Page 213 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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RECENT DEVELOPMENTS ON PRESERVERS (MS-38)
that a continuous linear map T : A → A∗∗ is anti-derivable at zero, where A is a C∗-algebra,
and also for linear maps on a zero product determined unital ∗-algebra to be anti-derivable at
zero.
We have been involved in the study of those linear maps on C∗-algebras which are deriva-
tions or triple derivations at zero or at the unit [2]. We shall revisit some of the main conclu-
sions on these kind of maps from the perspective of preservers. We have further explored in
[1] whether a full characterization of those (continuous) linear maps on a C∗-algebra which are
(∗-)anti-derivable at zero can be given in pure algebraic terms. In this talk we shall present the
latest advances in [1], which provide a complete solution to this problem.
References
[1] D.A. Abulhamail, F.B. Jamjoom, A.M. Peralta, Linear maps which are anti-derivable at
zero, to appear in Bull. Malays. Math. Sci. Soc.. arXiv: 1911.04134v2
[2] A.B.A. Essaleh, A.M. Peralta, Linear maps on C∗-algebras which are derivations or triple
derivations at a point, Linear Algebra and its Applications 538, 1-21 (2018).
[3] B. Fadaee, H. Ghahramani, Linear maps behaving like derivations or anti-derivations
at orthogonal elements on C∗-algebras, to appear in Bull. Malays. Math. Sci. Soc.
https://doi.org/10.1007/s40840-019-00841-6. arXiv:1907.03594v1
[4] H. Ghahramani, Z. Pan, Linear maps on ∗-algebras acting on orthogonal elements like
derivations or anti-derivations, Filomat 32, no. 13, 4543-4554 (2018).
Singularity preserving maps on matrix algebras
Valentin Promyslov, valentin.promyslov@gmail.com
Lomonosov Moscow State University,
Moscow Center for Fundamental and Applied Mathematics, Russian Federation
Coauthor: Artem Maksaev
The talk is based on the joined work with Alexander Guterman and Artem Maksaev.
The first result on linear preservers was obtained by Ferdinand Georg Frobenius, who char-
acterized linear maps on complex matrix algebra preserving the determinant.
Let Mn(F) be the n × n matrix algebra over a field F and Y be a subset of Mn(F). We say
that a transformation T : Y → Mn(F) is of a standard form if there exist non-singular matrices
P, Q such that
T (A) = P AQ or T (A) = P AT Q for all A ∈ Y. (1)
Frobenius [1] proved that if T : Mn(C) → Mn(C) is linear and preserves the determinant, i. e.,
det(T (A)) = det(A) for all A ∈ Mn(C), then T is of the standard form (1) with det(P Q) =
1. In 1949 Jean Dieudonné [2] generalized this result for an arbitrary field F. He replaced
the determinant preserving condition by the singularity preserving condition and proved the
corresponding result for a bijective map T .
In 2002 Gregor Dolinar and Peter Šemrl [3] modified the classical result of Frobenius by
removing the linearity and replacing the determinant preserving condition by
det(A + λB) = det(T (A) + λT (B)) for all A, B ∈ Mn(F) and all λ ∈ F (2)
211
that a continuous linear map T : A → A∗∗ is anti-derivable at zero, where A is a C∗-algebra,
and also for linear maps on a zero product determined unital ∗-algebra to be anti-derivable at
zero.
We have been involved in the study of those linear maps on C∗-algebras which are deriva-
tions or triple derivations at zero or at the unit [2]. We shall revisit some of the main conclu-
sions on these kind of maps from the perspective of preservers. We have further explored in
[1] whether a full characterization of those (continuous) linear maps on a C∗-algebra which are
(∗-)anti-derivable at zero can be given in pure algebraic terms. In this talk we shall present the
latest advances in [1], which provide a complete solution to this problem.
References
[1] D.A. Abulhamail, F.B. Jamjoom, A.M. Peralta, Linear maps which are anti-derivable at
zero, to appear in Bull. Malays. Math. Sci. Soc.. arXiv: 1911.04134v2
[2] A.B.A. Essaleh, A.M. Peralta, Linear maps on C∗-algebras which are derivations or triple
derivations at a point, Linear Algebra and its Applications 538, 1-21 (2018).
[3] B. Fadaee, H. Ghahramani, Linear maps behaving like derivations or anti-derivations
at orthogonal elements on C∗-algebras, to appear in Bull. Malays. Math. Sci. Soc.
https://doi.org/10.1007/s40840-019-00841-6. arXiv:1907.03594v1
[4] H. Ghahramani, Z. Pan, Linear maps on ∗-algebras acting on orthogonal elements like
derivations or anti-derivations, Filomat 32, no. 13, 4543-4554 (2018).
Singularity preserving maps on matrix algebras
Valentin Promyslov, valentin.promyslov@gmail.com
Lomonosov Moscow State University,
Moscow Center for Fundamental and Applied Mathematics, Russian Federation
Coauthor: Artem Maksaev
The talk is based on the joined work with Alexander Guterman and Artem Maksaev.
The first result on linear preservers was obtained by Ferdinand Georg Frobenius, who char-
acterized linear maps on complex matrix algebra preserving the determinant.
Let Mn(F) be the n × n matrix algebra over a field F and Y be a subset of Mn(F). We say
that a transformation T : Y → Mn(F) is of a standard form if there exist non-singular matrices
P, Q such that
T (A) = P AQ or T (A) = P AT Q for all A ∈ Y. (1)
Frobenius [1] proved that if T : Mn(C) → Mn(C) is linear and preserves the determinant, i. e.,
det(T (A)) = det(A) for all A ∈ Mn(C), then T is of the standard form (1) with det(P Q) =
1. In 1949 Jean Dieudonné [2] generalized this result for an arbitrary field F. He replaced
the determinant preserving condition by the singularity preserving condition and proved the
corresponding result for a bijective map T .
In 2002 Gregor Dolinar and Peter Šemrl [3] modified the classical result of Frobenius by
removing the linearity and replacing the determinant preserving condition by
det(A + λB) = det(T (A) + λT (B)) for all A, B ∈ Mn(F) and all λ ∈ F (2)
211
