Page 211 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 211
RECENT DEVELOPMENTS ON PRESERVERS (MS-38)
Maps on positive definite cones of C∗-algebras preserving the Wasserstein
mean
Lajos Molnar, molnarl@math.u-szeged.hu
University of Szeged, Hungary
In the past years, we have obtained some structural results for bijective maps between positive
definite cones of operator algebras which preserve some specific Kubo-Ando means (power
means, geometric mean).
Recently, Bhatia, Jain, and Lim have introduced a new mean for positive definite matrices
called Wasserstein mean. Its importance lies in its close connection to the Bures-Wasserstein
metric.
In this talk, we give the complete description of all (continuous) isomorphisms between
positive definite cones of C∗-algebras with respect to the operation of the Wasserstein mean and
present a result concerning the nonexistence of nonconstant such morphisms into the positive
reals. Comments on the algebraic properties of the Wasserstein mean relating associativity are
also made.
Loewner’s theorem for maps on operator domains
Michiya Mori, michiya.mori@riken.jp
RIKEN, Japan
The classical Loewner’s theorem states that operator monotone functions on real intervals are
described by holomorphic functions on the upper half-plane. We characterize local order iso-
morphisms on operator domains by biholomorphic automorphisms of the generalized upper
half-plane, which is the collection of all operators with positive invertible imaginary part. We
describe such maps in an explicit manner, and examine properties of local order isomorphisms.
Moreover, in the finite-dimensional case, we explain that every order embedding of a matrix
domain is a homeomorphic order isomorphism onto another matrix domain. This is joint work
with Peter Šemrl (University of Ljubljana).
Surjective isometries on Banach algebras of Lipschitz maps taking values
in a unital C∗-algebra
Shiho Oi, shiho-oi@math.sc.niigata-u.ac.jp
Department of Mathematics, Faculty of Science, Niigata University, Japan
Let A be a unital C∗-algebra. If its center is trivial, i.e. A ∩ A = C1, we call it a unital factor
C∗-algebra. In this talk, we consider unital surjective complex linear isometries on Lip(X, A)
with · L = · ∞ + L(·). The celebrated Kadison theorem yields that unital surjective linear
isometries between unital C∗-algebras are Jordan ∗-isomorphisms. We consider surjective lin-
ear isometries on Banach algebras of continuous maps taking values in a C∗-algebra and derive
extensions of the Kadison theorem.
In [O. Hatori, K. Kawamura and S. Oi, Hermitian operators and isometries on injective
tensor products of uniform algebras and C∗-algebras, JMAA, 2019], we proved that if Ai is a
unital factor C∗-algebra for i = 1, 2, every surjective linear isometry U from C(K1, A1) onto
209
Maps on positive definite cones of C∗-algebras preserving the Wasserstein
mean
Lajos Molnar, molnarl@math.u-szeged.hu
University of Szeged, Hungary
In the past years, we have obtained some structural results for bijective maps between positive
definite cones of operator algebras which preserve some specific Kubo-Ando means (power
means, geometric mean).
Recently, Bhatia, Jain, and Lim have introduced a new mean for positive definite matrices
called Wasserstein mean. Its importance lies in its close connection to the Bures-Wasserstein
metric.
In this talk, we give the complete description of all (continuous) isomorphisms between
positive definite cones of C∗-algebras with respect to the operation of the Wasserstein mean and
present a result concerning the nonexistence of nonconstant such morphisms into the positive
reals. Comments on the algebraic properties of the Wasserstein mean relating associativity are
also made.
Loewner’s theorem for maps on operator domains
Michiya Mori, michiya.mori@riken.jp
RIKEN, Japan
The classical Loewner’s theorem states that operator monotone functions on real intervals are
described by holomorphic functions on the upper half-plane. We characterize local order iso-
morphisms on operator domains by biholomorphic automorphisms of the generalized upper
half-plane, which is the collection of all operators with positive invertible imaginary part. We
describe such maps in an explicit manner, and examine properties of local order isomorphisms.
Moreover, in the finite-dimensional case, we explain that every order embedding of a matrix
domain is a homeomorphic order isomorphism onto another matrix domain. This is joint work
with Peter Šemrl (University of Ljubljana).
Surjective isometries on Banach algebras of Lipschitz maps taking values
in a unital C∗-algebra
Shiho Oi, shiho-oi@math.sc.niigata-u.ac.jp
Department of Mathematics, Faculty of Science, Niigata University, Japan
Let A be a unital C∗-algebra. If its center is trivial, i.e. A ∩ A = C1, we call it a unital factor
C∗-algebra. In this talk, we consider unital surjective complex linear isometries on Lip(X, A)
with · L = · ∞ + L(·). The celebrated Kadison theorem yields that unital surjective linear
isometries between unital C∗-algebras are Jordan ∗-isomorphisms. We consider surjective lin-
ear isometries on Banach algebras of continuous maps taking values in a C∗-algebra and derive
extensions of the Kadison theorem.
In [O. Hatori, K. Kawamura and S. Oi, Hermitian operators and isometries on injective
tensor products of uniform algebras and C∗-algebras, JMAA, 2019], we proved that if Ai is a
unital factor C∗-algebra for i = 1, 2, every surjective linear isometry U from C(K1, A1) onto
209
