| Let H be a complex Hilbert space,effect algebra E（H）is a positive bounded linear contraction on a Hilbert space H.We say oλ is an convex sequential product on E（H）for some λ ∈[0,1]defined by Aολ B=λA1/2BA1/2+（1-λ）B1/2AB1/2,VA,B∈E（H）.In this paper,we study the algebraic properties of convex sequential product on E（H）.We characterize the automorphism of the effect algebra with respect to convex sequential product,and further research the structure of the mapping that preserving the norm of the convex sequential product.The main results of this paper are as follows:In the first part,we use Uhlhorn’s theorem to characterize the automorphism of convex sequential product on complex Hilbert space effect algebras whose dimension is less than 3.Let cp be a bijection on E（H）,if φ（Aολ B）=φ（A）ολφ（B）for any A,B ∈ E（H）,then there is a unitary or an anti-unitary U such that φ（A）=UAU*for any A ∈ E（H）.In the second part,we discuss the mapping φ of preserving norms with respect to convex sequential product on complex Hilbert space effect algebras whose dimension is less than 2,that is,cp satisfies the property that ‖φ（A）ολφ（B）=‖AολB‖for any A,B∈ E（H）.Then we discuss the basic properties of this mapping and prove the continuous bijection of preserving the norm of convex sequential product is automorphism,that is,there exists a unitary or an anti-unitary U,such thatφ（A）=UAU*for any A ∈ E（H）. |
