constant κ ≥ 0 suchthat for every n ∈ N and every x1, . . . xn ∈ X we have∫ 1
0||
n∑k=1
rk(t)xk ||2 dt ≤ κn∑
k=1
||xk ||2.
A Banach space X has cotype 2 if there is a constant κ ≥ 0 suchthat for every n ∈ N and every x1, . . . xn ∈ X we have
n∑k=1
||xk ||2 ≤ κ∫ 1
0||
n∑k=1
rk(t)xk ||2 dt
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Lemma
Let X and Y be Banach spaces. Suppose that X has type 2 and Yhas cotype 2. Then K(X ,Y ) = A(X ,Y ).
Proof.
Let T ∈ K(X ,Y ). Kwapien’s theorem states that every operatorfrom a type 2 space to a cotype 2 space factors through a Hilbertspace H. Thus T = BA as in the diagram
X Y
H
T
A B
Since H is reflexive and has the AP we have T ∈ A(X ,Y ) by[Joh90].
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Lemma
Let X and Y be Banach spaces. Suppose that X has type 2 and Yhas cotype 2. Then K(X ,Y ) = A(X ,Y ).
Proof.
Let T ∈ K(X ,Y ). Kwapien’s theorem states that every operatorfrom a type 2 space to a cotype 2 space factors through a Hilbertspace H. Thus T = BA as in the diagram
X Y
H
T
A B
Since H is reflexive and has the AP we have T ∈ A(X ,Y ) by[Joh90].
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Example (Non-trivial ideals)
Let 1 < p < 2 < q <∞. Let X be a closed subspace of `q withAX 6= {0}, and let Y be a closed subspace of `p such thatAY 6= {0}. Then AX⊕Y contains two non-trivial closed ideals.
Proof.
Firstly, note that such spaces X and Y exists due to Bachelis[Bac76]. Furthermore, X has type 2 and Y cotype 2 and so byLemma 1 we have K(X ,Y ) = A(X ,Y ). Thus, as discussed above,
I =
(A(X ) K(Y ,X )A(X ,Y ) K(Y )
)and J =
(K(X ) K(Y ,X )A(X ,Y ) A(Y )
)define two distinct closed ideals of K(X ⊕ Y ) strictly betweenA(X ⊕ Y ) and K(X ⊕ Y ). Thus QI and QJ are non-trivialclosed ideals of AX⊕Y .
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Example (Non-trivial ideals)
Let 1 < p < 2 < q <∞. Let X be a closed subspace of `q withAX 6= {0}, and let Y be a closed subspace of `p such thatAY 6= {0}. Then AX⊕Y contains two non-trivial closed ideals.
Proof.
Firstly, note that such spaces X and Y exists due to Bachelis[Bac76]. Furthermore, X has type 2 and Y cotype 2 and so byLemma 1 we have K(X ,Y ) = A(X ,Y ). Thus, as discussed above,
I =
(A(X ) K(Y ,X )A(X ,Y ) K(Y )
)and J =
(K(X ) K(Y ,X )A(X ,Y ) A(Y )
)define two distinct closed ideals of K(X ⊕ Y ) strictly betweenA(X ⊕ Y ) and K(X ⊕ Y ). Thus QI and QJ are non-trivialclosed ideals of AX⊕Y .
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
? Suppose we can in addition choose X and Y so thatK(Y ,X ) 6= A(Y ,X ). Then Q(I ∩ J ) is also a non-trivialideal of AX⊕Y
? Question: Is there a Banach space X with type 2 and Y withcotype 2 such that K(Y ,X ) 6= A(Y ,X )?
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
? Suppose we can in addition choose X and Y so thatK(Y ,X ) 6= A(Y ,X ). Then Q(I ∩ J ) is also a non-trivialideal of AX⊕Y
? Question: Is there a Banach space X with type 2 and Y withcotype 2 such that K(Y ,X ) 6= A(Y ,X )?
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Another potential candidate for a non-trivial closed ideal
Definition
An operator T ∈ L(X ,Y ) is called compactly approximable,T ∈ CA(X ,Y ), if for every ε > 0 and every compact subsetK ⊂ X there is S ∈ F(X ,Y ) such that supx∈K ||Tx − Sx || < ε.
? CA(X ) = L(X ) if and only if X has AP.
? CA is a closed Banach operator ideal. Thus
A(X ) ⊂ CA(X ) ∩ K(X ) ⊂ K(X ).
? CA(X ) ∩ K(X ) ( K(X ) whenever AX has a non-trivialproduct.
? However, CA(X ) ∩ K(X ) = A(X ) whenever X is reflexive.(Godefroy & Saphar and Valdivia)
? Is there a non-reflexive X such that A(X ) ( CA(X ) ∩ K(X )?
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Another potential candidate for a non-trivial closed ideal
Definition
An operator T ∈ L(X ,Y ) is called compactly approximable,T ∈ CA(X ,Y ), if for every ε > 0 and every compact subsetK ⊂ X there is S ∈ F(X ,Y ) such that supx∈K ||Tx − Sx || < ε.
? CA(X ) = L(X ) if and only if X has AP.
? CA is a closed Banach operator ideal. Thus
A(X ) ⊂ CA(X ) ∩ K(X ) ⊂ K(X ).
? CA(X ) ∩ K(X ) ( K(X ) whenever AX has a non-trivialproduct.
? However, CA(X ) ∩ K(X ) = A(X ) whenever X is reflexive.(Godefroy & Saphar and Valdivia)
? Is there a non-reflexive X such that A(X ) ( CA(X ) ∩ K(X )?
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Another potential candidate for a non-trivial closed ideal
Definition
An operator T ∈ L(X ,Y ) is called compactly approximable,T ∈ CA(X ,Y ), if for every ε > 0 and every compact subsetK ⊂ X there is S ∈ F(X ,Y ) such that supx∈K ||Tx − Sx || < ε.
? CA(X ) = L(X ) if and only if X has AP.
? CA is a closed Banach operator ideal. Thus
A(X ) ⊂ CA(X ) ∩ K(X ) ⊂ K(X ).
? CA(X ) ∩ K(X ) ( K(X ) whenever AX has a non-trivialproduct.
? However, CA(X ) ∩ K(X ) = A(X ) whenever X is reflexive.(Godefroy & Saphar and Valdivia)
? Is there a non-reflexive X such that A(X ) ( CA(X ) ∩ K(X )?
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Another potential candidate for a non-trivial closed ideal
Definition
An operator T ∈ L(X ,Y ) is called compactly approximable,T ∈ CA(X ,Y ), if for every ε > 0 and every compact subsetK ⊂ X there is S ∈ F(X ,Y ) such that supx∈K ||Tx − Sx || < ε.
? CA(X ) = L(X ) if and only if X has AP.
? CA is a closed Banach operator ideal. Thus
A(X ) ⊂ CA(X ) ∩ K(X ) ⊂ K(X ).
? CA(X ) ∩ K(X ) ( K(X ) whenever AX has a non-trivialproduct.
? However, CA(X ) ∩ K(X ) = A(X ) whenever X is reflexive.(Godefroy & Saphar and Valdivia)
? Is there a non-reflexive X such that A(X ) ( CA(X ) ∩ K(X )?
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
Another potential candidate for a non-trivial closed ideal
Definition
An operator T ∈ L(X ,Y ) is called compactly approximable,T ∈ CA(X ,Y ), if for every ε > 0 and every compact subsetK ⊂ X there is S ∈ F(X ,Y ) such that supx∈K ||Tx − Sx || < ε.
? CA(X ) = L(X ) if and only if X has AP.
? CA is a closed Banach operator ideal. Thus
A(X ) ⊂ CA(X ) ∩ K(X ) ⊂ K(X ).
? CA(X ) ∩ K(X ) ( K(X ) whenever AX has a non-trivialproduct.
? However, CA(X ) ∩ K(X ) = A(X ) whenever X is reflexive.(Godefroy & Saphar and Valdivia)
? Is there a non-reflexive X such that A(X ) ( CA(X ) ∩ K(X )?
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
? Other natural ideals to consider:A(X ) ⊂ · · · ⊂ K (X )n ⊂ · · · ⊂ K (X )3 ⊂ K (X )2 ⊂ K (X ).Here
K (X )n = span{T1 · · ·Tn | T1, . . . ,Tn ∈ K (X )}
? [Dal13] Can we have K (X )2 ( K (X )?
? There is an example of Read outlined in Dales (2000) whereK (X )2 6= K (X ).
Thank you!
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
? Other natural ideals to consider:A(X ) ⊂ · · · ⊂ K (X )n ⊂ · · · ⊂ K (X )3 ⊂ K (X )2 ⊂ K (X ).Here
K (X )n = span{T1 · · ·Tn | T1, . . . ,Tn ∈ K (X )}
? [Dal13] Can we have K (X )2 ( K (X )?
? There is an example of Read outlined in Dales (2000) whereK (X )2 6= K (X ).
Thank you!
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
? Other natural ideals to consider:A(X ) ⊂ · · · ⊂ K (X )n ⊂ · · · ⊂ K (X )3 ⊂ K (X )2 ⊂ K (X ).Here
K (X )n = span{T1 · · ·Tn | T1, . . . ,Tn ∈ K (X )}
? [Dal13] Can we have K (X )2 ( K (X )?
? There is an example of Read outlined in Dales (2000) whereK (X )2 6= K (X ).
Thank you!
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
? Other natural ideals to consider:A(X ) ⊂ · · · ⊂ K (X )n ⊂ · · · ⊂ K (X )3 ⊂ K (X )2 ⊂ K (X ).Here
K (X )n = span{T1 · · ·Tn | T1, . . . ,Tn ∈ K (X )}
? [Dal13] Can we have K (X )2 ( K (X )?
? There is an example of Read outlined in Dales (2000) whereK (X )2 6= K (X ).
Thank you!
Henrik Wirzenius BAA2019
Introduction Closed ideals Example References
References
[Bac76] G.F. Bachelis, A factorization theorem for compactoperators, Illinois J. Math. 20 (1976), 626–629.
[Dal13] H.G. Dales, A Banach algebra related to a Banach spacewithout AP, notes, 2013.
[Joh90] K. John, On the compact nonnuclear operator problem,Math. Ann. 287 (1990), 509–514.
[TW19] H.-O. Tylli and H. Wirzenius, The quotient algebra ofcompact-by-approximable operators on Banach spacesfailing the approximation property, J. Aust. Math. Soc.(2019), 1–23 (Published online ahead of print).
Henrik Wirzenius BAA2019