On some automorphisms of the set of effects on Hilbert space

Local automorphisms of operator algebras on Banach spaces

In this paper we extend a result of Semrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms of matrix algebras of which statement we present a short proof. The need to get such a proof was formulated in Semrl's paper.

Coexistency on Hilbert space effect algebras and a characterisation of its symmetry transformations

The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig's formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper's first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the…

Some Calkin algebras have outer automorphisms

We consider various quotients of the C*-algebra of bounded operators on a nonseparable Hilbert space, and prove in some cases that, consistently, there are many outer automorphisms.

All automorphisms of all Calkin algebras

The Proper Forcing Axiom implies all automorphisms of every Calkin algebra associated with an infinite-dimensional complex Hilbert space and the ideal of compact operators are inner. As a means of the proof we introduce the notion of Polish $ω_1$-trees and cohrerent families of Polish spaces and prove some uniformization results.

Hilbert von Neumann modules

We introduce a way of regarding Hilbert von Neumann modules as spaces of operators between Hilbert space, not unlike [Skei], but in an apparently much simpler manner and involving far less machinery. We verify that our definition is equivalent to that of [Skei], by verifying the `Riesz lemma' or what is called `self-duality' in [Skei]. An advantage with our approach is that we can totally side-step the need to go through $C^*$-modules and avoid the two stages of completion - first in norm, then in the strong operator topology - involved in the former approach. We establish the analogue of the …

Some automorphism invariance properties for multicontractions

In the theory of row contractions on a Hilbert space, as initiated by Popescu, two important objects are the Poisson kernel and the characteristic function. We determine their behaviour with respect to the action of the group of unitarily implemented automorphisms of the algebra generated by creation operators on the Fock space. The case of noncommutative varieties, introduced recently by Popescu, is also discussed.

Characteristic functions for multicontractions and automorphisms of the unit ball

A \emph{multicontraction} on a Hilbert space $\HH$ is an $n$-tuple of operators $T=(T_1,...,T_n)$ acting on $\HH$, such that $\sum_{i=1}^n T_i T_i^*\le \1_\HH$. We obtain some results related to the characteristic function of a commuting multicontraction, most notably discussing its behaviour with respect to the action of the analytic automorphisms of the unit ball.

On certain automorphisms of sets of partial isometries

Under the mild condition of continuity at a single point we describe all the bijections of the set of all partial isometries on a Hilbert space which preserve the order and the orthogonality in both directions. Moreover, we present a natural analogue of Wigner's theorem on quantum mechanical symmetries for the set of all rank-1 partial isometries.

等变 Tannaka-Krein 重构与离散结构的量子自同构群

Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures

本文通过推广 Tannaka-Krein 重构定理，为一大类离散结构构造量子自同构群，并给出从连通局部有限离散结构得到相应范畴的方法，以量子 Cayley 图为例加以说明。

What type of dynamics arise in E_0-dilations of commuting quantum Markov process?

Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $ϕ$ and $θ$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $α$ and $β$ such that $ϕ_s \circ θ_t (P_H A P_H) = P_H α_s \circ β_t (A) P_H$ for all s,t and all A in B(K). In this note we prove that if $ϕ$ is not an automorphism semigroup then $α$ is cocycle conjugate to the minimal *-endomorphic dilation of $ϕ$, and that if $ϕ$ is an automorphism semigroup then $α$ is also an automorphism semigroup. In particular, we conclude that if $ϕ$ is not an automorphism semigroup a…

Bilocal *-automorphisms of B(H) satisfying the 3-local property

We prove that, for a complex Hilbert space $H$ with dimension bigger or equal than three, every linear mapping $T: B(H)\to B(H)$ satisfying the 3-local property is a $^*$-monomorphism, that is, every linear mapping $T: B(H) \to B(H)$ satisfying that for every $a$ in $B(H)$ and every $ξ,η$ in $H$, there exists a $^*$-automorphism $π_{a,ξ,η}: B(H)\to B(H)$, depending on $a$, $ξ$, and $η$, such that $$T(a) (ξ) = π_{a,ξ,η} (a) (ξ), \hbox{ and } T(a) (η) = π_{a,ξ,η} (a) (η),$$ is a $^*$-monomorphism. This solves a question posed by L. Molnár in [\emph{Arch. Math.} \textbf{102}, 83-89 (2014)].

Topological entropy of some automorphisms of reduced amalgamated free product C*-algebras

Certain classes of automorphisms of recued amalgamated free products of C*-algebras are shown to have Brown-Voiculescu topological entropy zero. Also, for automorphisms of exact C*-algebras, the Connes-Narnhofer-Thirring entropy is shown to be bounded above by the Brown-Voiculescu entropy. These facts are applied to generalize Stormer's result about entropy of automorphisms of the II_1-factor of a free group.

Some characterizations of the automorphisms of $B(H)$ and $C(X)$

We present some nonlinear characterizations of the automorphisms of the operator algebra $B(H)$ and the function algebra $C(X)$ by means of their spectrum preserving properties.

Unique ergodicity of free shifts and some other automorphisms of C*-algebras

A notion of unique ergodicity relative to the fixed-point subalgebra is defined for automorphisms of unital C*-algebras. It is proved that the free shift on any reduced amalgamated free product C*-algebra is uniquely ergodic relative to its fixed-point subalgebra, as are autormorphisms of reduced group C*-algebras arising from certain automorphisms of groups. A generalization of Haagerup's inequality, yielding bounds on the norms of certain elements in reduced amalgamated free product C*-algebras, is proved.

A reflexivity problem concerning the $C^*$-algebra $C(X)\otimes B(H)$

Let X be a compact Hausdorff space and let H be a separable Hilbert space. We prove that the group of all order automorphisms of the $C^*$-algebra $C(X)\otimes B(H)$ is algebraically reflexive.