ON WEIGHTED REVERSE ORDER LAWS FOR 4 THE MOORE-PENROSE INVERSE AND 1 0 K 2 -INVERSES n a J Enrico Boasso, Dragana S. Cvetkovi´c-Ili´c and Robin Harte 0 3 ] A Abstract R The main objective of this article is to study several generalizations of the reverse . h order law for the Moore-Penrose inverse in ring with involution. t a Keywords: Moore-Penrose inverse, K-inverse, reverse order law, ring with involution, m prime ring. [ 2000 Mathematics Subject Classification: 15A09. 1 v 7 1. Introduction 9 7 7 Given a complex matrix a, the Moore-Penrose inverse of a is the unique complex matrix . 1 b satisfying the following Penrose equations (Penrose (1955)): 0 4 1 (1)a = aba, (2)b = bab, (3)(ab)∗ = ab, (4)(ba)∗ = ba. : v This generalization of the inverse of a non-singular square matrix was first introduced by E. i X H. Moore, but remained unknown mainly because of Moore’s special notation (see Moore r (1920)). The equations (1)-(4) were formulated by Penrose, and they characterize the same a object considered by E. H. Moore. T. N. Greville first characterized when the product of two complex matrices a and b satisfies the so-called reverse order law for the Moore-Penrose inverse, that is when (ab)† = b†a†, where c† denotes the Moore-Penrose of a complex matrix c (Greville (1966)); note that the proofs in Greville (1966) remain valid for pairs a,b of Moore-Penrose invertible C∗-algebra elements whose product ab also has a Moore-Penrose inverse. In this context, in Boasso (2006) several other conditions equivalent to the reverse order law for the Moore-Penrose inverse were proved. In the framework of rings with involution, J. J. Koliha, D. S. Djordjevi´c and D. S. Cvetkovi´c extended the characterization in Greville (1966) under the additional assump- tion of the ∗-left cancellation property of a particular element of the ring (Koliha, Djordjevi´c, Cvetkovi´c (2007)). 1 BOASSO ET AL. 2 The first objective of the present article is to study the following generalization of the reverse order law for the Moore-Penrose inverse: given a ring with involution R, elements in R for which a, b and ab are Moore-Penrose invertible, and an element c∈ R which commutes with b and b∗, characterize when the following identity holds: (ab)† = cb†a†. This identity and others presented in section 2 will be called weighted reverse order laws for the Moore-Penrose inverse. Naturally, when c = e a characterization of the usual reverse order law is obtained. Fur- thermore, other similar generalizations of thereverse order law for the Moore-Penrose inverse in rings with involution and in complex algebras with involution will be also considered, see next section. Note that no additional assumption such as the ∗-cancellation property for elements of the ring is needed. On the other hand, given a C∗-algebra A, an element a ∈ A and a subsetK ⊆ {1,2,3,4}, an element x ∈ A is said to be a K-inverse of a, if x satisfies the Penrose equation (j) for each j ∈ K. Several reverse order laws for K-inverses of products of two C∗-algebra elements werecharacterized by D.S.Cvetkovi´c-Ili´c andR.E.Harte. Thesecond objective of thiswork is to extend some of the results in Cvetkovi´c-Ili´c, Harte (2011) to weighted reverse order laws in rings with involution, see section 3. Before going on, several definitions and some notation will be recalled. Let R bean associative ringwith unit element e. Thering R is said to bea prime ring, if whenever elements a and b ∈ R satisfy aRb = {0}, then 0 ∈ {a,b} (see McCoy (1949)). For example, given n ∈ N, the ring of square matrices Cn×n is prime, see Lemma 3 in Baksalary, Baksalary (2005). It is not difficult to prove that the same is true when A ⊆ L(X) is a subalgebra of the Banach algebra of all bounded operators defined on the Banach space X which contains the ideal of finite rank operators. In the case of general Banach algebras, prime, ultraprimeand spectrally primealgebras were considered in Harte, Hern´andez (1998). An element a ∈ R is said to be group invertible if there exists b ∈ R such that aba = a, bab = b, ab = ba. It is well known that if a ∈ R is group invertible, then there is only one group inverse of a (Mosi´c, Djordjevi´c (2009)), which will be denoted by a♯. An involution ∗: R → R is an anti-isomorphism of degree 2, that is (a∗)∗ = a, (a+b)∗ = a∗+b∗, (ab)∗ = b∗a∗. Given R a ring with involution, an element a ∈ R is said to be Hermitian if a = a∗, and a is said to be Moore-Penrose invertible if there exists b ∈ R such that a and b satisfy the Penrose equations presented above. It is well known that given a ∈ R, there is at most one Moore-Penrose inverse of a, see Roch, Silbermann (1999). When the Moore-Penrose inverse of a ∈ R exists, it will be denoted,asbefore,bya†. Inaddition,R† willstandforthesetofallMoore-Penroseinvertible elements of a ∈ R. Note that if a ∈ R†, then aa† and a†a are hermitian idempotents. What is more, if a ∈ R†, then a† ∈ R† and (a†)† = a. Moreover, it is easy to prove that a ∈ R† if WEIGHTED REVERSE ORDER LAWS 3 and only if a∗ ∈ R†. Furthermore, in this case, (a∗)† = (a†)∗. In what follows (a†)∗ will be denoted by a†∗. Given a ∈ R and K ⊆ {1,2,3,4}, x ∈ R will be said to be a K-inverse of a, if x satisfies the same condition recalled above for C∗-algebra elements. The set of all K-inverses of a given a ∈ R will be denoted by aK. Finally, if p and q are idempotents in R, then an arbitrary x ∈ R can be represented as a 2×2 matrix over R; specifically x x x = 1 2 , (cid:20) x3 x4 (cid:21)p,q where x = pxq, x = px(e − q), x = (e − p)xq and x = (e − p)x(e − q). Note that 1 2 3 4 x = x +x +x +x . 1 2 3 4 2. Weighted reverse order laws for the Moore-Penrose inverse We begin by presenting an equivalent formulation for the Moore-Penrose inverse. Al- thoughitsproofisnotdifficult(Penrose(1955)), itwillbeusedbelow,andhencewereproduce it here. Proposition 2.1. Let R be a ring with involution and consider a ∈ R. Then, the following statements are equivalent: (i) b ∈ R is the Moore-Penrose inverse of a; (ii) a = aa∗b∗ and b∗ = abb∗. Proof. If b = a† then, since (ab)∗ = ab, a = aba= a(ba) = a(ba)∗ = aa∗b∗, b∗ = (bab)∗ = (ab)∗b∗ = abb∗. On the other hand, if statement (ii) holds, then ba = baa∗b∗ = ba(ba)∗, (ab)∗ = b∗a∗ = abb∗a∗ = ab(ab)∗, equivalently, ab and ba are hermitian idempotents. However, according to statement (ii), a = a(ba)∗ = aba, b = b(ab)∗ = bab. The following proposition will extend to rings with involution a well known result con- cerning the Moore-Penrose inverse of C∗-algebra elements, see Theorem 7 in Harte, Mbekhta (1992). Proposition 2.2. Let R be a ring with involution and consider a ∈ R† and c ∈ R. Necessary and sufficient condition for c to commute with a and a∗ is that c commutes with a† and a†∗. BOASSO ET AL. 4 Proof. Let a ∈R†. Then, according to Theorem 5.3 in Koliha, Patr´ıcio (2002), (a∗a)♯ exists. Moreover, a† = (a∗a)♯a∗. If c commutes with a and a∗, then c commutes with a∗a. In addition, since a∗a is group invertible, c commutes with (a∗a)♯, see Mosi´c, Djordjevi´c (2009). Therefore c commutes with a† = (a∗a)♯a∗. In addition, since a∗ ∈ R† and (a∗)∗ = a, according to what has been proved, c commutes with (a∗)† = a†∗. On the other hand, if c commutes with a† and a†∗, then since (a†)† = a and (a†)†∗ = a∗, c commutes with a and a∗. Let R be a ring with involution and consider a,b ∈R†. Define p = bb†, q = a†a†∗, r = bb∗, s = a†a. Clearly p, q, r and s are hermitian elements. Moreover, according to Proposition 2.1, a = as, a†∗ = aq, b = rb†∗, b†∗ = pb†∗. Note that p, q, r and s are blanket notations for this section. In the following theorems several weighted reverse order laws for the Moore-Penrose in- verse will be presented. Note that when c = e, then a characterization of the usual reverse order law in rings with involution is obtained. Theorem 2.3. Let R be a ring with involution. Consider a,b ∈ R† such that ab ∈ R†, and c ∈R such that c commutes with b and b∗. Then, the following statements are equivalent: (i) (ab)† = cb†a†; (ii) a(cpq−qp)b†∗c∗ = 0 and a(rsc∗−sr)b†∗ = 0; (iii) scpqpc∗ = qpc∗ and srspc∗ = sr. Proof. In first place, note that according to Proposition 2.1, a = aa∗a†∗, b = bb∗b†∗, ab= ab(ab)∗(ab)†∗, a†∗ = aa†a†∗, b†∗ = bb†b†∗, (ab)†∗ = ab(ab)†(ab)†∗. (i) ⇒ (ii). Suppose that (ab)† = cb†a†. Then, since (ab)∗ = b∗a∗ and (ab)†∗ = (cb†a†)∗ = a†∗b†∗c∗, ab= abb∗a∗a†∗b†∗c∗, a†∗b†∗c∗ = abcb†a†a†∗b†∗c∗, which, since c and b commute, can be written as asrb†∗ = arsb†∗c∗, aqpb†∗c∗ = acpqb†∗c∗. However, according to Proposition 2.2 these identities are equivalent to a(cpq−qp)b†∗c∗ = 0, a(rsc∗−sr)b†∗ = 0. WEIGHTED REVERSE ORDER LAWS 5 (ii) ⇒ (iii). If the second statement holds, then a†acpqb†∗c∗b∗ = a†aqpb†∗c∗b∗, a†arsc∗b†∗b∗ = a†asrb†∗b∗. However, since c commutes with b and b†, a†acpqb†∗b∗c∗ = a†aqpb†∗b∗c∗, a†arsb†∗b∗c∗ = a†asrb†∗b∗. What is more, according again to Proposition 2.1 and to the fact that s = s∗ and p = p∗, these equations can be rewritten as scpqpc∗ = a†(aa†a†∗)b(b†b†∗b∗)c∗ = a†a†∗bb†c∗ = qpc∗, srspc∗ = (a†aa∗)a†∗(bb∗b†∗)b∗ = a∗a†∗bb∗ = sr. (iii) ⇒ (i). Suppose that statement (iii) holds. Then, since p = p∗, s = s∗ and b and c commute, a†abcb†a†a†∗b†∗b∗c∗ = a†a†∗bb†c∗, a†abb∗a∗a†∗b†∗b∗c∗ = a∗a†∗bb∗. Moreover, since c and b† commute, (aa†a)bcb†a†a†∗(b†∗b∗b†∗)c∗ = (aa†a†∗)(bb†b†∗)c∗, (aa†a)bb∗a∗a†∗(b†∗b∗b†∗)c∗ = (aa∗a†∗)(bb∗b†∗). However, according to Proposition 2.1, these equations are equivalent to ab(cb†a†)(cb†a†)∗ = (cb†a†)∗, ab(ab)∗(cb†a†)∗ = ab. Therefore, (ab)† = cb†a†. As an application of Theorem 2.3, other generalizations of the reverse order law can be characterized. Theorem 2.4. Let R be a ring with involution. Consider a,b ∈ R† such that ab ∈ R†, and c ∈R such that c commutes with a and a∗. Then, the following statements are equivalent: (i) (ab)† = b†a†c; (ii) b∗(c∗sr†−r†s)a†c= 0 and b∗(q†pc−pq†)a† = 0; (iii) pc∗sr†sc= r†sc and pq†psc= pq†. BOASSO ET AL. 6 Proof. Recallthatgiven h∈ R, necessaryandsufficientforhtobelongtoR† isthath∗ ∈ R†, (see Theorem 5.4 in Koliha, Patr´ıcio (2002)). Moreover, in this case (h∗)† = (h†)∗. It is not difficult to prove that the identity (ab)† = b†a†c is equivalent to (b∗a∗)† = c∗(a∗)†(b∗)†. On the other hand, denote by p , q , r and s the elements of R corresponding to p, q, 1 1 1 1 r and s defined using b∗a∗ instead of ab. Then, it is easy to prove that p = s, s = p. 1 1 In addition, according to the proof of Theorem 5.3 in Koliha, Patr´ıcio (2002), q = r†, r = q†. 1 1 To conclude the proof, apply Theorem 2.3 to b∗, a∗, b∗a∗ and c∗ in place of a, b, ab and c. Theorem 2.5. Let R be a ring with involution. Consider a,b ∈ R† and c ∈ R such that cab ∈ R†. Then, if c commutes with a and a∗, the following statements are equivalent: (i) (cab)† = b†a†; (ii) b†(csr−rs)a∗c∗ = 0 and b†(qpc∗−pq)a∗ = 0; (iii) pcsrsc∗ = rsc∗ and pqpsc∗ = pq. Proof. Note that cab∈ R† and (cab)† =b†a† if and only if b†a† ∈ R† and (b†a†)† = cab. As in the proof of Theorem 2.4, p , q , r and s denote the elements of R corresponding 2 2 2 2 to p, q, r and s defined using b†a† instead of ab. Then, it is easy to prove that p = s, q = r, r = q, s = p. 2 2 2 2 To conclude the proof, apply Theorem 2.3 to b†, a†, b†a† and c in place of a, b, ab and c. Theorem 2.6. Let R be a ring with involution. Consider a,b ∈ R† and c ∈ R such that abc ∈ R†. Then, if c commutes with b and b∗, the following statements are equivalent: (i) (abc)† = b†a†; (ii) a†∗(c∗pq†−q†p)bc = 0 and a†∗(r†sc−sr†)b = 0; (iii) sc∗pq†pc= q†pc and sr†spc= sr†. Proof. It is easy to prove that the first statement is equivalent to (a†∗b†∗)† = c∗b∗a∗. As in Theorem 2.4 and Theorem 2.5, denote by p , q , r and s the elements of R 3 3 3 3 corresponding to p, q, r and s defined using a†∗b†∗ instead of ab. Then, using the proof of Theorem 5.3 in Koliha, Patr´ıcio (2002), we prove that p = p, q = q†, r = r†, s = s. 3 3 3 3 To conclude the proof, apply Theorem 2.3 to a†∗, b†∗, a†∗b†∗ and c∗ in place of a, b, ab and c. WEIGHTED REVERSE ORDER LAWS 7 Specializing to the case of an algebra with involution over the complex numbers C (R = A), λ ∈ C will stand for the complex conjugate of λ ∈ C. Note that (λa)∗ = λa∗ for any a in the algebra. In particular we can now allow the element c ∈ A be a scalar multiple of the identity, i.e., c = λe. Corollary 2.7. Let A be an algebra with involution over C. Consider a,b ∈ A† such that ab∈ A†, and λ ∈C. Then, the following statements are equivalent: (i) (ab)† = λb†a†; (ii) a(λpq−qp)b†∗ = 0 and a(rsλ−sr)b†∗ =0; (iii) λspqp = qp and λsrsp= sr. Proof. Apply Theorem 2.3. Remark 2.8. Let A be a C∗-algebra and consider a,b ∈ A† such that ab ∈ A†. Let p, q, r and s be the elements of A defined before Theorem 2.3. Recall that, according to Remark 3.5 in Boasso (2006) or Greville (1966), (ab)† = b†a† if and only if rs= sr and pq = qp. Note that according to Theorem 7 in Harte, Mbekhta (1992), p and q commute (respectively r and s commute) if and only if p and q† commute (respectively s and r† commute). What is more, these statements are equivalent to the at first sight weaker conditions of Theorems 3.1-3.4 in Boasso (2006). WhenRaringwithinvolution, a,b ∈ R† and(e−a†a)bisleft∗-cancellable, necessaryand sufficient for ab to belong to R† and (ab)† = b†a† is that rs = sr and pq† = q†p (see Theorem 3 in Koliha, Djordjevi´c, Cvetkovi´c (2007)). Note also that according to Proposition 2.2, p and q commute (respectively r and s commute) if and only if p and q† commute (respectively s and r† commute). Therefore, considering c = e, the conditions presented in Theorem 2.3 are weaker than the ones in Theorem 3 in Koliha, Djordjevi´c, Cvetkovi´c (2007) and to prove them the cancellation property is not necessary. In particular, while all the aforementioned results are equivalent in C∗-algebras, in the case of rings with involution, according to the characterization of Theorem 2.3, if the reverse order law is satisfied by a and b, the identities rs= sr andpq† = q†pneednottobesatisfied. AccordingtoTheorem3inKoliha, Djordjevi´c, Cvetkovi´c (2007), these equalities are satisfied when the cancellation property is assumed. 3. Weighted reverse order laws for K-inverses in prime rings In this section, R will be a primering with involution and K ⊆ {1,2,3,4}. For a,b ∈ R, several weighted reverse order laws for K-inverses of ab will be characterized. First we will present some preliminary facts. Remark 3.1. Consider a,b ∈ R† and c ∈ R such that c commutes with a and a∗. Let b 0 a a p = bb†, q = b†b and r = aa†. We have that b = and a = 1 2 . An (cid:20) 0 0 (cid:21) (cid:20) 0 0 (cid:21) p,q r,p b† 0 arbitrary b(1,3) ∈ b{1,3} has the form b(1,3) = , for some u ∈ (e − q)Ap and (cid:20) u v (cid:21) q,p v ∈ (e−q)A(e−p), and an arbitrary a(1,3) has the form a(1,3) = a† +(e−a†a)x, for some x x z z x = 1 2 ∈ R, i.e., a(1,3) = 1 2 , where (cid:20) x3 x4 (cid:21)p,r (cid:20) z3 z4 (cid:21)p,r BOASSO ET AL. 8 z = a∗d†+(e−a∗d†a )x −a∗d†a x , 1 1 1 1 1 1 2 3 z = (e−a∗d†a )x −a∗d†a x , 2 1 1 2 1 2 4 (3.1) z = a∗d†−a∗d†a x +(e−a∗d†a )x , 3 2 2 1 1 2 2 3 z = −a∗d†a x +(e−a∗d†a )x . 4 2 1 2 2 2 4 a∗d† 0 Also, a† = a∗(aa∗)† = 1 (Theorem 5.3 in Koliha, Patr´ıcio (2002)) where (cid:20) a∗d† 0 (cid:21) 2 p,r d = aa∗ = a a∗+a a∗ and d† = (aa∗)†. 1 1 2 2 c 0 If c commutes with a and a∗, it follows that c= 1 . (cid:20) 0 c (cid:21) 2 r,r Theorem 3.2. Let R be a prime ring with involution. Consider a,b ∈ R† and c ∈ R such that c commutes with a and a∗. Then, following statements are equivalent: (i) b{1,3}·a{1,3}·c ⊆ (ab){1,3}; (ii) b†a†c ∈ab{1,3}, b†a† ∈ ab{1} and a† ∈ a(e−bb†){1}. Proof. Note that the case ab = 0 is trivial. Hence, we will consider the case ab6= 0. Clearly b{1,3}·a{1,3}·c ⊆ (ab){1,3} is equivalent to the fact that for any a(1,3) ∈ a{1,3} and any b(1,3) ∈ b{1,3}: b(1,3)a(1,3)c∈ (ab){1,3}. (3.2) Using the matrix representations considered in Remark 3.1, we have that (3.2) is equiva- lent to (i)a z c a = a , (ii)a z c = 0, (iii)(a z c )∗ = a z c , (3.3) 1 1 1 1 1 1 2 2 1 1 1 1 1 1 where z and z are defined by (3.1). Now, using (3.1), the identity (3.3)(i) is equivalent to 1 2 a a∗d†c a +(a −a a∗d†a )x c a −a a∗d†a x c a = a . 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 1 1 1 Since, x and x are arbitrary elements from appropriate subalgebras, (3.3)(i) is equivalent 1 3 to (i) a a∗d†c a = a , (ii) (a −a a∗d†a )x c a = 0, (iii) a a∗d†a x c a = 0. (3.4) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 1 1 What is more, since a = rap, x = pxr and c = rcr, (3.4)(ii) is equivalent to 1 1 1 (a −a a∗d†a )xc a = 0, 1 1 1 1 1 1 where x ∈ R is arbitrary. However, since R is a prime ring, a −a a∗d†a = 0 or c a = 0. 1 1 1 1 1 1 Similarly, from (3.4)(iii), we get that a a∗d†a = 0 or c a = 0. 1 1 2 1 1 Note that the case c a = 0 implies that ab= 0, which is not possible. 1 1 Therefore, c a 6= 0, and equation (3.3)(i) is equivalent to 1 1 (i)a a∗d†c a = a , (ii)a −a a∗d†a =0, (iii)a a∗d†a = 0, (3.5) 1 1 1 1 1 1 1 1 1 1 1 2 WEIGHTED REVERSE ORDER LAWS 9 i.e., (i)b†a†c ∈ ab{1}, (ii)b†a† ∈ ab{1}, (iii)a† ∈ a(e−bb†){1}. (3.6) Now, (3.5) imply that a z = 0 and the fact that a z c = a a∗d†c is hermitian is 1 2 1 1 1 1 1 1 equivalent to the fact that abb†a†c is hermitian, i.e. b†a†c ∈ ab{3}. From the proof of Theorem 3.2 it follows that under the assumption a = a(e−bb†) ∈ R†, 2 condition (3.5)(ii) implies condition (3.5)(iii): a −a a∗d†a = 0⇒ a −dd†a +a a∗d†a = 0 ⇒ a∗d†a = 0, 1 1 1 1 1 1 2 2 1 2 1 so we get the following corollary. Corollary 3.3. Let R be a prime ring with involution. Consider a,b ∈ R† such that a(e− bb†)∈ R† and let c ∈ R such that c commutes with a and a∗. Then, following statements are equivalent: (i) b{1,3}·a{1,3}·c ⊆ (ab){1,3}; (ii) b†a†c ∈ab{1,3}, b†a† ∈ ab{1}. In the following theorem, for given M ⊆ R, M∗ will stand for the set of all adjoint elements of M, i.e., M∗ = {x∗: x ∈ M}. Theorem 3.4. Let R be a prime ring with involution. Consider a,b ∈ R† and c ∈ R such that c commutes with b and b∗. Then, following statements are equivalent: (i) c·b{1,4}·a{1,4} ⊆ (ab){1,4}; (ii) cb†a† ∈ab{1,4}, b†a† ∈ ab{1} and b† ∈(e−a†a)b{1}. Proof. Note that for given x ∈ R, (x{1,4})∗ = x∗{1,3}. Therefore, the first statement is equivalent to a∗{1,3}·b∗{1,3}·c∗ ⊆ (b∗a∗){1,3}. Now apply Theorem 3.2. As in the case of Theorem 3.2, the following corollary can be deduced from Theorem 3.4. Corollary 3.5. Let R be a prime ring with involution. Consider a,b ∈ R† such that (e− a†a)b ∈ R† and let c ∈ R such that c commutes with b and b∗. Then, following statements are equivalent: (i) c·b{1,4}·a{1,4} ⊆ (ab){1,4}; (ii) cb†a† ∈ab{1,4}, b†a† ∈ ab{1} . Theorem 3.6. Let R be a prime ring with involution. Consider a,b ∈ R† and c ∈ R such that c commutes with a and a∗. Then, following statements are equivalent: (i) b{1,3}·a{1,3} ⊆ (cab){1,3}; (ii) b†a† ∈ (cab){1,3}, cab= cabb†a†ab and ca(e−bb†)a†a(e−bb†) = ca(e−bb†). BOASSO ET AL. 10 Proof. Using arguments similar to the ones in the proof of Theorem 3.2, it is not difficult to prove that the first statement of the theorem is equivalent to the following equations. (i)c a a∗d†c a = c a , (ii)c a a∗d† = (c a a∗d†)∗, 1 1 1 1 1 1 1 1 1 1 1 1 1 (iii)c a −c a a∗d†a = 0, (iv)c a a∗d†a = 0. 1 1 1 1 1 1 1 1 1 2 The first two equations are equivalent to b†a† ∈ (cab){1,3}, the third to cabb†a†ab = cab and the fourth to ca(e−bb†)a†a(e−bb†) = ca(e−bb†). Theorem 3.7. Let R be a prime ring with involution. Consider a,b ∈ R† and c ∈ R such that c commutes with b and b∗. Then, following statements are equivalent: (i) b{1,4}·a{1,4} ⊆ (abc){1,4}; (ii) b†a† ∈ (abc){1,4}, ab= abb†a†abc and (e−a†a)bc = (e−a†a)bb†(e−a†a)bc. Proof. As in Theorem 3.4, since given x ∈ R, (x{1,4})∗ = x∗{1,3}, the first statement is equivalent to a∗{1,3}·b∗{1,3} ⊆ (c∗b∗a∗){1,3}. Now apply Theorem 3.6. Next some characterizations of reverse order laws for K-inverses in C∗-algebras will be extended to the context of the present work. Theorem 3.8. Let R be a ring with involution. Consider a,b ∈ R† such that ab,abb†,a(e− bb†) ∈ R†. Let c ∈ R such that c commutes with a and a∗, cab = ab and c∗ab = ab. Then, the following statements are equivalent: (i) bb†a∗ab= a∗ab; (ii) b{1,3}·a{1,3}·c⊆ (ab){1,3}; (iii) b†a†c ∈ (ab){1,3}; (iv) b†a†c ∈(ab){1,2,3}. Proof. Undertheconditionsofthetheorem,usingthematrixrepresentationsgiveninRemark 3.1, it is not difficult to prove that b†a†c = b†a† and that necessary and sufficient condition for (ii) to holds is the fact that b{1,3}·a{1,3} ⊆ (ab){1,3}. In particular, it is enough to prove the equivalences among statements (i)-(iv) for the case c = e. Now, the proof of this case follows by Theorem 3.1 in Cvetkovi´c-Ili´c, Harte (2011), where the same conditions of statements (i)-(iv) were considered for a, b two C∗-algebra elements and c= e. However, for the sake of completeness the proof of the case c= e will be presented. We will show that (i) ⇒ (ii) ⇒ (iii) ⇒ (i) and then (i) ⇒ (iv) ⇒ (iii). Note that the notation of Remark 3.1 will be used. In particular, under the hypothesis of the theorem, a , 1 a ∈ R†. 2 (i) ⇒ (ii). Suppose that bb†a∗ab = a∗ab, which is equivalent to a∗a = 0, i.e., a∗a = 0. 2 1 1 2 For arbitrary a(1,3),b(1,3) we have that a z a b 0 abb(1,3)a(1,3)ab = 1 1 1 . (cid:20) 0 0 (cid:21) r,q