Page 212 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 212
RECENT DEVELOPMENTS ON PRESERVERS (MS-38)
C(K2, A2) is a weighted composition operator of the form U F (y) = uVy(F (ϕ(y))), where
ϕ : K2 → K1 is a homeomorphism, {Vy}y∈K2 is a strongly continuous family of Jordan ∗-
isomorphisms from A1 onto A2, and u ∈ C(K2, A2) is a unitary element. A main result in
this talk is a version of the Banach algebras of all Lipschitz maps of the theorem. Recently in
[S. Oi, Hermitian operators and isometries on algebras of matrix-valued Lipschitz maps, Linear
Multilinear Algebra, 2020], we gave a complete description of surjective linear isometries on
Lip(X, Mn(C)), where Mn(C) is the Banach algebra of complex matrices of order n. Hence
the main result is the generalization of it.
In the course of the proof, we characterize hermitian operators on Lip(X, E) with ·
L for any Banach space E. Note that similar results characterizing hermitian operators on
Lip(X, E) with · M = max{ · ∞, L(·)} have been already obtained in [F. Botelho, J. Jami-
son, A. Jiménez-Vargas and M. Villegas-Vallecillos, Hermitian operators on Lipschitz function
spaces, Studia Math., 2013].
This work is supported by JSPS KAKENHI Grant Number JP21K13804.
Free functions preserving certain partial orders of operators
Miklós Pálfia, palfiam@skku.edu
Sungkyunkwan University, South Korea, and University of Szeged, Hungary
Recently free analysis has been a very active topic of study in operator and function theory. In
particular free functions that preserve partial orders of operators have been studied by a number
of authors, in connection with Loewner’s theorem. Also operator concave and convex free
functions naturally get into the picture as we study the positive definite order preserving free
functions. We will go through recent results and open problems in this field, and we will cover
some recent joint work with M. Gaál on real operator monotone functions.
Linear maps which are (triple) derivable or anti-derivable at a point
Antonio M. Peralta, aperalta@ugr.es
Universidad de Granada, Spain
A typical challenge in the setting of preservers asks whether a linear map T from a C∗-algebra
A into a Banach A-bimodule X behaving like a derivation (i.e. D(ab) = D(a)b + aD(b)) or
like an anti-derivation (D(ab) = D(b)a + bD(a)) only on those pairs of elements (a, b) in a
proper subset D ⊂ A2 is in fact a derivation or an anti-derivation. A protagonist role is played
by sets of the form Dz := {(a, b) ∈ A2 : ab = z}, where z is a fixed point in A. A linear
map T : A → X is said to be a derivation or an anti-derivation at a point z ∈ A if it behaves
like a derivation or like an anti-derivation on pairs (a, b) ∈ Dz. These maps are usually called
derivable or anti-derivable at z. Let us simply observe that applying a similar method to define
linear maps which are homomorphisms at zero, we find a natural link with the fruitful line of
results on zero products preservers.
A recent study developed by B. Fadaee and H. Ghahramani in [3] characterizes continuous
linear maps from a C∗-algebra A into its bidual which are derivable at zero. A similar problem
was considered by H. Ghahramani and Z. Pan for linear maps on a complex Banach algebra
which is zero product determined [4]. These authors also find necessary conditions to guarantee
210
C(K2, A2) is a weighted composition operator of the form U F (y) = uVy(F (ϕ(y))), where
ϕ : K2 → K1 is a homeomorphism, {Vy}y∈K2 is a strongly continuous family of Jordan ∗-
isomorphisms from A1 onto A2, and u ∈ C(K2, A2) is a unitary element. A main result in
this talk is a version of the Banach algebras of all Lipschitz maps of the theorem. Recently in
[S. Oi, Hermitian operators and isometries on algebras of matrix-valued Lipschitz maps, Linear
Multilinear Algebra, 2020], we gave a complete description of surjective linear isometries on
Lip(X, Mn(C)), where Mn(C) is the Banach algebra of complex matrices of order n. Hence
the main result is the generalization of it.
In the course of the proof, we characterize hermitian operators on Lip(X, E) with ·
L for any Banach space E. Note that similar results characterizing hermitian operators on
Lip(X, E) with · M = max{ · ∞, L(·)} have been already obtained in [F. Botelho, J. Jami-
son, A. Jiménez-Vargas and M. Villegas-Vallecillos, Hermitian operators on Lipschitz function
spaces, Studia Math., 2013].
This work is supported by JSPS KAKENHI Grant Number JP21K13804.
Free functions preserving certain partial orders of operators
Miklós Pálfia, palfiam@skku.edu
Sungkyunkwan University, South Korea, and University of Szeged, Hungary
Recently free analysis has been a very active topic of study in operator and function theory. In
particular free functions that preserve partial orders of operators have been studied by a number
of authors, in connection with Loewner’s theorem. Also operator concave and convex free
functions naturally get into the picture as we study the positive definite order preserving free
functions. We will go through recent results and open problems in this field, and we will cover
some recent joint work with M. Gaál on real operator monotone functions.
Linear maps which are (triple) derivable or anti-derivable at a point
Antonio M. Peralta, aperalta@ugr.es
Universidad de Granada, Spain
A typical challenge in the setting of preservers asks whether a linear map T from a C∗-algebra
A into a Banach A-bimodule X behaving like a derivation (i.e. D(ab) = D(a)b + aD(b)) or
like an anti-derivation (D(ab) = D(b)a + bD(a)) only on those pairs of elements (a, b) in a
proper subset D ⊂ A2 is in fact a derivation or an anti-derivation. A protagonist role is played
by sets of the form Dz := {(a, b) ∈ A2 : ab = z}, where z is a fixed point in A. A linear
map T : A → X is said to be a derivation or an anti-derivation at a point z ∈ A if it behaves
like a derivation or like an anti-derivation on pairs (a, b) ∈ Dz. These maps are usually called
derivable or anti-derivable at z. Let us simply observe that applying a similar method to define
linear maps which are homomorphisms at zero, we find a natural link with the fruitful line of
results on zero products preservers.
A recent study developed by B. Fadaee and H. Ghahramani in [3] characterizes continuous
linear maps from a C∗-algebra A into its bidual which are derivable at zero. A similar problem
was considered by H. Ghahramani and Z. Pan for linear maps on a complex Banach algebra
which is zero product determined [4]. These authors also find necessary conditions to guarantee
210
