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Journal ArticleDOI

Star, left-star, and right-star partial orders in Rickart ∗-rings

01 Feb 2015-Linear & Multilinear Algebra (Taylor & Francis)-Vol. 63, Iss: 2, pp 343-365
TL;DR: In this paper, the star, the left-star and the right-star partial orders of a ring admitting involution are studied and conditions under which these orders are equivalent to the minus partial order are obtained.
Abstract: Let be a unital ring admitting involution. We introduce an order on and show that in the case when is a Rickart -ring, this order is equivalent to the well-known star partial order. The notion of the left-star and the right-star partial orders is extended to Rickart -rings. Properties of the star, the left-star and the right-star partial orders are studied in Rickart -rings and some known results are generalized. We found matrix forms of elements and when , where is one of these orders. Conditions under which these orders are equivalent to the minus partial order are obtained. The diamond partial order is also investigated.

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Citations
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Journal ArticleDOI
TL;DR: In this article, the generalized concept of order relations in Rickart rings and rings which was proposed by Semrl and which covers the star partial order, the left-star partial order (LPO), the right-star LPO, and the minus partial order was considered.
Abstract: We consider the generalized concept of order relations in Rickart rings and Rickart -rings which was proposed by Semrl and which covers the star partial order, the left-star partial order, the right-star partial order and the minus partial order. We show that on Rickart rings the definitions of orders introduced by Jones and Semrl are equivalent. We also connect the generalized concept of order relations with the sharp order and prove that the sharp order is a partial order on the subset of elements in a ring with identity which have the group inverse. Properties of the sharp partial order in are studied and some known results are generalized.

17 citations


Cites background from "Star, left-star, and right-star par..."

  • ...In [15], authors also generalized the left-star and the right-star partial orders from B(H) to Rickart ∗-rings....

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  • ...In [15], equivalent definitions of the star partial order ≤ ∗ on A...

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Journal ArticleDOI
TL;DR: In this paper, a generalization of a formula by Fill and Fishkind, regarding the Moore-Penrose inverse of the sum of two matrices, in the setting of arbitrary Hilbert spaces, is presented.
Abstract: Recently, in (Arias ML, Corach G, Maestripieri A. Range additivity, shorted operator and the Sherman–Morrison–Woodburry formula, Linear Algebra Appl. 2015;467), authors gave a generalization of a formula by Fill and Fishkind, regarding the Moore–Penrose inverse of the sum of two matrices, in the setting of arbitrary Hilbert spaces. We consider this formula under some weaker assumptions and derive certain conclusions generalizing the mentioned result. We also extend a formula connecting the infimum of two orthogonal projections and their parallel sum to a formula connecting the star-infimum and the parallel sum of operators which need not be positive, using the concept of parallel sum that was introduced in (Antezana J, Corach G, Stojanoff D. Bilateral shorted operators and parallel sums. Linear Algebra Appl. 2006;414).

14 citations

Journal ArticleDOI
TL;DR: In this article, the star order in a star-ordered Rickart ∗-ring has been investigated, and it is shown that every pair of elements bounded from above has a meet and also a join.
Abstract: Various authors have investigated properties of the star order (introduced by Drazin in 1978) on algebras of matrices and of bounded linear operators on a Hilbert space. Rickart involution rings (∗-rings) are a certain algebraic analogue of von Neumann algebras, which cover these particular algebras. In 1983, Janowitz proved, in particular, that, in a star-ordered Rickart ∗-ring, every pair of elements bounded from above has a meet and also a join. However, the latter conclusion seems to be based on some wrong assumption. We show that the conclusion is nevertheless correct, and provide equational descriptions of joins and meets for this case. We also present various general properties of the star order in Rickart∗-rings, give several necessary and sufficient conditions (again, equational) for a pair of elements to have a least upper bound of a special kind, and discuss the question when a star-ordered Rickart ∗-ring is a lower semilattice.

13 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the results of Hartwig and Drazin (1982) and Janowitz (1983) require much simpler conditions, such as necessary and sufficient conditions for the existence of star supremum.

11 citations

References
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Journal ArticleDOI
TL;DR: In this paper, it was shown that the proper *-semigroup axioms allow the simultaneous development of a surprisingly rich common theory of these special cases of groups and inverse semigroups.
Abstract: Let S be any semigroup, i.e. a set furnished with an associative binary operation, denoted by juxtaposition. An involution on S will mean a bijection u —• u* of S onto itself, satisfying (a*)* = a, (ab)* = b*a*. Such an involution is called proper (cf. [3, p. 74] ) iff a*a = a*b = b*a = b*b implies a = b. A proper ̂ semigroup will mean a pair (5, *) where 5 is a semigroup and * is a specified proper involution of S; in practice, we write S as an abbreviation for (S, *). From now on, 5 will denote an arbitrary proper *-semigroup. Obvious natural special cases are (i) all proper *-rings, with \"properness\" (Herstein [4, p. 794] prefers to say \"positive deflniteness\") as customarily defined (cf. [5, p. 31], [1, p. 10]) via u*u = 0 implying u = 0 (in particular, with the obvious choices for *, all commutative rings with no nonzero nilpotent elements, all Boolean rings, the ring B(S) of all bounded linear operators on any complex Hubert space H, and the ring Mn(C) of all n x n complex matrices), and, only slightly less trivially, (ii) all inverse semigroups (in particular, all groups). We make a start here towards showing that the proper *-semigroup axioms allow the simultaneous development of a surprisingly rich common theory of these special cases. While there is clearly little likelihood of learning anything new about groups or Boolean rings by such an approach, none of the results about proper *-semigroups which we state below has previously been noted even for n x n matrices (still less for B(H) or for *-rings), and most provide new information also about inverse semigroups. We recall that an element a E S is called regular iff a E aSa, and *-regular iff there is an x E S with

199 citations


"Star, left-star, and right-star par..." refers background in this paper

  • ...It is known (see [4] or [12]) that in proper involutory rings, an element a is a ∗-regular element if and only if aa∗ and a∗a are both regular....

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  • ...For example, Drazin [4] defined the star partial order in the following way: A ≤ ∗ B when A∗ A = A∗ B and AA∗ = B A∗, (1) A, B ∈ B(H)....

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  • ...Since any Rickart ∗-ring is a proper involutory ring and since Drazin introduced the star partial order (1) in a broader sense of proper ∗-semigroups, the next theorem is a direct consequence of Theorem 1 ((i) ⇔ (iii))....

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  • ...For example, Drazin [4] defined the star partial order in the following way:...

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  • ...Drazin introduced in [4] the star partial order in a broader sense of proper ∗-semigroups....

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Journal ArticleDOI
01 Mar 1986
TL;DR: In this paper, it was shown that for any semigroup (S, ) the relation a < b iff a = xb = by, xa = a for some x,y E S1, is a partial order.
Abstract: A partial order on a semigroup (S, ) is called natural if it is defined by means of the multiplication of S. It is shown that for any semigroup (S, ) the relation a < b iff a = xb = by, xa = a for some x,y E S1, is a partial order. It coincides with the well-known natural partial order for regular semigroups defined by Hartwig [4] and Nambooripad [10]. Similar relations derived from the natural partial order on the regular semigroup (Tx, o) of all maps on the set X are investigated.

156 citations


"Star, left-star, and right-star par..." refers background in this paper

  • ...Since bb†a = ab†b = ab†a = a, we may conclude from Lemma 1 in [17] (compare items (iv) and (v)) that there exists x ∈ A such that axa = a, xax = x , xa = xb and ax = bx ....

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Journal ArticleDOI
TL;DR: In this article, the authors considered the problems of inheriting certain properties under a given ordering, preserving an ordering under some matrix multiplications, relationships between an ordering among direct (or Kronecker) and Hadamard products and the corresponding orderings between the factors involved, and ordering between generalized inverses of a given matrix.

78 citations


"Star, left-star, and right-star par..." refers result in this paper

  • ...Many of these results are motivated by the corresponding results for complex matrices.[7,8] Even in the case of matrices, the proofs are not elementary....

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Journal ArticleDOI
TL;DR: The main result of as mentioned in this paper is that Drazin's "star" partial ordering holds if and only if A ∠ B and B −A −A =(B−A)

76 citations


"Star, left-star, and right-star par..." refers result in this paper

  • ...Many of these results are motivated by the corresponding results for complex matrices.[7,8] Even in the case of matrices, the proofs are not elementary....

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  • ...The result is, in some way, analogous to Theorem 1 in [7] where the case of complex matrices is considered....

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Journal ArticleDOI
TL;DR: In this paper, two partial orderings in the set of complex matrices are introduced by combining each of the conditions A*A = A*B and AA* = BA*, which define the star partial ordering.

75 citations


"Star, left-star, and right-star par..." refers methods in this paper

  • ...Inspired by a paper of Baksalary and Mitra [9], Liu, Benítez and Zhong generalized in [13] the left-star and the right-star partial orders from the set of all complex m × n matrices to a C† in the following way....

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  • ...Baksalary and Mitra proved that for A, B ∈ Mm,n , A ≤∗ B implies A ∗≤ B and A ≤∗ B, and A ∗≤ B or A ≤∗ B implies A −≤ B....

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  • ...The left-star and the right-star partial orders were introduced by Baksalary and Mitra in [9] in the following way....

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