Chi-Wai Leung

SEPARATING LINEAR MAPS OF CONTINUOUS FIELDS OF BANACH SPACES

In this paper, we give a complete description of the structure of separating linear maps between continuous fields of Banach spaces. Some automatic continuity results are obtained.

Property (T) for non-unital C∗-algebras

Abstract Inspired by the recent work of Bekka, we study two reasonable analogues of property ( T ) for not necessarily unital C ∗ -algebras. The stronger one of the two is called “property ( T ) ” and the weaker one is called “property ( T e ) .” It is shown that all non-unital C ∗ -algebras do not have property ( T ) (neither do their unitalizations). Moreover, all non-unital σ -unital C ∗ -algebras do not have property ( T e ) .

Transition probabilities of normal states determine the Jordan structure of a quantum system

Let Φ : 𝔖(M1) → 𝔖(M2) be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras M1 and M2, respectively. This paper concerns with the situation when Φ preserves (or partially preserves) one of the following three notions of “transition probability” on the normal state spaces: the transition probability PU introduced by Uhlmann [Rep. Math. Phys. 9, 273-279 (1976)], the transition probability PR introduced by Raggio [Lett. Math. Phys. 6, 233-236 (1982)], and an “asymmetric transition probability” P0 (as introduced in this article). It is shown that the two systems are isomorphic, i.e., M1 and M2 are Jordan ∗-isomorphic, if Φ preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states μ and ν, we have PΦ(μ),Φ(ν)=0 if and only if P(μ, ν) = 0, where P stands for PU (respectively, PR or P0). Furthermore, as an extension of Wi...

On a notion of closeness of groups

Enlightened by the notion of perturbation of C∗-algebras, we introduce and study briefly in this article, a notion of closeness of groups. We show that if two groups are “close enough” to each other, and one of them has the property that the orders of its elements have a uniform finite upper bound, then the two groups are isomorphic (but in general they are not). We also study groups that are close to abelian groups, as well as an equivalence relation induced by closeness.