Left-star and right-star partial orderings

Some results on partial ordering and reverse order law of elements of C∗-algebras

“…Inspired by a paper of Baksalary and Mitra [1], we define left-star and right-star partial ordering of Moore-Penrose invertible elements a, b of a C * -algebra by a * b ⇐⇒ a * a = a * b and b π r a = 0, and a * b ⇐⇒ aa * = ba * and ab π l = 0, Furthermore, we will consider the minus ordering defined in [16]. An extension to A † of an equivalent form of this ordering (see [18] or [9]) is the following:…”

“…Obviously, when a ∈ A EP , then a π l = a π r . Matrix partial orderings have been an area of intense research in the past few years (see [1][2][3][4]). Analogously to the definition introduced by Drazin [11], we define the star ordering in an arbitrary C * -algebra by a * b ⇐⇒ a * a = a * b and aa * = ba * .…”

On partial orders in Rickart rings

“…Namely, A ≤ * B if and only if there exist self-adjoint idempotent operators P, Q ∈ B(H ) such that Im P = Im A, Ker A = Ker Q, P A = P B and AQ = B Q. Two orders which are related to the minus and the star partial orders, named the left-star partial order and the right-star partial order, were introduced by Baksalary and Mitra [11] on the set of m × n complex matrices M m,n . The definition of the left-star partial order is as follows.…”

Some results on matrix partial orderings and reverse order law

“…In 1991, Baksalary and Mitra [5] defined the left-star and right-star orderings characterized as respectively.…”