Star order automorphisms on the poset of type 1 operators

[摘要] Let H be a complex infinite dimensional Hilbert space and B(H) the algebra of all bounded linear operators on H. The star partial order is defined by A <=* B if and only if A*A = A* B and AA* = AB* for any A and B in B(H). We give a type decomposition of operators with respect to star order. For any A is an element of B(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 under star order such that A(i) <=* 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 B(H). (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-15

[关键词] Star order;Type decomposition;Automorphism