Compatible complex structures on Kalton-Peck space
Wilson A. Cu´ellar
Universidade de S˜ao Paulo
joint work with professors: J. M. F. Castillo, V. Ferenczi, Y. Moreno.
August 25, 2014
First Brazilian Workshop in Geometry of Banach Spaces
Complex structures
A real Banach spaceX is said to admit acomplex structureif there exists an operatorI:X→X such thatI2=−Id.
The complex spaceXI is the spaceX with theC-linear structure: if α, β∈Randx∈X, then
(α+iβ)x=αx+βI(x). Endowed with the norm:
k|x|k= sup
0≤θ≤2π
kcosθx+ sinθIxk.
Example:The operatorJ(x, y) = (−y, x)onX⊕X satisfiesJ2=−Id. The complex structureX⊕XJ is called thecomplexificationofX and is denoted byX⊕CX.
Definition
Two complex structuresXI andXJ onX are said to be equivalent if there exists aR-linear automorphismT :X →X such that T IT−1=J. T :XI →XJ is a C-isomorphism.
Complex structures
A real Banach spaceX is said to admit acomplex structureif there exists an operatorI:X→X such thatI2=−Id.
The complex spaceXI is the spaceX with theC-linear structure: if α, β∈Randx∈X, then
(α+iβ)x=αx+βI(x).
Endowed with the norm:
k|x|k= sup
0≤θ≤2π
kcosθx+ sinθIxk.
Example:The operatorJ(x, y) = (−y, x)onX⊕X satisfiesJ2=−Id. The complex structureX⊕XJ is called thecomplexificationofX and is denoted byX⊕CX.
Definition
Two complex structuresXI andXJ onX are said to be equivalent if there exists aR-linear automorphismT :X →X such that T IT−1=J. T :XI →XJ is a C-isomorphism.
Complex structures
A real Banach spaceX is said to admit acomplex structureif there exists an operatorI:X→X such thatI2=−Id.
The complex spaceXI is the spaceX with theC-linear structure: if α, β∈Randx∈X, then
(α+iβ)x=αx+βI(x).
Endowed with the norm:
k|x|k= sup
0≤θ≤2π
kcosθx+ sinθIxk.
Example:The operatorJ(x, y) = (−y, x)onX⊕X satisfiesJ2=−Id. The complex structureX⊕XJ is called thecomplexificationofX and is denoted byX⊕CX.
Definition
Two complex structuresXI andXJ onX are said to be equivalent if there exists aR-linear automorphismT :X →X such that T IT−1=J. T :XI →XJ is a C-isomorphism.
Complex structures
A real Banach spaceX is said to admit acomplex structureif there exists an operatorI:X→X such thatI2=−Id.
The complex spaceXI is the spaceX with theC-linear structure: if α, β∈Randx∈X, then
(α+iβ)x=αx+βI(x).
Endowed with the norm:
k|x|k= sup
0≤θ≤2π
kcosθx+ sinθIxk.
Example:The operatorJ(x, y) = (−y, x)onX⊕X satisfiesJ2=−Id.
The complex structureX⊕XJ is called thecomplexificationofX and is denoted byX⊕CX.
Definition
Two complex structuresXI andXJ onX are said to be equivalent if there exists aR-linear automorphismT :X →X such that T IT−1=J. T :XI →XJ is a C-isomorphism.
Complex structures
A real Banach spaceX is said to admit acomplex structureif there exists an operatorI:X→X such thatI2=−Id.
The complex spaceXI is the spaceX with theC-linear structure: if α, β∈Randx∈X, then
(α+iβ)x=αx+βI(x).
Endowed with the norm:
k|x|k= sup
0≤θ≤2π
kcosθx+ sinθIxk.
Example:The operatorJ(x, y) = (−y, x)onX⊕X satisfiesJ2=−Id.
The complex structureX⊕XJ is called thecomplexificationofX and is denoted byX⊕CX.
Definition
Two complex structuresXI andXJ onX are said to be equivalent if there exists aR-linear automorphismT :X →X such that T IT−1=J.
T :XI →XJ is a C-isomorphism.
Complex structures
A real Banach spaceX is said to admit acomplex structureif there exists an operatorI:X→X such thatI2=−Id.
The complex spaceXI is the spaceX with theC-linear structure: if α, β∈Randx∈X, then
(α+iβ)x=αx+βI(x).
Endowed with the norm:
k|x|k= sup
0≤θ≤2π
kcosθx+ sinθIxk.
Example:The operatorJ(x, y) = (−y, x)onX⊕X satisfiesJ2=−Id.
The complex structureX⊕XJ is called thecomplexificationofX and is denoted byX⊕CX.
Definition
Two complex structuresXI andXJ onX are said to be equivalent if there exists aR-linear automorphismT :X →X such that T IT−1=J. T:XI →XJ is a C-isomorphism.
Complex structures
Theorem (Kalton, 2009)
LetX be a real Banach space such thatX⊕CX is primary, thenX has at most one complex structure.
Example: The classical spaces`p (1≤p≤ ∞),c0,C[0,1]andLp (1≤p≤ ∞) has unique complex structure.
All complex structures on`2 areC-isomorphic to`u22, where u2(x1, x2, x3, x4, . . .) = (−x2, x1,−x4, x3, . . .)
Definition
For a complex Banach spaceZ, itscomplex conjugateZ, is defined to be the spaceZ equipped with the law of multiplication by scalars: λz:=λz, for everyλ∈Candz∈Z.
• Z andZ are isometric as real spaces.
• XI =X−I.
• S. Szarek (86), Bourgain (86), Kalton (95)There exist spaces not isomorphic to their complex conjugate.
These spaces admit at least two complex structures.
Complex structures
Theorem (Kalton, 2009)
LetX be a real Banach space such thatX⊕CX is primary, thenX has at most one complex structure.
Example: The classical spaces`p (1≤p≤ ∞),c0,C[0,1]andLp
(1≤p≤ ∞) has unique complex structure.
All complex structures on`2 areC-isomorphic to`u22, where u2(x1, x2, x3, x4, . . .) = (−x2, x1,−x4, x3, . . .)
Definition
For a complex Banach spaceZ, itscomplex conjugateZ, is defined to be the spaceZ equipped with the law of multiplication by scalars: λz:=λz, for everyλ∈Candz∈Z.
• Z andZ are isometric as real spaces.
• XI =X−I.
• S. Szarek (86), Bourgain (86), Kalton (95)There exist spaces not isomorphic to their complex conjugate.
These spaces admit at least two complex structures.
Complex structures
Theorem (Kalton, 2009)
LetX be a real Banach space such thatX⊕CX is primary, thenX has at most one complex structure.
Example: The classical spaces`p (1≤p≤ ∞),c0,C[0,1]andLp
(1≤p≤ ∞) has unique complex structure.
All complex structures on`2 areC-isomorphic to`u22, where u2(x1, x2, x3, x4, . . .) = (−x2, x1,−x4, x3, . . .)
Definition
For a complex Banach spaceZ, itscomplex conjugateZ, is defined to be the spaceZ equipped with the law of multiplication by scalars: λz:=λz, for everyλ∈Candz∈Z.
• Z andZ are isometric as real spaces.
• XI =X−I.
• S. Szarek (86), Bourgain (86), Kalton (95)There exist spaces not isomorphic to their complex conjugate.
These spaces admit at least two complex structures.
Complex structures
Theorem (Kalton, 2009)
LetX be a real Banach space such thatX⊕CX is primary, thenX has at most one complex structure.
Example: The classical spaces`p (1≤p≤ ∞),c0,C[0,1]andLp
(1≤p≤ ∞) has unique complex structure.
All complex structures on`2 areC-isomorphic to`u22, where u2(x1, x2, x3, x4, . . .) = (−x2, x1,−x4, x3, . . .)
Definition
For a complex Banach spaceZ, itscomplex conjugateZ, is defined to be the spaceZ equipped with the law of multiplication by scalars: λz:=λz, for everyλ∈Candz∈Z.
• Z andZ are isometric as real spaces.
• XI =X−I.
• S. Szarek (86), Bourgain (86), Kalton (95)There exist spaces not isomorphic to their complex conjugate.
These spaces admit at least two complex structures.
Complex structures
Theorem (Kalton, 2009)
LetX be a real Banach space such thatX⊕CX is primary, thenX has at most one complex structure.
Example: The classical spaces`p (1≤p≤ ∞),c0,C[0,1]andLp
(1≤p≤ ∞) has unique complex structure.
All complex structures on`2 areC-isomorphic to`u22, where u2(x1, x2, x3, x4, . . .) = (−x2, x1,−x4, x3, . . .)
Definition
For a complex Banach spaceZ, itscomplex conjugateZ, is defined to be the spaceZ equipped with the law of multiplication by scalars: λz:=λz, for everyλ∈Candz∈Z.
• Z andZ are isometric as real spaces.
• XI =X−I.
• S. Szarek (86), Bourgain (86), Kalton (95)There exist spaces not isomorphic to their complex conjugate.
These spaces admit at least two complex structures.
Complex structures
Theorem (Kalton, 2009)
LetX be a real Banach space such thatX⊕CX is primary, thenX has at most one complex structure.
Example: The classical spaces`p (1≤p≤ ∞),c0,C[0,1]andLp
(1≤p≤ ∞) has unique complex structure.
All complex structures on`2 areC-isomorphic to`u22, where u2(x1, x2, x3, x4, . . .) = (−x2, x1,−x4, x3, . . .)
Definition
For a complex Banach spaceZ, itscomplex conjugateZ, is defined to be the spaceZ equipped with the law of multiplication by scalars: λz:=λz, for everyλ∈Candz∈Z.
• Z andZ are isometric as real spaces.
• XI =X−I.
• S. Szarek (86), Bourgain (86), Kalton (95)There exist spaces not isomorphic to their complex conjugate.
These spaces admit at least two complex structures.
Complex structures
Theorem (Kalton, 2009)
LetX be a real Banach space such thatX⊕CX is primary, thenX has at most one complex structure.
Example: The classical spaces`p (1≤p≤ ∞),c0,C[0,1]andLp
(1≤p≤ ∞) has unique complex structure.
All complex structures on`2 areC-isomorphic to`u22, where u2(x1, x2, x3, x4, . . .) = (−x2, x1,−x4, x3, . . .)
Definition
For a complex Banach spaceZ, itscomplex conjugateZ, is defined to be the spaceZ equipped with the law of multiplication by scalars: λz:=λz, for everyλ∈Candz∈Z.
• Z andZ are isometric as real spaces.
• XI =X−I.
• S. Szarek (86), Bourgain (86), Kalton (95)There exist spaces not isomorphic to their complex conjugate.
These spaces admit at least two complex structures.
Complex structures
Theorem (Kalton, 2009)
LetX be a real Banach space such thatX⊕CX is primary, thenX has at most one complex structure.
Example: The classical spaces`p (1≤p≤ ∞),c0,C[0,1]andLp
(1≤p≤ ∞) has unique complex structure.
All complex structures on`2 areC-isomorphic to`u22, where u2(x1, x2, x3, x4, . . .) = (−x2, x1,−x4, x3, . . .)
Definition
For a complex Banach spaceZ, itscomplex conjugateZ, is defined to be the spaceZ equipped with the law of multiplication by scalars: λz:=λz, for everyλ∈Candz∈Z.
• Z andZ are isometric as real spaces.
• XI =X−I.
• S. Szarek (86), Bourgain (86), Kalton (95)There exist spaces not isomorphic to their complex conjugate.
These spaces admit at least two complex structures.
Twisted sums
Definition
LetX andY be two Banach spaces. Atwisted sumofX andY is a quasi-Banach spaceZ which contains a subspaceX0⊆Z isomorphic to X such that the quotientZ/X0 is isomorphic toY.
Equivalently,
0→X →j Z →q Y →0.
Every twisted sum is equivalent to one of the typeX⊕ΩY for a quasi-linear operatorΩ :Y →X.
X⊕ΩY is the spaceX×Y endowed with the quasi-norm: k(x, y)kΩ=kx−Ωyk+kyk
The twisted sum is trivial if it is equivalent toX⊕Y ⇐⇒ Ω =B+L. F. Cabello S´anchez, J. Castillo, J. Su´arez (2012): The quotient mapqis strictly singular iff the restriction ofΩto every infinite dimensional closed subspace is never trivial. In this caseΩis said to besingular.
Twisted sums
Definition
LetX andY be two Banach spaces. Atwisted sumofX andY is a quasi-Banach spaceZ which contains a subspaceX0⊆Z isomorphic to X such that the quotientZ/X0 is isomorphic toY. Equivalently,
0→X →j Z →q Y →0.
Every twisted sum is equivalent to one of the typeX⊕ΩY for a quasi-linear operatorΩ :Y →X.
X⊕ΩY is the spaceX×Y endowed with the quasi-norm: k(x, y)kΩ=kx−Ωyk+kyk
The twisted sum is trivial if it is equivalent toX⊕Y ⇐⇒ Ω =B+L. F. Cabello S´anchez, J. Castillo, J. Su´arez (2012): The quotient mapqis strictly singular iff the restriction ofΩto every infinite dimensional closed subspace is never trivial. In this caseΩis said to besingular.
Twisted sums
Definition
LetX andY be two Banach spaces. Atwisted sumofX andY is a quasi-Banach spaceZ which contains a subspaceX0⊆Z isomorphic to X such that the quotientZ/X0 is isomorphic toY. Equivalently,
0→X →j Z →q Y →0.
Every twisted sum is equivalent to one of the typeX⊕ΩY for a quasi-linear operatorΩ :Y →X.
X⊕ΩY is the spaceX×Y endowed with the quasi-norm:
k(x, y)kΩ=kx−Ωyk+kyk
The twisted sum is trivial if it is equivalent toX⊕Y ⇐⇒ Ω =B+L. F. Cabello S´anchez, J. Castillo, J. Su´arez (2012): The quotient mapqis strictly singular iff the restriction ofΩto every infinite dimensional closed subspace is never trivial. In this caseΩis said to besingular.
Twisted sums
Definition
LetX andY be two Banach spaces. Atwisted sumofX andY is a quasi-Banach spaceZ which contains a subspaceX0⊆Z isomorphic to X such that the quotientZ/X0 is isomorphic toY. Equivalently,
0→X →j Z →q Y →0.
Every twisted sum is equivalent to one of the typeX⊕ΩY for a quasi-linear operatorΩ :Y →X.
X⊕ΩY is the spaceX×Y endowed with the quasi-norm:
k(x, y)kΩ=kx−Ωyk+kyk The twisted sum is trivial if it is equivalent toX⊕Y
⇐⇒ Ω =B+L. F. Cabello S´anchez, J. Castillo, J. Su´arez (2012): The quotient mapqis strictly singular iff the restriction ofΩto every infinite dimensional closed subspace is never trivial. In this caseΩis said to besingular.
Twisted sums
Definition
LetX andY be two Banach spaces. Atwisted sumofX andY is a quasi-Banach spaceZ which contains a subspaceX0⊆Z isomorphic to X such that the quotientZ/X0 is isomorphic toY. Equivalently,
0→X →j Z →q Y →0.
Every twisted sum is equivalent to one of the typeX⊕ΩY for a quasi-linear operatorΩ :Y →X.
X⊕ΩY is the spaceX×Y endowed with the quasi-norm:
k(x, y)kΩ=kx−Ωyk+kyk
The twisted sum is trivial if it is equivalent toX⊕Y ⇐⇒ Ω =B+L.
F. Cabello S´anchez, J. Castillo, J. Su´arez (2012): The quotient mapqis strictly singular iff the restriction ofΩto every infinite dimensional closed subspace is never trivial. In this caseΩis said to besingular.
Twisted sums
Definition
LetX andY be two Banach spaces. Atwisted sumofX andY is a quasi-Banach spaceZ which contains a subspaceX0⊆Z isomorphic to X such that the quotientZ/X0 is isomorphic toY. Equivalently,
0→X →j Z →q Y →0.
Every twisted sum is equivalent to one of the typeX⊕ΩY for a quasi-linear operatorΩ :Y →X.
X⊕ΩY is the spaceX×Y endowed with the quasi-norm:
k(x, y)kΩ=kx−Ωyk+kyk
The twisted sum is trivial if it is equivalent toX⊕Y ⇐⇒ Ω =B+L.
F. Cabello S´anchez, J. Castillo, J. Su´arez (2012): The quotient mapqis strictly singular iff the restriction ofΩto every infinite dimensional closed subspace is never trivial.
In this caseΩis said to besingular.
Twisted sums
Definition
LetX andY be two Banach spaces. Atwisted sumofX andY is a quasi-Banach spaceZ which contains a subspaceX0⊆Z isomorphic to X such that the quotientZ/X0 is isomorphic toY. Equivalently,
0→X →j Z →q Y →0.
Every twisted sum is equivalent to one of the typeX⊕ΩY for a quasi-linear operatorΩ :Y →X.
X⊕ΩY is the spaceX×Y endowed with the quasi-norm:
k(x, y)kΩ=kx−Ωyk+kyk
The twisted sum is trivial if it is equivalent toX⊕Y ⇐⇒ Ω =B+L.
F. Cabello S´anchez, J. Castillo, J. Su´arez (2012): The quotient mapqis strictly singular iff the restriction ofΩto every infinite dimensional closed subspace is never trivial. In this caseΩis said to besingular.
Z
2Kalton-Peck space
Z2=`2⊕Ω2`2 is the twisted Hilbert space obtained by considering the non-trivial quasi-linear map (defined on finitely supported sequences)
Ω2(x)(n) =x(n) log kxk
|x(n)|.
Properties
• Ω2 is singular.
• Z2has a 2-dimensional unconditional decomposition generated by the subspacesEn=span{(en,0),(0, en)}.
• Z2has no unconditional basis. Moreover, it does not admit G.L−l.u.st
• Every infinite-dimensional complemented subspace ofZ2 contains a complemented subspace isomorphic to Z2.
• Z2is isomorphic to its square.
Question
Z2is isomorphic to its hyperplanes?
Z
2Kalton-Peck space
Z2=`2⊕Ω2`2 is the twisted Hilbert space obtained by considering the non-trivial quasi-linear map (defined on finitely supported sequences)
Ω2(x)(n) =x(n) log kxk
|x(n)|.
Properties
• Ω2 is singular.
• Z2has a 2-dimensional unconditional decomposition generated by the subspacesEn=span{(en,0),(0, en)}.
• Z2has no unconditional basis. Moreover, it does not admit G.L−l.u.st
• Every infinite-dimensional complemented subspace ofZ2 contains a complemented subspace isomorphic to Z2.
• Z2is isomorphic to its square.
Question
Z2is isomorphic to its hyperplanes?
Z
2Kalton-Peck space
Z2=`2⊕Ω2`2 is the twisted Hilbert space obtained by considering the non-trivial quasi-linear map (defined on finitely supported sequences)
Ω2(x)(n) =x(n) log kxk
|x(n)|.
Properties
• Ω2 is singular.
• Z2has a 2-dimensional unconditional decomposition generated by the subspacesEn=span{(en,0),(0, en)}.
• Z2has no unconditional basis. Moreover, it does not admit G.L−l.u.st
• Every infinite-dimensional complemented subspace ofZ2 contains a complemented subspace isomorphic to Z2.
• Z2is isomorphic to its square.
Question
Z2is isomorphic to its hyperplanes?
Z
2Kalton-Peck space
Z2=`2⊕Ω2`2 is the twisted Hilbert space obtained by considering the non-trivial quasi-linear map (defined on finitely supported sequences)
Ω2(x)(n) =x(n) log kxk
|x(n)|.
Properties
• Ω2 is singular.
• Z2has a 2-dimensional unconditional decomposition generated by the subspacesEn=span{(en,0),(0, en)}.
• Z2has no unconditional basis. Moreover, it does not admit G.L−l.u.st
• Every infinite-dimensional complemented subspace ofZ2 contains a complemented subspace isomorphic to Z2.
• Z2is isomorphic to its square.
Question
Z2is isomorphic to its hyperplanes?
Z
2Kalton-Peck space
Z2=`2⊕Ω2`2 is the twisted Hilbert space obtained by considering the non-trivial quasi-linear map (defined on finitely supported sequences)
Ω2(x)(n) =x(n) log kxk
|x(n)|.
Properties
• Ω2 is singular.
• Z2has a 2-dimensional unconditional decomposition generated by the subspacesEn=span{(en,0),(0, en)}.
• Z2has no unconditional basis. Moreover, it does not admit G.L−l.u.st
• Every infinite-dimensional complemented subspace ofZ2 contains a complemented subspace isomorphic to Z2.
• Z2is isomorphic to its square.
Question
Z2is isomorphic to its hyperplanes?
Z
2Kalton-Peck space
Z2=`2⊕Ω2`2 is the twisted Hilbert space obtained by considering the non-trivial quasi-linear map (defined on finitely supported sequences)
Ω2(x)(n) =x(n) log kxk
|x(n)|.
Properties
• Ω2 is singular.
• Z2has a 2-dimensional unconditional decomposition generated by the subspacesEn=span{(en,0),(0, en)}.
• Z2has no unconditional basis. Moreover, it does not admit G.L−l.u.st
• Every infinite-dimensional complemented subspace ofZ2 contains a complemented subspace isomorphic to Z2.
• Z2is isomorphic to its square.
Question
Z2is isomorphic to its hyperplanes?
Z
2Kalton-Peck space
Z2=`2⊕Ω2`2 is the twisted Hilbert space obtained by considering the non-trivial quasi-linear map (defined on finitely supported sequences)
Ω2(x)(n) =x(n) log kxk
|x(n)|.
Properties
• Ω2 is singular.
• Z2has a 2-dimensional unconditional decomposition generated by the subspacesEn=span{(en,0),(0, en)}.
• Z2has no unconditional basis. Moreover, it does not admit G.L−l.u.st
• Every infinite-dimensional complemented subspace ofZ2 contains a complemented subspace isomorphic to Z2.
• Z2is isomorphic to its square.
Question
Z2is isomorphic to its hyperplanes?
Complex structures on Z
2Proposition
The following complex spaces are isomorphic.
• Z2(u2,u2), where (u2, u2)(x, y) = (u2x, u2y).
• Z2⊕CZ2.
• Z2(C) =`2(C)⊕ΩC 2`2(C).
Corollary
For any complex structurewonZ2
• The space Z2w is isomorphic to a complemented subspace ofZ2(C).
• The space Z2w isZ2(C)-complementably saturated and
`2(C)-saturated.
• IfZ2w is isomorphic to its square then it is isomorphic toZ2(C).
Question
DoesZ2admit unique complex structure?
Complex structures on Z
2Proposition
The following complex spaces are isomorphic.
• Z2(u2,u2), where (u2, u2)(x, y) = (u2x, u2y).
• Z2⊕CZ2.
• Z2(C) =`2(C)⊕ΩC 2`2(C).
Corollary
For any complex structurewonZ2
• The space Z2w is isomorphic to a complemented subspace ofZ2(C).
• The space Z2w isZ2(C)-complementably saturated and
`2(C)-saturated.
• IfZ2w is isomorphic to its square then it is isomorphic toZ2(C).
Question
DoesZ2admit unique complex structure?
Complex structures on Z
2Proposition
The following complex spaces are isomorphic.
• Z2(u2,u2), where (u2, u2)(x, y) = (u2x, u2y).
• Z2⊕CZ2.
• Z2(C) =`2(C)⊕ΩC 2 `2(C).
Corollary
For any complex structurewonZ2
• The space Z2w is isomorphic to a complemented subspace ofZ2(C).
• The space Z2w isZ2(C)-complementably saturated and
`2(C)-saturated.
• IfZ2w is isomorphic to its square then it is isomorphic toZ2(C).
Question
DoesZ2admit unique complex structure?
Complex structures on Z
2Proposition
The following complex spaces are isomorphic.
• Z2(u2,u2), where (u2, u2)(x, y) = (u2x, u2y).
• Z2⊕CZ2.
• Z2(C) =`2(C)⊕ΩC 2 `2(C).
Corollary
For any complex structurewonZ2
• The space Z2w is isomorphic to a complemented subspace ofZ2(C).
• The space Z2w isZ2(C)-complementably saturated and
`2(C)-saturated.
• IfZ2w is isomorphic to its square then it is isomorphic toZ2(C).
Question
DoesZ2admit unique complex structure?
Complex structures on Z
2Proposition
The following complex spaces are isomorphic.
• Z2(u2,u2), where (u2, u2)(x, y) = (u2x, u2y).
• Z2⊕CZ2.
• Z2(C) =`2(C)⊕ΩC 2 `2(C).
Corollary
For any complex structurewonZ2
• The space Z2w is isomorphic to a complemented subspace ofZ2(C).
• The space Z2w isZ2(C)-complementably saturated and
`2(C)-saturated.
• IfZ2w is isomorphic to its square then it is isomorphic toZ2(C).
Question
DoesZ2admit unique complex structure?
Complex structures on Z
2Proposition
The following complex spaces are isomorphic.
• Z2(u2,u2), where (u2, u2)(x, y) = (u2x, u2y).
• Z2⊕CZ2.
• Z2(C) =`2(C)⊕ΩC 2 `2(C).
Corollary
For any complex structurewonZ2
• The space Z2w is isomorphic to a complemented subspace ofZ2(C).
• The space Z2w isZ2(C)-complementably saturated and
`2(C)-saturated.
• IfZ2w is isomorphic to its square then it is isomorphic toZ2(C).
Question
DoesZ2admit unique complex structure?
Complex structures on Z
2Proposition
The following complex spaces are isomorphic.
• Z2(u2,u2), where (u2, u2)(x, y) = (u2x, u2y).
• Z2⊕CZ2.
• Z2(C) =`2(C)⊕ΩC 2 `2(C).
Corollary
For any complex structurewonZ2
• The space Z2w is isomorphic to a complemented subspace ofZ2(C).
• The space Z2w isZ2(C)-complementably saturated and
`2(C)-saturated.
• IfZ2w is isomorphic to its square then it is isomorphic toZ2(C).
Question
DoesZ2admit unique complex structure?
Compatible complex structures on Z
2and its hyperplanes
Theorem
No complex structure on`2 can be extended to a complex structure on the hyperplane`2⊕Ω2iH.
Theorem
There exists a complex structureU on`2 that can not be extended to any operator onZ2.
An essential element to prove this is the following result:
Theorem (V. Ferenczi, E. Galego, 2007)
LetT, ube complex structures on, respectively, an infinite dimensional Banach spaceX and some hyperplaneH ofX. Then the operator T|H−uis not strictly singular.
Compatible complex structures on Z
2and its hyperplanes
Theorem
No complex structure on`2 can be extended to a complex structure on the hyperplane`2⊕Ω2iH.
Theorem
There exists a complex structureU on`2 that can not be extended to any operator onZ2.
An essential element to prove this is the following result:
Theorem (V. Ferenczi, E. Galego, 2007)
LetT, ube complex structures on, respectively, an infinite dimensional Banach spaceX and some hyperplaneH ofX. Then the operator T|H−uis not strictly singular.
Compatible complex structures on Z
2and its hyperplanes
Theorem
No complex structure on`2 can be extended to a complex structure on the hyperplane`2⊕Ω2iH.
Theorem
There exists a complex structureU on`2 that can not be extended to any operator onZ2.
An essential element to prove this is the following result:
Theorem (V. Ferenczi, E. Galego, 2007)
LetT, ube complex structures on, respectively, an infinite dimensional Banach spaceX and some hyperplaneH ofX. Then the operator T|H−uis not strictly singular.
Compatible complex structures on Z
2and its hyperplanes
Definition
The pair of operators(α, β)is said to be compatible withΩif there exists an operatorγ such that the diagram is commutative.
0 //X //
α
X⊕ΩY //Y //
β
0
0 //X //X⊕ΩY //Y //0
Proposition. The pair(α, β)is compatible withΩiff αΩ−Ωβ is trivial.
Example. The pair(u2, u2)is compatible withΩ2.
Compatible complex structures on Z
2and its hyperplanes
Definition
The pair of operators(α, β)is said to becompatiblewithΩif there exists an operatorγ such that the diagram is commutative.
0 //X //
α
X⊕ΩY //
γ
Y //
β
0
0 //X //X⊕ΩY //Y //0
Proposition. The pair(α, β)is compatible withΩiff αΩ−Ωβ is trivial.
Example. The pair(u2, u2)is compatible withΩ2.
Compatible complex structures on Z
2and its hyperplanes
Definition
The pair of operators(α, β)is said to becompatiblewithΩif there exists an operatorγ such that the diagram is commutative.
0 //X //
α
X⊕ΩY //
γ
Y //
β
0
0 //X //X⊕ΩY //Y //0
Proposition. The pair(α, β)is compatible withΩiff αΩ−Ωβ is trivial.
Example. The pair(u2, u2)is compatible withΩ2.
Compatible complex structures on Z
2and its hyperplanes
Definition
The pair of operators(α, β)is said to becompatiblewithΩif there exists an operatorγ such that the diagram is commutative.
0 //X //
α
X⊕ΩY //
γ
Y //
β
0
0 //X //X⊕ΩY //Y //0
Proposition. The pair(α, β)is compatible withΩiff αΩ−Ωβ is trivial.
Example. The pair(u2, u2)is compatible withΩ2.
Compatible complex structures on Z
2and its hyperplanes
Proposition
For every operatorT :`2→`2, and for every block subspaceW of`2, the commutatorΩ2T −TΩ2 is trivial on some block subspace ofW.
Proposition
Let(T, U)be a pair of compatible operators onZ2. ThenT−U is compact.
Sketch of the proof: Letube a complex structure on`2. Suppose that can be extended toU on`2⊕Ω2iH
0 //`2 //
u
`2⊕Ω2iH //
U
H //
v
0
0 //`2 //`2⊕Ω2iH //H //0
Extendingvto a complex structure V on`2, we have that(u, V)is compatible withΩ2. Thenu−V is compact.
On the other side, it follows from Ferenczi-Galego theorem thatu|H−v is not strictly singular. So we get a contradiction.
Compatible complex structures on Z
2and its hyperplanes
Proposition
For every operatorT :`2→`2, and for every block subspaceW of`2, the commutatorΩ2T −TΩ2 is trivial on some block subspace ofW.
Proposition
Let(T, U)be a pair of compatible operators onZ2. ThenT−U is compact.
Sketch of the proof: Letube a complex structure on`2. Suppose that can be extended toU on`2⊕Ω2iH
0 //`2 //
u
`2⊕Ω2iH //
U
H //
v
0
0 //`2 //`2⊕Ω2iH //H //0
Extendingvto a complex structure V on`2, we have that(u, V)is compatible withΩ2. Thenu−V is compact.
On the other side, it follows from Ferenczi-Galego theorem thatu|H−v is not strictly singular. So we get a contradiction.
Compatible complex structures on Z
2and its hyperplanes
Proposition
For every operatorT :`2→`2, and for every block subspaceW of`2, the commutatorΩ2T −TΩ2 is trivial on some block subspace ofW.
Proposition
Let(T, U)be a pair of compatible operators onZ2. ThenT−U is compact.
Sketch of the proof: Letube a complex structure on`2. Suppose that can be extended toU on`2⊕Ω2iH
0 //`2 //
u
`2⊕Ω2iH //
U
H //
v
0
0 //`2 //`2⊕Ω2iH //H //0
Extendingvto a complex structure V on`2, we have that(u, V)is compatible withΩ2. Thenu−V is compact.
On the other side, it follows from Ferenczi-Galego theorem thatu|H−v is not strictly singular. So we get a contradiction.
Compatible complex structures on Z
2and its hyperplanes
Proposition
For every operatorT :`2→`2, and for every block subspaceW of`2, the commutatorΩ2T −TΩ2 is trivial on some block subspace ofW.
Proposition
Let(T, U)be a pair of compatible operators onZ2. ThenT−U is compact.
Sketch of the proof: Letube a complex structure on`2. Suppose that can be extended toU on`2⊕Ω2iH
0 //`2 //
u
`2⊕Ω2iH //
U
H //
v
0
0 //`2 //`2⊕Ω2iH //H //0
Extendingvto a complex structure V on`2, we have that(u, V)is compatible withΩ2. Thenu−V is compact.
On the other side, it follows from Ferenczi-Galego theorem thatu|H−v is not strictly singular. So we get a contradiction.
Compatible complex structures on Z
2and its hyperplanes
Proposition
For every operatorT :`2→`2, and for every block subspaceW of`2, the commutatorΩ2T −TΩ2 is trivial on some block subspace ofW.
Proposition
Let(T, U)be a pair of compatible operators onZ2. ThenT−U is compact.
Sketch of the proof: Letube a complex structure on`2. Suppose that can be extended toU on`2⊕Ω2iH
0 //`2 //
u
`2⊕Ω2iH //
U
H //
v
0
0 //`2 //`2⊕Ω2iH //H //0
Extendingvto a complex structure V on`2, we have that(u, V)is compatible withΩ2. Thenu−V is compact.
On the other side, it follows from Ferenczi-Galego theorem thatu|H−v is not strictly singular. So we get a contradiction.
References
J. Bourgain.
Real isomorphic complex Banach spaces need not be complex isomorphic
Proc. Amer. Math. Soc. 96 (2)(1986), 221–226.
F. Cabello S´anchez, J.M.F. Castillo, J. Su´arez.
On strictly singular nonlinear centralizers Nonlinear Anal. 75 (2012), 3313–3321.
V. Ferenczi, E. Galego.
Even infinite-dimensional real Banach spaces J. Funct. Anal 253 (2)(2007), 534–549.
N. J. Kalton.
An elementary example of a Banach space not isomorphic to its complex conjugate
Canad. Math. Bull. (38)(1995), 218–222.
References
N.J. Kalton, N.T. Peck.
Twisted sums of sequence spaces and the three space problem Trans. Amer. Math. Soc. 255(1979) 1–30.
J. Lindenstrauss and L. Tzafriri
Classical Banach spaces I, sequence spaces Ergeb. Math. 92, Springer-Verlag(1977).
S. Szarek.
On the existence and uniqueness of complex structure and spaces with ‘few’ operators
Trans. Amer. Math. Soc. 293 (1)(1986), 339–353.
