arXiv subjects
Publications and source records attributed to Michiya Mori.
We give a characterization of linear isometries with finite corank between two arbitrary unital C$^*$-algebras in terms of Jordan $^*$-homomorphisms and ``finite-dimensional remainders''. Also, the corresponding results are given for linear isometries and linear order embeddings between self-adjoint parts of C$^*$-algebras. In the finite-dimensional case, we give new examples of linear isometries between matrix algebras.
We study lightlikeness preserving mappings from the $4$-dimensional Minkowski spacetime $\mathcal{M}_4$ to itself under no additional regularity assumptions like continuity, surjectivity, or injectivity. We prove that such a mapping $ϕ$ satisfies one of the following three conditions. (1) The mapping $ϕ$ can be written as a composition of a Lorentz transformation, a multiplication by a positive scalar, and a translation. (2) There is an event $r\in \mathcal{M}_4$ such that $ϕ(\mathcal{M}_4\setminus\{r\})$ is contained in one light cone. (3) There is a lightlike line $\ell$ such that $ϕ(\mathcal{M}_4\setminus \ell)$ is contained in another lightlike line. Here, a line that is contained in some light cone in $\mathcal{M}_4$ is called a lightlike line. We also give several similar results on mappings defined on a certain subset of $\mathcal{M}_4$ or the compactification of $\mathcal{M}_4$.
In this note, we study the distance from an arbitrary nonzero projection $P$ to the set of nilpotents in a factor $\mathcal{M}$ equipped with a normal faithful tracial state $τ$. We prove that the distance equals $(2\cos \frac{τ(P)π}{1+2τ(P)})^{-1}$. This is new even in the case where $\mathcal{M}$ is the matrix algebra. The special case settles a conjecture posed by Z. Cramer.
We study surjective maps between the sets of all self-adjoint elements of unital $C^*$-algebras which satisfy the multiplicatively spectrum-preserving property. We show that such maps are characterized by Jordan isomorphisms and central symmetries. This is an answer to a problem posed by Molnár.
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 $a_{ij}=τ(U_j^*U_i)$ for some $n$-tuple $U_1, U_2, \ldots, U_n$ 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 Rørdam which states the same for $n\geq 11$.
Problem 155 of the Scottish Book asks whether every bijection $U\colon X\to Y$ between two Banach spaces $X, Y$ with the property that, each point of $X$ has a neighborhood on which $U$ is isometric, is globally isometric on $X$. We prove that this is true under the additional assumption that $X$ is separable and the weaker assumption of surjectivity instead of bijectivity.
We prove that the distance from an $n\times n$ complex matrix $M$ to the set of nilpotents is at least $\frac{1}{2}\sec\fracπ{n+2}$ if there is a nonzero projection $P$ such that $PMP=M$ and $M^*M\geq P$. 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.
Wigner's theorem characterizes isometries of the set of all rank one projections on a Hilbert space. In metric geometry nonexpansive maps and noncontractive maps are well studied generalizations of isometries. We show that under certain conditions Wigner symmetries can be characterized as nonexpansive or noncontractive maps on the set of all projections of rank one. The assumptions required for such characterizations are injectivity or surjectivity and they differ in the finite and the infinite-dimensional case. Motivated by a recently obtained optimal version of Uhlhorn's generalization of Wigner's theorem, we also give a description of nonexpansive maps which satisfy a condition that is much weaker than surjectivity. Such maps do not need to be Wigner symmetries. The optimality of all presented results is shown by counterexamples.
We study (von Neumann) regular $^*$-subalgebras of $B(H)$, which we call R$^*$-algebras. The class of R$^*$-algebras coincides with that of "E$^*$-algebras that are pre-C$^*$-algebras" in the sense of Z. Szűcs and B. Takács. We give examples, properties and questions of R$^*$-algebras. We observe that the class of unital commutative R$^*$-algebras has a canonical one-to-one correspondence with the class of Boolean algebras. This motivates the study of R$^*$-algebras as that of noncommutative Boolean algebras. We explain that seemingly unrelated topics of functional analysis, like AF C$^*$-algebras and incomplete inner product spaces, naturally arise in the investigation of R$^*$-algebras. We obtain a number of interesting results on R$^*$-algebras by applying various famous theorems in the literature.
Let $H$ be a Hilbert space and $P(H)$ be the projective space of all quantum pure states. Wigner's theorem states that every bijection $ϕ\colon P(H)\to P(H)$ that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator $U\colon H\to H$. Uhlhorn's theorem generalises this result for bijective maps $ϕ$ that are only assumed to preserve the quantum angle $\fracπ{2}$ (orthogonality) in both directions. Recently, two papers, written by Li--Plevnik--Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $α$ in both directions, provided that $0 < α\leq \fracπ{4}$ holds. In this paper we solve the remaining structural problem for quantum angles $α$ that satisfy $\fracπ{4} < α< \fracπ{2}$, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner's.
Generalizing von Neumann's result on type II$_1$ von Neumann algebras, we characterize lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, we give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.
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.
Let $H$ be a finite-dimensional Hilbert space, $\dim H \ge 2$. We prove that every continuous coexistency preserving map on the effect algebra $E(H)$ is either a standard automorphism of $E(H)$, or a standard automorphism of $E(H)$ composed with the orthocomplementation. We present examples showing the optimality of the result.
We prove that, if the closed unit ball of a normed space $X$ has sufficiently many extreme points, then every mapping $Φ$ 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 $Φ$ 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.
We prove that every unital C*-algebra $A$ has the Mazur--Ulam property. Namely, every surjective isometry from the unit sphere $S_A$ of $A$ onto the unit sphere $S_Y$ of another normed space $Y$ extends to a real linear map. This extends the result of A. M. Peralta and F. J. Fernandez-Polo who have proved the same under the additional assumption that both $A$ and $Y$ are von Neumann algebras. In the course of the proof, we strengthen Mankiewicz's theorem and prove that every surjective isometry from a closed unit ball with enough extreme points onto an arbitrary convex subset of a normed space is necessarily affine.
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 I$_1$ 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.