因子von Neumann代数正锥上的凸序列积自同构

Convex Sequential Product Automorphisms on the Positive Cone of a Factor von Neumann Algebra

设H是复Hilbert空间,M是H上维数大于1的因子von Neumann代数,M+是M的正锥.设λ∈[0,1],定义Ao_λ=λA1/2BA1/2+(1-λ)B1/2AB1/2,?A,B∈M+,称o_λ为M+上的凸序列积.本文证明了M+上的凸序列积自同构是由M的一个*-同构或*-反同构实现.

Let M be a factor von Neumann algebra on a complex Hilbert space H with dim M>1 and M_+the positive cone of M.We consider automorphisms of M_+with respect to convex sequential product o_λon M_+for someλ∈[0,1]defined by A o_λB=λA~(1/2)BA~(1/2)+(1-λ)B~(1/2)AB~(1/2)for any A,B∈M_+.We show that an automorphism of M_+with respect to convex sequential product is implemented by a*-isomorphism or an anti-*-isomorphism of M.