Article
Ufa Mathematical Journal
Volume 15, Number 1, pp. 34-42
Partial orders on ∗-regular rings
Kudaybergenov К.К., Nurjanov B.O.
DOI:10.13108/2023-15-1-34
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In this work we consider some new partial orders on ∗-regular rings. Let A be a ∗-regular ring, P(A) be the lattice of all projectors in A and μ be a sharp normal normalized measure on P(A). Suppose that (A,ρ) is a complete metric ∗-ring with respect to the rank metric ρ on A defined as ρ(x,y)=μ(l(x−y))=μ(r(x−y)), x,y∈A,
where l(a), r(a) is respectively the left and right support of an element a. On A we define the following three partial orders:
a≺sb⟺b=a+c, a⊥c; a≺lb⟺l(a)b=a;
a≺rb⟺br(a)=a, a⊥c means algebraic orthogonality, that is,
ac=ca=a∗c=ac∗=0. We prove that the order topologies associated with these partial orders are stronger than the topology generated by the metric ρ. We consider the restrictions of these partial orders on the subsets of projectors, unitary operators and partial isometries of ∗-regular algebra A. In particular, we show that these three orders coincide with the usual order ≤ on the lattice of the projectors of ∗-regular algebra. We also show that the ring isomorphisms of ∗-regular rings preserve partial orders
≺l and ≺r.