Generalized Jordan Semitriple Maps on Hilbert Space Effect Algebras

Quan Yuan ,Kai He

doi:10.1155/2014/216713

Abstract

Letℰ(H)be the Hilbert space effect algebra on a Hilbert spaceHwithdim⁡H≥3,α,βtwo positive numbers with2α+β≠1andΦ:ℰ(H)→ℰ(H)a bijective map. We show that ifΦ(AαBβAα)=Φ(A)αΦ(B)βΦ(A)αholds for allA,B∈ℰ(H), then there exists a unitary or an antiunitary operatorUonHsuch thatΦ(A)=UAU*for everyA∈ℰ(H).

Highlights

Introduction and NotationsLet H be a Hilbert space
Let H be a Hilbert space with dim H ≥ 3 and Φ : E(H) → E(H) a bijective map. It seems that [5] is the first paper discussing the problem of characterizing the Jordan semitriple maps on Hilbert effect algebras
We prove that all bijective generalized Jordan semitriple maps on Hilbert space effect algebras, that is, the maps satisfying (4), are implemented by a unitary or an antiunitary operator on H

Summary

Introduction and Notations

Let H be a Hilbert space with dim H ≥ 3 and Φ : E(H) → E(H) a bijective map. It seems that [5] is the first paper discussing the problem of characterizing the Jordan semitriple maps on Hilbert effect algebras. Let H be a Hilbert space with dim H ≥ 3, α, β two positive real numbers with 2α + β ≠ 1, and Φ : E(H) → E(H) a bijective map satisfying. We prove that all bijective generalized Jordan semitriple maps on Hilbert space effect algebras, that is, the maps satisfying (4), are implemented by a unitary or an antiunitary operator on H

Orthoadditivity of Generalized Jordan Semitriple Maps

Characterization of Generalized Jordan Semitriple Maps