Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a real linear surjective isometry between the whole spaces. In this ...paper, we give an affirmative answer to Tingley's problem when both spaces are preduals of von Neumann algebras, the spaces of self-adjoint operators in von Neumann algebras or the spaces of self-adjoint normal functionals on von Neumann algebras. We also show that every surjective isometry between the unit spheres of unital C⁎-algebras restricts to a bijection between their unitary groups. In addition, we show that every surjective isometry between the normal state spaces or the normal quasi-state spaces of two von Neumann algebras extends to a linear surjective isometry.
We prove that if the closed unit ball of a normed space X has sufficiently many extreme points, then every mapping \Phi from X into itself with the following property is affine: For any pair of ...points in X, there exists a (not necessarily linear) surjective isometry on X that coincides with \Phi at the two points. We also consider surjectivity of such a mapping in some special cases including C ^*-algebras.
We give a complete description of order isomorphisms between operator intervals in general von Neumann algebras. For the description, we use Jordan
∗
-isomorphisms and locally measurable operators. ...Our results generalize several works by L. Molnár and P. Šemrl on type I factors.
Generalizing von Neumann’s result on type II
$_1$
von Neumann algebras, I characterise lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms ...between the algebras of locally measurable operators. Moreover, I give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.
We prove that the distance from an n×n complex matrix M to the set of nilpotents is at least 12secπn+2 if there is a nonzero projection P such that PMP=M and M⁎M≥P. In the particular case where M ...equals P, this verifies a conjecture by G.W. MacDonald in 1995. We also confirm a related conjecture in D.A. Herrero's book.
We show that every ring isomorphism between the algebras of locally measurable operators for type II∞$_\infty$ von Neumann algebras is similar to a real ∗^*$‐isomorphism. This together with previous ...results by the author and Ayupov–Kudaybergenov completely describes ring isomorphisms between the algebras of locally measurable operators as well as lattice isomorphisms between the projection lattices for a general pair of von Neumann algebras without finite type I direct summands.
For a positive integer $n$, we study the collection $\mathcal {F}_{\mathrm {fin}}(n)$ formed of all $n\times n$ matrices whose entries $a_{ij}$, $1\leq i,j\leq n$, can be written as $a_{ij}=\tau ...(U_j^*U_i)$ for some $n$-tuple $U_1, U_2, …, U_n$ of unitaries in a finite-dimensional von Neumann algebra $\mathcal {M}$ with tracial state τ. We show that $\mathcal {F}_{\mathrm {fin}}(n)$ is not closed for every $n\geq 8$. This improves a result by Musat and R{ø}rdam which states the same for $n\geq 11$.
We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan ⁎-isomorphisms. In particular, we prove ...that two von Neumann algebras without type I1 direct summands are Jordan ⁎-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Gehér and P. Šemrl, which is a generalization of Wigner's theorem.
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 isomorphisms 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 maximal local order isomorphisms. Moreover, in the finite-dimensional case, we prove that every order embedding of a matrix domain is a homeomorphic order isomorphism onto another matrix domain.
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