Star order automorphisms on the poset of type 1 operators

Abstract

Let $\mathcal{H}$ be a complex infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear operators on $\mathcal{H}$. The star partial order is defined by $A\stackrel{\ast }{\le }B$ if and only if $A^{\ast }A=A^{\ast }B$ and $A{A}^{\ast }=A{B}^{\ast }$ for any $A$ and $B$ in $\mathcal{B}(\mathcal{H})$. We give a type decomposition of operators with respect to star order. For any $A\in \mathcal{B}(\mathcal{H})$, there are unique type 1 operator $A_1$ which is $0$ or the supremum of those rank 1 operators less than $A_1$ and type 2 operator $A_2$ which is not greater than any rank 1 operator in star order such that $A_i\stackrel{\ast }{\le }A$$(i=1,2)$ and $A=A_1+A_2$. Moreover, we determine all automorphisms on the poset of type 1 operators. As a consequence, we characterize continuous automorphisms on $\mathcal{B}(\mathcal{H})$.