invitation to linear preserver problems i · mostafa mbekhta invitation to linear preserver...

INVITATION TO LINEAR PRESERVERPROBLEMS I

Mostafa Mbekhta

Lille 1

R.T.O.T.A. 3-5 May 2018Memphis

Mostafa Mbekhta INVITATION TO LINEAR PRESERVER PROBLEMS I

I. IntroductionLinear preserver problems is an active research area in Matrix,Operator Theory and Banach Algebras. These problems involvecertain linear transformations on spaces of matrices, operators orBanach algebras... that will be designates by A in what follows.

First, we start by listing the type of problems that are related to LinearPreserved Problems topic.We choose some well-known results to give examples in each typicalproblem. We do not pretend to make a historical study of thisproblem.

I. IntroductionLinear preserver problems is an active research area in Matrix,Operator Theory and Banach Algebras. These problems involvecertain linear transformations on spaces of matrices, operators orBanach algebras... that will be designates by A in what follows.First, we start by listing the type of problems that are related to LinearPreserved Problems topic.

We choose some well-known results to give examples in each typicalproblem. We do not pretend to make a historical study of thisproblem.

I. IntroductionLinear preserver problems is an active research area in Matrix,Operator Theory and Banach Algebras. These problems involvecertain linear transformations on spaces of matrices, operators orBanach algebras... that will be designates by A in what follows.First, we start by listing the type of problems that are related to LinearPreserved Problems topic.We choose some well-known results to give examples in each typicalproblem. We do not pretend to make a historical study of thisproblem.

Problem I

Problem I. Let F be a (scalar-valued, vector-valued, or set-valued)given function on A. We would like to characterize those lineartransformations φ on A which satisty

F(φ(X)) = F(X) (X ∈ A).

Examples : (i) scalar-valued : A = Mn(C) and F(X) = det(X)

Theorem [G. Frobenius 1897]A linear map φ fromMn(C) intoMn(C) preserves the determinant(i.e. det(φ(X)) = det(X)) if and only if it takes one of the followingforms

φ(X) = MXN or φ(X) = MXtrN, for all X ∈Mn(C)

where M,N ∈Mn(C) are non-singular matrices such thatdet(MN) = 1, and Xtr stands for the transpose of X.

Problem I

F(φ(X)) = F(X) (X ∈ A).

Problem I

F(φ(X)) = F(X) (X ∈ A).

Problem I

F(φ(X)) = F(X) (X ∈ A).

Problem I

F(φ(X)) = F(X) (X ∈ A).

Theorem [G. Dolinar and P.Semrl 2002]If φ :Mn(C)→Mn(C) is a surjective mapping satisfying

det(X + λY) = det(φ(X) + λφ(Y)) for all X,Y ∈Mn(C), λ ∈ C,

then φ takes one of the following forms

Problem I

(ii) A = C(K) and F(X) = ||X||∞ (scalar-valued).

Let K be a compact metric space and C(K) the Banach space ofcontinuos real valued functions defined on K, with the supremunnorm.

Theorem [Banach 1932]Let φ : C(K)→ C(K) be a surjective linear map. Thenφ is an isometry (i.e. ||φ(f )||∞ = ||f ||∞) if and only if

φ(f (t)) = h(t)f (ϕ(t)), t ∈ K

where |h(t)| = 1 and ϕ is a homeomorphism of K onto itself.

Problem I

(ii) A = C(K) and F(X) = ||X||∞ (scalar-valued).Let K be a compact metric space and C(K) the Banach space ofcontinuos real valued functions defined on K, with the supremunnorm.

φ(f (t)) = h(t)f (ϕ(t)), t ∈ K

Problem I

(ii) A = C(K) and F(X) = ||X||∞ (scalar-valued).Let K be a compact metric space and C(K) the Banach space ofcontinuos real valued functions defined on K, with the supremunnorm.

φ(f (t)) = h(t)f (ϕ(t)), t ∈ K

Problem I

Kadison (1951) described the surjective linear isometries ofC∗-algebras.

Theorem [Kadison 1951]Let A and B be C∗-algebras and φ : A → B be a surjective linearmap. Thenφ is an isometry if and only if there is unitary u ∈ B and aC∗-isomorphism ϕ : A → B such that φ = uϕ.

It follows from this general result that the surjective linear isometriesof the Banach space B(H) of all bounded linear operators acting onHilbert space H can be described in the following way.

Problem I

TheoremThe surjective linear map φ : B(H)→ B(H) is an isometry if andonly if there are unitary operators U,V on H such that φ is either ofthe form:

φ(X) = UXV or φ(X) = UXtrV (X ∈ B(H)),

where T tr is the transpose of T with respect to an arbitrary but fixedorthonormal basis of H,

An excellent reference on isometries:

R. Fleming, J. Jamison, Isometries on Banach Spaces: FunctionSpaces, Chapman and Hall/CRC, Boca Raton, 2003.

Problem I

(iii) vector-valued : A C∗-algebra and F : A → A, F(X) = |X|, theabsolute value of X (i.e. |X| = (X∗X)1/2).

TheoremLet A be a C∗-algebra and φ : A → A be a surjective linear map.Then |φ(x)| = |x| for all x ∈ A if and only if there exists an unitaryelement u ∈ A such that φ(x) = ux for all x ∈ A.

(iv) set-valued : A Banach algebra and F(X) = σ(X).

Problem I

Problem II

Problem II. Let S be a given subset of A. We would like tocharacterize those linear map φ on A wich satisfy :

X ∈ S =⇒ φ(X) ∈ S (X ∈ A)

or satisfyX ∈ S ⇐⇒ φ(X) ∈ S (X ∈ A).

Examples :(i) S = {T ∈ B(H); T∗T = TT∗} the set of normal operators. We have

Theorem [Bresar and Semrel 1997]Let H be a Hilbert space and φ : B(H)→ B(H) be a bijective linearmap. Then φ preserves normal operators if and only if there is aunitary operator U in B(H) and 0 6= α ∈ C such that φ takes one ofthe following formsφ(T) = αUTU∗ + f (T)I or φ(T) = αUT trU∗ + f (T)I for all T,where f is a linear functional on B(H).

Problem II

X ∈ S =⇒ φ(X) ∈ S (X ∈ A)

Problem II

X ∈ S =⇒ φ(X) ∈ S (X ∈ A)

Problem II

X ∈ S =⇒ φ(X) ∈ S (X ∈ A)

Problem II

X ∈ S =⇒ φ(X) ∈ S (X ∈ A)

(ii) S = {T ∈ B(H); T2 = T} the set of idempotent operators. Wehave.

TheoremLet H be a Hilbert space and φ : B(H)→ B(H) be a bijective linearmap. Then φ preserves idempotent operators if and only if there is ainvertible operator A in B(H) such that φ takes one of the followingforms φ(T) = ATA−1 or φ(T) = AT trA−1 for all T,

(iii) S = A−1 = {X ∈ A; X invertible}(iv) S = {X ∈ A;∃Y ∈ A; XYX = X}, the set of all the elements of Ahaving a generalized inverse.The two previous examples, where the first one is related toKaplansky problem, will be developed in detail in this presentation,later on.

(iii) S = A−1 = {X ∈ A; X invertible}

(iv) S = {X ∈ A;∃Y ∈ A; XYX = X}, the set of all the elements of Ahaving a generalized inverse.The two previous examples, where the first one is related toKaplansky problem, will be developed in detail in this presentation,later on.

(iii) S = A−1 = {X ∈ A; X invertible}

(iv) S = {X ∈ A;∃Y ∈ A; XYX = X}, the set of all the elements of Ahaving a generalized inverse.The two previous examples, where the first one is related toKaplansky problem, will be developed in detail in this presentation,later on.

(iii) S = A−1 = {X ∈ A; X invertible}(iv) S = {X ∈ A;∃Y ∈ A; XYX = X}, the set of all the elements of Ahaving a generalized inverse.

The two previous examples, where the first one is related toKaplansky problem, will be developed in detail in this presentation,later on.

(iii) S = A−1 = {X ∈ A; X invertible}(iv) S = {X ∈ A;∃Y ∈ A; XYX = X}, the set of all the elements of Ahaving a generalized inverse.The two previous examples, where the first one is related toKaplansky problem, will be developed in detail in this presentation,later on.

Problem III

Problem III. LetR be a relation on A. We would like to Characterizethose linear transformations φ on A which satisfy

X R Y =⇒ φ(X) R φ(Y), (X,Y ∈ A)

or satisfy

X R Y ⇐⇒ φ(X) R φ(Y), (X,Y ∈ A).

Examples :(i)R could be commutativity. We have

TheoremSuppose that the dimension of H is greater than 2 andφ : B(H)→ B(H) be a bijective linear map. Then φ preservescommutativity if and only if there is a invertible operator A in B(H)and 0 6= α ∈ C such that φ takes one of the following formsφ(T) = αATA−1 + f (T)I or φ(T) = αAT trA−1 + f (T)I for all T,where f is a linear functional on B(H).

Problem III

X R Y =⇒ φ(X) R φ(Y), (X,Y ∈ A)

or satisfy

X R Y ⇐⇒ φ(X) R φ(Y), (X,Y ∈ A).

Problem III

X R Y =⇒ φ(X) R φ(Y), (X,Y ∈ A)

or satisfy

X R Y ⇐⇒ φ(X) R φ(Y), (X,Y ∈ A).

(ii)R could be similarityWe will say that T and S are similar if there is W invertible operatorsuch that T = WSW−1.We will also say that φ preserves similarity if T, S similar impliesφ(T), φ(S) similar.

TheoremA linear map φ onMn preserves similarity if and only if there existα, β ∈ C and an invertible A ∈Mn such that φ is of the form

φ(T) = αATA−1 +βtr(T)I or φ(T) = αAT trA−1 +βtr(T)I for all T,

or, there exists a fixed B ∈Mn such that φ is of the form

φ(T) = (tr(T))B for all T.

where tr(T) is the trace of T .

(ii)R could be similarityWe will say that T and S are similar if there is W invertible operatorsuch that T = WSW−1.We will also say that φ preserves similarity if T, S similar impliesφ(T), φ(S) similar.

TheoremA linear map φ onMn preserves similarity if and only if there existα, β ∈ C and an invertible A ∈Mn such that φ is of the form

φ(T) = αATA−1 +βtr(T)I or φ(T) = αAT trA−1 +βtr(T)I for all T,

or, there exists a fixed B ∈Mn such that φ is of the form

φ(T) = (tr(T))B for all T.

where tr(T) is the trace of T .

(iii) “zero product preserving mappings” i.e. φ(X)φ(Y) = 0 for allX,Y ∈ A with XY = 0. We have

Let X be a Hausdorff compact topological space and E a real orcomplex Banach space. Let us the notationC(X,E) = {f : X → E; continuous}, with the norm‖f‖ = sup{‖f (t)‖; t ∈ X}.In this context, a linear map φ : C(X,E)→ C(X,E) preserves zeroproducts, if ‖f (t)‖‖g(t)‖ = 0 for every t ∈ X implies that‖φ(f (t))‖‖φ(g(t))‖ = 0.

Theorem[J.Jamison and M.Rajagopalan 1988]

φ : C(X,E)→ C(X,E) a bounded linear map preserving zeroproducts if and only if

φ(f )(t) = W(t)(f ◦ ϕ)(t) for every f ∈ C(X,E) and t ∈ X

where ϕ : X → X continuous and W : X → B(E) continuous.

(iii) “zero product preserving mappings” i.e. φ(X)φ(Y) = 0 for allX,Y ∈ A with XY = 0. We haveLet X be a Hausdorff compact topological space and E a real orcomplex Banach space. Let us the notationC(X,E) = {f : X → E; continuous}, with the norm‖f‖ = sup{‖f (t)‖; t ∈ X}.

In this context, a linear map φ : C(X,E)→ C(X,E) preserves zeroproducts, if ‖f (t)‖‖g(t)‖ = 0 for every t ∈ X implies that‖φ(f (t))‖‖φ(g(t))‖ = 0.

(iii) “zero product preserving mappings” i.e. φ(X)φ(Y) = 0 for allX,Y ∈ A with XY = 0. We haveLet X be a Hausdorff compact topological space and E a real orcomplex Banach space. Let us the notationC(X,E) = {f : X → E; continuous}, with the norm‖f‖ = sup{‖f (t)‖; t ∈ X}.In this context, a linear map φ : C(X,E)→ C(X,E) preserves zeroproducts, if ‖f (t)‖‖g(t)‖ = 0 for every t ∈ X implies that‖φ(f (t))‖‖φ(g(t))‖ = 0.

In case Hilbert space, we have the following

TheoremLet H1 and H2 be infinite-dimensional complex Hilbert spaces andφ : B(H1)→ B(H2) a bijective additive map preserving zeroproducts. Then

φ(T) = αATA−1 for all T ∈ B(H)1,

where α is a nonzero scalar and A is an invertible bounded linearoperator from H2 onto H1.

In case Hilbert space, we have the following

TheoremLet H1 and H2 be infinite-dimensional complex Hilbert spaces andφ : B(H1)→ B(H2) a bijective additive map preserving zeroproducts. Then

φ(T) = αATA−1 for all T ∈ B(H)1,

where α is a nonzero scalar and A is an invertible bounded linearoperator from H2 onto H1.

Problem IV

Problem IV. Given a map F : A → A. We would like to characterizethose linear transformations φ on A which satisfy

F(φ(X)) = φ(F(X)) (X ∈ A).

Examples :(i) A = Mn(C) and F(X) = X†

(X† stands for the Moore-Penrose inverses of X).

(ii) F(X) = |X|, the absolute value of X (i.e. |X| = (X∗X)1/2).R. V. Kadison proved the following theorem

Theorem[Kadison 52]Let A and B be C∗-algebras and φ : A → B be linear map, whichφ(1) = 1. If |φ(x)| = φ(|x|) then φ is a C∗-Jordan homomorphism.

(iii) F(X) = Xn, n being a fixed integer not less than 2. When n = 2,φ is a Jordan homomorphism.

Problem IV

F(φ(X)) = φ(F(X)) (X ∈ A).

Problem IV

F(φ(X)) = φ(F(X)) (X ∈ A).

(ii) F(X) = |X|, the absolute value of X (i.e. |X| = (X∗X)1/2).

R. V. Kadison proved the following theorem

Problem IV

F(φ(X)) = φ(F(X)) (X ∈ A).

Problem IV

F(φ(X)) = φ(F(X)) (X ∈ A).

Problem IV

F(φ(X)) = φ(F(X)) (X ∈ A).

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,

or equivalently, φ(a2) = φ(a)2 for all a in A.We say that φ is unital if φ(1) = 1.Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,or equivalently, φ(a2) = φ(a)2 for all a in A.

We say that φ is unital if φ(1) = 1.Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,or equivalently, φ(a2) = φ(a)2 for all a in A.We say that φ is unital if φ(1) = 1.

Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,or equivalently, φ(a2) = φ(a)2 for all a in A.We say that φ is unital if φ(1) = 1.Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.

(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,or equivalently, φ(a2) = φ(a)2 for all a in A.We say that φ is unital if φ(1) = 1.Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.

(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,or equivalently, φ(a2) = φ(a)2 for all a in A.We say that φ is unital if φ(1) = 1.Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.

(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,or equivalently, φ(a2) = φ(a)2 for all a in A.We say that φ is unital if φ(1) = 1.Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Kaplansky’s problem

Problem Assuming that φ : A → B is a linear map having a certainpreserving property, we would like to know whether such a map hasto be a Jordan homomorphism?

One of the most famous problems in this direction is

Kaplansky’s problem :Let φ be a unital surjective linear map between two semi-simpleBanach algebras A and B which preserves invertibility (i.e.,φ(x) ∈ B−1 whenever x ∈ A−1). Is it true that φ is a Jordanhomomorphism ?

This problem has been first solved in the finite-dimensional case:In 1897, Frobenius showed that a linear map φ from Mn(C) intoMn(C) which preserves the determinant is the composition of anautomorphism or an anti-automorphism with a left multiplication by amatrix of determinant 1.

Problem Assuming that φ : A → B is a linear map having a certainpreserving property, we would like to know whether such a map hasto be a Jordan homomorphism?One of the most famous problems in this direction is

This problem has been first solved in the finite-dimensional case:

In 1897, Frobenius showed that a linear map φ from Mn(C) intoMn(C) which preserves the determinant is the composition of anautomorphism or an anti-automorphism with a left multiplication by amatrix of determinant 1.

Theorem [J. Dieudonné, 1943]Let φ : Mn(C)→ Mn(C) be a surjective, unital linear map. Ifσ(φ(x)) ⊆ σ(x) for all x ∈ Mn(C), then there is an invertible elementa ∈ Mn(C) such that φ takes one of the following forms:

φ(x) = axa−1 or φ(x) = axtra−1.

In the commutative case the well-known Gleason-Kahane-Zelazkotheorem provides an affirmative answer.

Theorem [Gleason-Kahan-Zelazko, 1968]Let A be a Banach algebra and B a semi-simple commutative Banachalgebra.If φ : A → B is a linear map such that σ(φ(x)) ⊆ σ(x) for all x ∈ A,then φ is an homomorphism.

In the non-commutative case, the best known results so far are due toAupetit and Sourour.

Theorem [Sourour, 1996]Let X,Y be two Banach spaces and let φ : B(X)→ B(Y) a linear,bijective and unital map. Then the following properties areequivalent:(i) σ(φ(T)) ⊆ σ(T) for all T ∈ B(X);(ii) φ is an isomorphism or an anti-isomorphism;(iii) either there is A : Y → X invertible such that φ(T) = A−1TA forall T ∈ B(X);or, there is B : Y → X∗ invertible such that φ(T) = B−1T∗B for allT ∈ B(X), and in this case X and Y are reflexive.

Theorem [Sourour, 1996]Let X,Y be two Banach spaces and let φ : B(X)→ B(Y) a linear,bijective and unital map. Then the following properties areequivalent:

(i) σ(φ(T)) ⊆ σ(T) for all T ∈ B(X);(ii) φ is an isomorphism or an anti-isomorphism;(iii) either there is A : Y → X invertible such that φ(T) = A−1TA forall T ∈ B(X);or, there is B : Y → X∗ invertible such that φ(T) = B−1T∗B for allT ∈ B(X), and in this case X and Y are reflexive.

Theorem [Sourour, 1996]Let X,Y be two Banach spaces and let φ : B(X)→ B(Y) a linear,bijective and unital map. Then the following properties areequivalent:

(i) σ(φ(T)) ⊆ σ(T) for all T ∈ B(X);(ii) φ is an isomorphism or an anti-isomorphism;(iii) either there is A : Y → X invertible such that φ(T) = A−1TA forall T ∈ B(X);or, there is B : Y → X∗ invertible such that φ(T) = B−1T∗B for allT ∈ B(X), and in this case X and Y are reflexive.

Theorem [Sourour, 1996]Let X,Y be two Banach spaces and let φ : B(X)→ B(Y) a linear,bijective and unital map. Then the following properties areequivalent:(i) σ(φ(T)) ⊆ σ(T) for all T ∈ B(X);

(ii) φ is an isomorphism or an anti-isomorphism;(iii) either there is A : Y → X invertible such that φ(T) = A−1TA forall T ∈ B(X);or, there is B : Y → X∗ invertible such that φ(T) = B−1T∗B for allT ∈ B(X), and in this case X and Y are reflexive.

Theorem [Sourour, 1996]Let X,Y be two Banach spaces and let φ : B(X)→ B(Y) a linear,bijective and unital map. Then the following properties areequivalent:(i) σ(φ(T)) ⊆ σ(T) for all T ∈ B(X);(ii) φ is an isomorphism or an anti-isomorphism;

(iii) either there is A : Y → X invertible such that φ(T) = A−1TA forall T ∈ B(X);or, there is B : Y → X∗ invertible such that φ(T) = B−1T∗B for allT ∈ B(X), and in this case X and Y are reflexive.

Theorem [Sourour, 1996]Let X,Y be two Banach spaces and let φ : B(X)→ B(Y) a linear,bijective and unital map. Then the following properties areequivalent:(i) σ(φ(T)) ⊆ σ(T) for all T ∈ B(X);(ii) φ is an isomorphism or an anti-isomorphism;(iii) either there is A : Y → X invertible such that φ(T) = A−1TA forall T ∈ B(X);or, there is B : Y → X∗ invertible such that φ(T) = B−1T∗B for allT ∈ B(X), and in this case X and Y are reflexive.

We will say that a C∗-algebra A is of real rank zero if the set formedby all the real linear combinations of (orthogonal) projections is densein the set of self-adjoint elements of A.

It is well known that this property is satisfied by every von Neumannalgebra, and in particular by the C∗-algebra B(H) of all boundedlinear operators on a Hilbert space H, and by the Calkin algebraC(H) = B(H)/K(H) .

Theorem [B. Aupetit, 2000]

Let A be a C∗-algebra of real rank zero and B a semi-simple Banachalgebra.If φ : A → B is a surjective linear map such that σ(φ(x)) = σ(x) forall x ∈ A, then φ is a Jordan isomorphism.

We will say that a C∗-algebra A is of real rank zero if the set formedby all the real linear combinations of (orthogonal) projections is densein the set of self-adjoint elements of A.It is well known that this property is satisfied by every von Neumannalgebra, and in particular by the C∗-algebra B(H) of all boundedlinear operators on a Hilbert space H, and by the Calkin algebraC(H) = B(H)/K(H) .

Let A be a C∗-algebra of real rank zero and B a semi-simple Banachalgebra.

If φ : A → B is a surjective linear map such that σ(φ(x)) = σ(x) forall x ∈ A, then φ is a Jordan isomorphism.

Let A be a C∗-algebra of real rank zero and B a semi-simple Banachalgebra.

If φ : A → B is a surjective linear map such that σ(φ(x)) = σ(x) forall x ∈ A, then φ is a Jordan isomorphism.

RemarkThe Kaplansky problem is still open when A,B are C∗-algebras.

I conclude this section with the following results of L. T. Gardner

Theorem[Gardner 1979]Let A and B be C∗-algebras and φ : A → B be linear map, whichφ(1) = 1. Suppose that f is a non-affine, real continuous function onthe real open interval ]a, b[ such that

(φ ◦ f )(x) = (f ◦ φ)(x), for all x = x∗ with σ(x) ⊆ [a, b].

Then φ is a C∗-Jordan homomorphism.

As a consequence of this theorem, for f (t) = t−1, t ∈]0, 1[, weobtain the following theorem of M. D. Choi.

Theorem[Choi 1974]Let A and B be C∗-algebras and φ : A → B be linear map, unital andpositive such that

φ(x−1) = φ(x)−1, for all x ∈ A−1+ .

Then φ is a C∗-Jordan homomorphism.where A−1

+ = {x ∈ A; x positive and invertible}.

As a consequence of this theorem, for f (t) = t−1, t ∈]0, 1[, weobtain the following theorem of M. D. Choi.

Theorem[Choi 1974]Let A and B be C∗-algebras and φ : A → B be linear map, unital andpositive such that

φ(x−1) = φ(x)−1, for all x ∈ A−1+ .

Then φ is a C∗-Jordan homomorphism.where A−1

+ = {x ∈ A; x positive and invertible}.

Maps preserving the nilpotency

Let N (X) be the set of all nilpotent operators i. e.N (X) = {A ∈ B(X); ∃ d ∈ N,Ad = 0} and

A ? B = AB (usual product) orA ? B = ABA (Jordan triple product), A,B ∈ B(X).

Theorem[C.-K Li, P. Semrl and N.-S. Sze 2007]Let X be an infinite dimensional Banach space. Then a surjective mapφ : B(X)→ B(X) satisfiesA ? B ∈ N (X) ⇐⇒ φ(A) ? φ(B) ∈ N (X), A,B ∈ B(X),If and only if(a) there is a bijective bounded linear or ani-linear operator S : X → Xsuch that φ(T) = S[f (T)T]S−1 T ∈ B(X), or(b) the space X is reflexive, and there exists a bijective bounded linearor ani-linear operator S : X∗ → X such thatφ(T) = S[f (T)T∗]S−1 T ∈ B(X),where f : B(X)→ C∗ is a map such that for every nonzero T ∈ B(X)the map λ 7→ λf (λT) is surjective on C.

Let N (X) be the set of all nilpotent operators i. e.N (X) = {A ∈ B(X); ∃ d ∈ N,Ad = 0} andA ? B = AB (usual product) orA ? B = ABA (Jordan triple product), A,B ∈ B(X).

Theorem[C.-K Li, P. Semrl and N.-S. Sze 2007]Let X be an infinite dimensional Banach space. Then a surjective mapφ : B(X)→ B(X) satisfiesA ? B ∈ N (X) ⇐⇒ φ(A) ? φ(B) ∈ N (X), A,B ∈ B(X),If and only if

(a) there is a bijective bounded linear or ani-linear operator S : X → Xsuch that φ(T) = S[f (T)T]S−1 T ∈ B(X), or(b) the space X is reflexive, and there exists a bijective bounded linearor ani-linear operator S : X∗ → X such thatφ(T) = S[f (T)T∗]S−1 T ∈ B(X),where f : B(X)→ C∗ is a map such that for every nonzero T ∈ B(X)the map λ 7→ λf (λT) is surjective on C.

Theorem[C.-K Li, P. Semrl and N.-S. Sze 2007]Let X be an infinite dimensional Banach space. Then a surjective mapφ : B(X)→ B(X) satisfiesA ? B ∈ N (X) ⇐⇒ φ(A) ? φ(B) ∈ N (X), A,B ∈ B(X),If and only if

(a) there is a bijective bounded linear or ani-linear operator S : X → Xsuch that φ(T) = S[f (T)T]S−1 T ∈ B(X), or(b) the space X is reflexive, and there exists a bijective bounded linearor ani-linear operator S : X∗ → X such thatφ(T) = S[f (T)T∗]S−1 T ∈ B(X),where f : B(X)→ C∗ is a map such that for every nonzero T ∈ B(X)the map λ 7→ λf (λT) is surjective on C.

Theorem[C.-K Li, P. Semrl and N.-S. Sze 2007]Let X be an infinite dimensional Banach space. Then a surjective mapφ : B(X)→ B(X) satisfiesA ? B ∈ N (X) ⇐⇒ φ(A) ? φ(B) ∈ N (X), A,B ∈ B(X),If and only if(a) there is a bijective bounded linear or ani-linear operator S : X → Xsuch that φ(T) = S[f (T)T]S−1 T ∈ B(X), or

(b) the space X is reflexive, and there exists a bijective bounded linearor ani-linear operator S : X∗ → X such thatφ(T) = S[f (T)T∗]S−1 T ∈ B(X),where f : B(X)→ C∗ is a map such that for every nonzero T ∈ B(X)the map λ 7→ λf (λT) is surjective on C.

Remark (1) Note that the map φ is not assumed to be linear.

(2) Question: I do not know if Jordan product was considered in theprevious theorem?we recall the definition of Jordan product. A ? B = 1

2 [AB + BA](Jordan product)

Some open problemsProblem 1. maps that preserve the set of algebraic elementsLet P be a complex polynomial of degree greater than 1. We say that alinear map φ : A → B preserves elements annihilated by P ifP(φ(a)) = 0 whenever P(a) = 0.R. Howard (1980) gives a characterization of invertible linear mapswhich preserve matrices annihilated by a polynomial.For the infinite dimensional case, let H be a Hilbert space and P acomplex polynomial of degree greater than 1 and φ : B(H)→ B(H)is a unital, surjective linear map. P. Semrl (1996) proved that if φpreserves operators annihilated by P in both directions, then φ iseither an automorphism or an anti-automorphism.

Remark (1) Note that the map φ is not assumed to be linear.

(2) Question: I do not know if Jordan product was considered in theprevious theorem?we recall the definition of Jordan product. A ? B = 1

Remark (1) Note that the map φ is not assumed to be linear.(2) Question: I do not know if Jordan product was considered in theprevious theorem?we recall the definition of Jordan product. A ? B = 1

Some open problemsProblem 1. maps that preserve the set of algebraic elements

Let P be a complex polynomial of degree greater than 1. We say that alinear map φ : A → B preserves elements annihilated by P ifP(φ(a)) = 0 whenever P(a) = 0.R. Howard (1980) gives a characterization of invertible linear mapswhich preserve matrices annihilated by a polynomial.For the infinite dimensional case, let H be a Hilbert space and P acomplex polynomial of degree greater than 1 and φ : B(H)→ B(H)is a unital, surjective linear map. P. Semrl (1996) proved that if φpreserves operators annihilated by P in both directions, then φ iseither an automorphism or an anti-automorphism.

Some open problemsProblem 1. maps that preserve the set of algebraic elements

Let P be a complex polynomial of degree greater than 1. We say that alinear map φ : A → B preserves elements annihilated by P ifP(φ(a)) = 0 whenever P(a) = 0.R. Howard (1980) gives a characterization of invertible linear mapswhich preserve matrices annihilated by a polynomial.For the infinite dimensional case, let H be a Hilbert space and P acomplex polynomial of degree greater than 1 and φ : B(H)→ B(H)is a unital, surjective linear map. P. Semrl (1996) proved that if φpreserves operators annihilated by P in both directions, then φ iseither an automorphism or an anti-automorphism.

Some open problemsProblem 1. maps that preserve the set of algebraic elementsLet P be a complex polynomial of degree greater than 1. We say that alinear map φ : A → B preserves elements annihilated by P ifP(φ(a)) = 0 whenever P(a) = 0.

R. Howard (1980) gives a characterization of invertible linear mapswhich preserve matrices annihilated by a polynomial.For the infinite dimensional case, let H be a Hilbert space and P acomplex polynomial of degree greater than 1 and φ : B(H)→ B(H)is a unital, surjective linear map. P. Semrl (1996) proved that if φpreserves operators annihilated by P in both directions, then φ iseither an automorphism or an anti-automorphism.

Some open problemsProblem 1. maps that preserve the set of algebraic elementsLet P be a complex polynomial of degree greater than 1. We say that alinear map φ : A → B preserves elements annihilated by P ifP(φ(a)) = 0 whenever P(a) = 0.R. Howard (1980) gives a characterization of invertible linear mapswhich preserve matrices annihilated by a polynomial.

For the infinite dimensional case, let H be a Hilbert space and P acomplex polynomial of degree greater than 1 and φ : B(H)→ B(H)is a unital, surjective linear map. P. Semrl (1996) proved that if φpreserves operators annihilated by P in both directions, then φ iseither an automorphism or an anti-automorphism.

Theorem J. Hou and S. Hou 2002]Let A be a von Neumann algebra and B a unital Banach algebra, andlet φ : A → B be a bounded linear map. Let P be a polynomial withno repeated roots and deg P ≥ 2 such that P(0) = 0. Then thefollowing statements are equivalent.

(1) φ commutes with P (i. e. φ(P(x)) = P(φ(x)));(2) φ preserves elements annihilated by P;(3) There exist an r-potent element B ∈ B with spectrum contained in{λ; λZ(P) = Z(P)} and a Jordan homomorphism ψ : A → B suchthat

φ = Bψ = ψB,

where Z(P) denote the set of zeros of P.

Theorem J. Hou and S. Hou 2002]Let A be a von Neumann algebra and B a unital Banach algebra, andlet φ : A → B be a bounded linear map. Let P be a polynomial withno repeated roots and deg P ≥ 2 such that P(0) = 0. Then thefollowing statements are equivalent.

(1) φ commutes with P (i. e. φ(P(x)) = P(φ(x)));(2) φ preserves elements annihilated by P;(3) There exist an r-potent element B ∈ B with spectrum contained in{λ; λZ(P) = Z(P)} and a Jordan homomorphism ψ : A → B suchthat

φ = Bψ = ψB,

Theorem J. Hou and S. Hou 2002]Let A be a von Neumann algebra and B a unital Banach algebra, andlet φ : A → B be a bounded linear map. Let P be a polynomial withno repeated roots and deg P ≥ 2 such that P(0) = 0. Then thefollowing statements are equivalent.(1) φ commutes with P (i. e. φ(P(x)) = P(φ(x)));

(2) φ preserves elements annihilated by P;(3) There exist an r-potent element B ∈ B with spectrum contained in{λ; λZ(P) = Z(P)} and a Jordan homomorphism ψ : A → B suchthat

φ = Bψ = ψB,

Theorem J. Hou and S. Hou 2002]Let A be a von Neumann algebra and B a unital Banach algebra, andlet φ : A → B be a bounded linear map. Let P be a polynomial withno repeated roots and deg P ≥ 2 such that P(0) = 0. Then thefollowing statements are equivalent.(1) φ commutes with P (i. e. φ(P(x)) = P(φ(x)));(2) φ preserves elements annihilated by P;

(3) There exist an r-potent element B ∈ B with spectrum contained in{λ; λZ(P) = Z(P)} and a Jordan homomorphism ψ : A → B suchthat

φ = Bψ = ψB,

Theorem J. Hou and S. Hou 2002]Let A be a von Neumann algebra and B a unital Banach algebra, andlet φ : A → B be a bounded linear map. Let P be a polynomial withno repeated roots and deg P ≥ 2 such that P(0) = 0. Then thefollowing statements are equivalent.(1) φ commutes with P (i. e. φ(P(x)) = P(φ(x)));(2) φ preserves elements annihilated by P;(3) There exist an r-potent element B ∈ B with spectrum contained in{λ; λZ(P) = Z(P)} and a Jordan homomorphism ψ : A → B suchthat

φ = Bψ = ψB,

Now, let P(X) be the set of algebric operators i. e.P(X) = {A ∈ B(X); ∃ 0 6= P ∈ C[X],P(A) = 0}

Conjecture. Let X be an infinite dimensional Banach space. Then asurjective linear map φ : B(X)→ B(X) satisfiesA ∈ P(X) ⇐⇒ φ(A) ∈ P(X), A ∈ B(X),if and only if(a) there is a bijective bounded linear operator S : X → X and α ∈ Csuch that φ(T) = αSTS−1 T ∈ B(X), or(b) the space X is reflexive, and there exists a bijective bounded linearoperator S : X∗ → X and α such that φ(T) = αST∗S−1 T ∈ B(X).

Conjecture. Let X be an infinite dimensional Banach space. Then asurjective linear map φ : B(X)→ B(X) satisfiesA ∈ P(X) ⇐⇒ φ(A) ∈ P(X), A ∈ B(X),if and only if

(a) there is a bijective bounded linear operator S : X → X and α ∈ Csuch that φ(T) = αSTS−1 T ∈ B(X), or(b) the space X is reflexive, and there exists a bijective bounded linearoperator S : X∗ → X and α such that φ(T) = αST∗S−1 T ∈ B(X).

Conjecture. Let X be an infinite dimensional Banach space. Then asurjective linear map φ : B(X)→ B(X) satisfiesA ∈ P(X) ⇐⇒ φ(A) ∈ P(X), A ∈ B(X),if and only if(a) there is a bijective bounded linear operator S : X → X and α ∈ Csuch that φ(T) = αSTS−1 T ∈ B(X), or

(b) the space X is reflexive, and there exists a bijective bounded linearoperator S : X∗ → X and α such that φ(T) = αST∗S−1 T ∈ B(X).

Problem 2.Characterization of maps that preserve the set of hyponormaloperators on Hilbert space ( i.e. T is hyponormal if TT∗ ≤ T∗T).

Conjecture. Let H be a Hilbert space and φ : B(H)→ B(H) be abijective linear map. Then φ preserves hyponormal operatorsif and only ifthere is a unitary operator U in B(H) and 0 6= α ∈ C such that φtakes one of the following formsφ(T) = αUTU∗ + f (T)I or φ(T) = αUT trU∗ + f (T)I for all T,where T tr is the transpose of T with respect to an arbitrary but fixedorthonormal basis of H, and f is a linear functional on B(H).

Conjecture. Let H be a Hilbert space and φ : B(H)→ B(H) be abijective linear map. Then φ preserves hyponormal operatorsif and only if

there is a unitary operator U in B(H) and 0 6= α ∈ C such that φtakes one of the following formsφ(T) = αUTU∗ + f (T)I or φ(T) = αUT trU∗ + f (T)I for all T,where T tr is the transpose of T with respect to an arbitrary but fixedorthonormal basis of H, and f is a linear functional on B(H).

Problem 3.Let K be a compact subset of C. Let us the notation

ΣK(X) = {T ∈ B(X) σ(T) = K}.

where X is a Banach space.

Conjecture. Let X be an infinite dimensional Banach space. Then asurjective linear map φ : B(X)→ B(X) satisfiesA ∈ ΣK(X) ⇐⇒ φ(A) ∈ ΣK(X), A ∈ B(X),if and only if(a) there is a bijective bounded linear operator S : X → X such thatφ(T) = STS−1 T ∈ B(X), or(b) the space X is reflexive, and there exists a bijective bounded linearoperator S : X∗ → X such that φ(T) = ST∗S−1 T ∈ B(X).

In particular if K = {0} then ΣK(X) is the set of quasi-nilpotentoperators.

ΣK(X) = {T ∈ B(X) σ(T) = K}.

Conjecture. Let X be an infinite dimensional Banach space. Then asurjective linear map φ : B(X)→ B(X) satisfiesA ∈ ΣK(X) ⇐⇒ φ(A) ∈ ΣK(X), A ∈ B(X),if and only if

(a) there is a bijective bounded linear operator S : X → X such thatφ(T) = STS−1 T ∈ B(X), or(b) the space X is reflexive, and there exists a bijective bounded linearoperator S : X∗ → X such that φ(T) = ST∗S−1 T ∈ B(X).

ΣK(X) = {T ∈ B(X) σ(T) = K}.

Conjecture. Let X be an infinite dimensional Banach space. Then asurjective linear map φ : B(X)→ B(X) satisfiesA ∈ ΣK(X) ⇐⇒ φ(A) ∈ ΣK(X), A ∈ B(X),if and only if(a) there is a bijective bounded linear operator S : X → X such thatφ(T) = STS−1 T ∈ B(X), or

(b) the space X is reflexive, and there exists a bijective bounded linearoperator S : X∗ → X such that φ(T) = ST∗S−1 T ∈ B(X).

ΣK(X) = {T ∈ B(X) σ(T) = K}.

invitation to linear preserver problems i · mostafa mbekhta invitation to linear preserver...

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