Page 207 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 207
RECENT DEVELOPMENTS ON PRESERVERS (MS-38)
Recent generalisations of classical theorems on quantum mechanical
symmetry transformations
György Pál Gehér, gehergyuri@gmail.com
University of Reading, United Kingdom
Symmetries of geometric and operator structures are important in the foundations of quan-
tum mechanics. One famous example is Wigner’s theorem on quantum mechanical symmetry
transformations, which describes all maps on the space of quantum pure states that leave the
transition probability invariant. In particular, Wigner’s theorem is an important step in obtaining
the general time-dependent Schrodinger equation from purely mathematical assumptions.
In my talk I’ll review some of these classical results, and will present some of their recently
obtained generalizations.
Part of my talk will be based on joint works with Peter Šemrl (University of Ljubljana,
Slovenia) and Michiya Mori (University of Tokyo, Japan).
Tingley’s problem on uniform algebras
Osamu Hatori, hatori@math.sc.niigata-u.ac.jp
Institute of Science and Technology, Niigata University, Japan
The Tingley’s problem asks if a surjective isometry between the unit spheres of two Banach
spaces is extended to a real-linear surjective isometry between whole of the Banach spaces.
Several progresses have been done by many authors, although I do not show a complete list
of the name of authors. Tanaka and Peralta initiated to study the case of algebras of matrices
and operators. Recently Mori and Ozawa gave a positive partial solution by proving that C∗-
algebras satisfy the Mazur-Ulam property. Several progresses are going on in this direction.
Very recently, Becerra-Guerrero, Cueto-Avellaneda, ,Ferenández-Polo and Peralta published
the results about JBW ∗-triples. On the other hand, the case of Banach spaces of analytic
functions, except Hiblert spaces, are still missing. I will give a talk in this case including
uniform algebras.
This is a joint work with Shiho Oi and Rumi Shindo Togashi.
Wigner’s theorem in normed spaces
Dijana Iliševic´, ilisevic@math.hr
University of Zagreb, Croatia
Coauthors: Aleksej Turnšek, Matjaž Omladicˇ
Let (H, (·, ·)) and (K, (·, ·)) be inner product spaces over F ∈ {R, C} and suppose that f : H →
K is a mapping satisfying
|(f (x), f (y))| = |(x, y)|, x, y ∈ H. (1)
Then the famous Wigner’s unitary–antiunitary theorem says that f is a solution of (1) if and
only if it is phase equivalent to a linear or an anti-linear isometry, say U , that is,
f (x) = σ(x)U x, x ∈ H,
205
Recent generalisations of classical theorems on quantum mechanical
symmetry transformations
György Pál Gehér, gehergyuri@gmail.com
University of Reading, United Kingdom
Symmetries of geometric and operator structures are important in the foundations of quan-
tum mechanics. One famous example is Wigner’s theorem on quantum mechanical symmetry
transformations, which describes all maps on the space of quantum pure states that leave the
transition probability invariant. In particular, Wigner’s theorem is an important step in obtaining
the general time-dependent Schrodinger equation from purely mathematical assumptions.
In my talk I’ll review some of these classical results, and will present some of their recently
obtained generalizations.
Part of my talk will be based on joint works with Peter Šemrl (University of Ljubljana,
Slovenia) and Michiya Mori (University of Tokyo, Japan).
Tingley’s problem on uniform algebras
Osamu Hatori, hatori@math.sc.niigata-u.ac.jp
Institute of Science and Technology, Niigata University, Japan
The Tingley’s problem asks if a surjective isometry between the unit spheres of two Banach
spaces is extended to a real-linear surjective isometry between whole of the Banach spaces.
Several progresses have been done by many authors, although I do not show a complete list
of the name of authors. Tanaka and Peralta initiated to study the case of algebras of matrices
and operators. Recently Mori and Ozawa gave a positive partial solution by proving that C∗-
algebras satisfy the Mazur-Ulam property. Several progresses are going on in this direction.
Very recently, Becerra-Guerrero, Cueto-Avellaneda, ,Ferenández-Polo and Peralta published
the results about JBW ∗-triples. On the other hand, the case of Banach spaces of analytic
functions, except Hiblert spaces, are still missing. I will give a talk in this case including
uniform algebras.
This is a joint work with Shiho Oi and Rumi Shindo Togashi.
Wigner’s theorem in normed spaces
Dijana Iliševic´, ilisevic@math.hr
University of Zagreb, Croatia
Coauthors: Aleksej Turnšek, Matjaž Omladicˇ
Let (H, (·, ·)) and (K, (·, ·)) be inner product spaces over F ∈ {R, C} and suppose that f : H →
K is a mapping satisfying
|(f (x), f (y))| = |(x, y)|, x, y ∈ H. (1)
Then the famous Wigner’s unitary–antiunitary theorem says that f is a solution of (1) if and
only if it is phase equivalent to a linear or an anti-linear isometry, say U , that is,
f (x) = σ(x)U x, x ∈ H,
205
