A projection–less approach to Rickart Jordan structures

Journal of Algebra 609 (2022) 567–605 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra A projection–less approach to Rickart Jordan structures Jorge J. Garcés a, Lei Li b, Antonio M. Peralta c,∗, Haifa M. Tahlawi d aDepartamento de Matemática Aplicada, ETSIDI, Universidad Poltécnica de Madrid, Madrid, Spain bSchool of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China cInstituto de Matemáticas de la Universidad de Granada (IMAG). Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain dDepartment of Mathematics, College of Science, King Saud University, P.O.Box 2455-5, Riyadh, 11451, Saudi Arabia a r t i c l e i n f o a b s t r a c t Article history: Received 16 February 2022 Available online 22 June 2022 Communicated by Alberto Elduque To the memory of Professor C.M. Edwards with admiration, affection, and respect MSC: primary 17C65, 16W10, 46L57 secondary 46L05, 46L60 Keywords: Rickart C∗-algebra, JB∗-algebra and JB∗-triple Baer C∗-algebra and JB∗-algebra Weakly order Rickart JB∗-triple Von Neumann regularity, Inner ideal The main goal of this paper is to introduce and explore an appropriate notion of weakly Rickart JB∗-triples. We introduce weakly and weakly order Rickart JB∗-triples, and we show that a C∗-algebra Ais a weakly (order) Rickart JB∗-triple precisely when it is a weakly Rickart C∗-algebra. We also prove that the Peirce-2 subspace associated with any tripotent in a weakly order Rickart JB∗-triple is a Rickart JB∗-algebra in the sense of Ayupov and Arzikulov. By extending a classical property of Rickart C∗-algebras, we prove that every weakly order Rickart JB∗-triple is generated by its tripotents. © 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). *Corresponding author. E-mail addresses: [email protected] (J.J. Garcés), [email protected] (L. Li), ap[email protected] (A.M. Peralta), h[email protected] (H.M. Tahlawi). https://doi.org/10.1016/j.jalgebra.2022.06.007 0021-8693/© 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 568 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 1. Introduction and preliminaries The reference [59]is the founding work of the fruitful theory of Rickart and Baer C∗- algebras. C. E. Rickart [59] stated that “Our general purpose is to study the structure of a B∗-algebra in terms of its projections. Such a study of course demands the existence of many projections .... a B∗-algebra is defined to be a B∗ p-algebra (now called a Rickart C∗-algebra) provided it contains, in a certain sense, “sufficiently many” projections.” The chosen notion was built around left and right annihilators. For each nonempty subset S of an associative ring A, the rightand left-annihilator of Sare defined by R(S)={x∈A:sx =0foralls∈S} and L(S)={x∈A:xs =0foralls∈S}, respectively. If Ais an associative ∗-ring, a projection pin Awill be a self-adjoint (p∗=p) idempotent (p2=p). A Rickart ∗-ring is an associative ∗-ring Asuch that, for each a ∈A, R({a}) =pA for a (unique) projection p(see [12, §3, Definition 2]). In such a case we have L({a}) =(R({a∗}))∗=(qA)∗=Aq for a suitable projection q. A Rickart C∗-algebra is a C∗-algebra which is also a Rickart ∗-ring (cf. [12, §3, Definition 3] and the original work by Rickart [59]). Each Rickart ∗-ring has a unity element and its involution is proper, i.e., xx∗=0 ⇒x =0(see [12, §3, Proposition 2]). The projections of a Rickart C∗-algebra form a lattice which is not necessarily complete (cf. [12, §3, Proposition 7 and Example 2]). A C∗-algebra Ais called weakly Rickart if for each x ∈Athere exists an annihilating right projection (briefly, ARP) of x, that is, a projection psatisfying xp =x, and xy =0 implies py =0. Let us observe that annihilating left projections (ALP) are similarly defined. The ARP and ALP of each element xare uniquely determined by x, and we shall denote them by RP(x)and LP(x), respectively. Every unital weakly Rickart C∗- algebra is a Rickart C∗-algebra, since for each x ∈Awe have R({x}) =(1 −RP(x))A. Rickart proved in [59, Theorem 2.10] that every Rickart C∗-algebra is generated by its projections. As seen before, the definition of a Rickart ∗-ring is given in terms of the annihilators of singletons. When singletons are replaced by general subsets we find the notion of Baer ∗-ring. Concretely, a Baer ∗-ring is an associative ∗-ring Asuch that, for every nonempty subset S⊂Awe have R(S) =pA for a suitable projection pin A(see [12, §4, Definition 1]). Baer ∗-rings are precisely those Rickart ∗-rings whose projections form a complete lattice, equivalently, every orthogonal family of projections has a supremum (cf. [12, §4, Proposition 1]). As introduced in the pioneering works of Kaplansky [46–48], an AW∗-algebra is a C∗-algebra that is a Baer ∗-ring (see [12, §4, Definition 2]). Since for each element ain a C∗-algebra Awe have R({a}) =R({a∗a}), in the definition of Rickart C∗-algebra we can restrict our attention to the right-annihilators J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 569 of positive elements. Similarly, in the definition of AW∗-algebras we can consider rightannihilators of sets of the form {a∗a :a ∈S}, where Sis any subset of the C∗-algebra under study. Each von Neumann algebra (i.e., a ∗-subalgebra of B(H)whose bicommutant coincides with itself, or equivalently, by Sakai’s theorem [64], a C∗-algebra which is also a dual Banach space) is an AW∗-algebra [12, §4, Proposition 9]. After Sakai’s theorem, von Neumann algebras are also known as W∗-algebras. Though AW∗-algebras were actually introduced with the aim of finding an algebraic characterization of von Neumann or W∗-algebras, it was soon shown by Dixmier that there exist commutative AW∗-algebras which cannot be represented as von Neumann algebras (see [25]or [12, §7, Exercises 2, 3]). Wright found in [68]examples of monotone complete factors which are not von Neumann algebras. The reader has probably realized that we take the references [59,46,12,62]as the basic bibliography on Rickart and AW∗-algebras. In the list of problems and future directions in [61, page 144], A. Rodríguez-Palacios somehow anticipated and suggested the study of Rickart Jordan algebras as those Jordan algebras for which “the annihilator of every element in Zelmanov sense is generated by an idempotent” (see subsection 1.1 for the basic theory on Jordan algebras). However, we have to wait until 2016 to find the first study on Rickart and Baer Jordan algebras by Sh. A. Ayupov and F. N. Arzikulov (see [7]). The (outer) quadratic annihilator of a subset Sin a Jordan algebra M–with product ◦– is defined as the set Ann(S)=S⊥q:= {a∈M:Ua(s)=2(a◦s)◦a−(a◦a)◦s=0,∀s∈S}. A Jordan algebra Mis called a Rickart Jordan algebra if for each element a ∈M2there exists an idempotent e ∈Msuch that {a}⊥q=Ue(M), where Ue(x) := 2(e ◦x) ◦e −e2◦x. If in the definition of Rickart Jordan algebra, the sets given by a single element a ∈M2 are replaced by arbitrary subsets S⊂M2, we get the notion of Baer Jordan algebra (cf. [7]). Rickart and Baer Jordan algebras are appropriate notions for JB-algebras, where we have projections and positive elements. It is shown by Ayupov and Arzikulov that for each C∗-algebra A, its self-adjoint part, Asa, is a Rickart (respectively, Baer) Jordan algebra if and only if Ais a Rickart (respectively, Baer) C∗-algebra [7,8]. A Rickart (respectively, Baer) JB∗-algebra is a JB∗-algebra Mwhose self-adjoint part is a Rickart (respectively, Baer) JB-algebra. The original aim in Rickart’s studies was completed in the case of JB-algebras by F. N. Arzikulov who proved that a JB-algebra Nis a Baer Jordan algebra if and only if N satisfies the following properties: (1) Every subset of pairwise orthogonal projections in the partially ordered set of projections has a least upper bound in this set; (2) Every maximal strongly associative subalgebra of Nis generated by its projections (see [4, Theorem 2.1]). 570 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 The available notions of Rickart and Baer Jordan algebras have a strong dependence on quadratic annihilators, projections and positive elements. However, if we are interested in developing these notions in more general Jordan structures, like JB∗-triples, where projections and positive elements do not make any sense, we need an alternative approach. This is the main goal of this paper. Section 2is devoted to revisit the main results on Rickart and weakly Rickart C∗- algebras with the aim of finding a characterization which can be stated without appealing to projections and positive elements. We shall show (see Propositions 2.5 and 2.10) that by mixing and extending a characterization due to G. K. Pedersen in [56]with key contributions by P. Ara and D. Goldstein [2,3,35], the following characterizations hold for every C∗-algebra A: (a)Ais a weakly Rickart C∗-algebra if, and only if, any of the following statements holds: (1) Given x ∈Aand an inner ideal J⊆Awhich is orthogonal to the inner ideal I=A(x)of Agenerated by x, there exists a partial isometry ein Asuch that I⊆A2(e)and J⊆A0(e). (2) Given x ∈Aand an inner ideal J⊆Awith I=A(x) ⊥J, there exists a partial isometry ein Asuch that I⊆A2(e), e∗e =RP(x), ee∗=LP(x), xis a positive element in the C∗-algebra (A2(e), •e, ∗e), A(x)is a C∗-subalgebra of the latter C∗-algebra and J⊆A0(e). (b)Ais a Rickart C∗-algebra if, and only if, Ais unital and for each x ∈Aand each inner ideal J⊆Awhich is orthogonal to I=A(x), there exists a partial isometry e in Asuch that I⊆A2(e)and J⊆A0(e). The advantage of the previous characterizations (especially the one in (a)(1)) relies on their independence of projections and positive elements, and can be therefore extended to wider settings. Before further extensions, in section 3we explore the notions of weakly Rickart and SAJBW-algebras, both in terms of projections and positive elements. For example, a JB-algebra Nis called a weakly Rickart JB-algebra if for each element a ∈N+ there exists a projection p ∈Nsuch that p ◦a =a, and for each z∈Nwith Uz(a) =0 we have p ◦z=0. In Proposition 3.14 we establish several characterizations of Baer or AJBW∗-algebras, (weakly) Rickart JB∗-algebras and SAJBW∗-algebras in terms of hereditary JB∗-subalgebras. After several technical conclusions in the line of classical results, we arrive to our main goal of section 3in Theorem 3.16, where it is proved that every weakly Rickart JB∗-algebra is generated by its projections. In section 4we introduce several definitions of Rickart, weakly Rickart and weakly order Rickart JB∗-triples. We show that, thanks to the characterization of the corresponding notions for C∗-algebras presented in section 2, the new definitions coincide with the classical notions in the setting of C∗-algebras. Special interest is received by weakly order Rickart JB∗-triples. This new notion agrees with the concept of Rickart C∗-algebra in the C∗- setting. A weakly order-Rickart JB∗-triple Eis a JB∗-triple satis- J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 571 fying that for each x ∈E, if we write E(x)for the inner ideal of Egenerated by x, then for each inner ideal J⊆Ewith I=E(x) ⊥J, there exists a tripotent ein Esuch that xis positive in E2(e), and J⊆E0(e). We prove in Proposition 4.4 that if Eis a weakly order Rickart JB∗-triple and e ∈E is a tripotent, then the Peirce-2 subspace E2(e)is a Rickart JB∗-algebra. This allows us to conclude that every weakly order Rickart JB∗-triple is generated by its tripotents (see Theorem 4.5). Finally, in section 5we explore the connections with von Neumann regularity, by showing that each inner ideal Iof a weakly order Rickart JB∗-triple Econtains a dense subset of von Neumann regular elements (cf. Theorem 5.4). 1.1. Background and basic definitions This subsection is aimed to provide a basic compendium on the Jordan structures studied in this note. The reader will find some brief historical introduction, definitions, notions and basic references. These contents are not really required to follow section 2, which has been written to be accessible with tools of C∗-algebras. The early contributions by Jordan, von Neumann and Wigner in the decade of 1930s led to the idea of employing non-associative structures, specially Jordan algebras, in quantum mechanics (see the interesting monograph [52]for a fantastic historical overview). A real or complex Jordan algebra is a non-necessarily associative algebra M whose product (denoted by ◦) is commutative and satisfies the Jordan-identity: (a◦b)◦a2=a◦(b◦a2)(a, b ∈M).(1) The Jordan algebra Mis called unital if there exists a unit element 1in Msuch that 1 ◦a =afor all a ∈M. Jordan algebras are power associative, that is, a subalgebra generated by a single element is associative. In other words, for each a ∈Mdefine a0:= 1if Mis unital, a1=aand an+1 =a ◦an(n ⩾1). Then an+m=an◦amfor all natural numbers mand n[39, Lemma 2.4.5]. For each a ∈Mwe shall denote by Tathe Jordan multiplication operator by the element a, that is, Ta(x) =a ◦x(x ∈M). An element ain a unital Jordan Banach algebra Mis called invertible whenever there exists b ∈Msatisfying a ◦b =1and a2◦b =a. The element bis unique and it will be denoted by a−1(cf. [39, 3.2.9] and [22, Definition 4.1.2]). We know from [22, Theorem 4.1.3] that an element a ∈Mis invertible if and only if Uais a bijective mapping, and in such a case U−1 a=Ua−1. As in the associative case, an involution on a Jordan algebra Mis a mapping a → a∗ satisfying (a∗)∗=aand (a ◦b)∗=a∗◦b∗for all a, b ∈M. The involution ∗is called proper if a ◦a∗= 0 implies a =0. A very special source of examples is provided by associative algebras. Namely, suppose Ais a real or complex associative algebra with product denoted by juxtaposition. Then the natural Jordan product a ◦b := 1 2(ab +ba) defines a structure of Jordan algebra on 572 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 A; Jordan algebras of this type are called special, as they are isomorphic to subalgebras of associative algebras equipped with a new multiplication (a term coined by Jordan, von Neumann & Wigner [45]). There are Jordan algebras which are not special (cf. [39, Corollary 2.8.5]), these algebras are called exceptional. Suppose that Ais a C∗-algebra. The (associative) product of two self-adjoint elements in Aneed not be, in general, self-adjoint. Another good property of the natural Jordan product assures that the Jordan product of two self-adjoint elements in Aalso is in Asa. Therefore, Asa is a real Jordan subalgebra of A, but not an associative subalgebra. A central notion in the study of Jordan algebras is the so-called U-mapping. Let a, b be two elements in a Jordan algebra M. The Ua,b mapping is the linear map on Mgiven by Ua,b(x)=(a◦x)◦b+(b◦x)◦a−(a◦b)◦x, for all x ∈M. The mapping Ua,a is usually denoted by Ua. The U-maps satisfy the following fundamental identity: UaUbUa=UUa(b),for all a, b in a Jordan algebra M, (2) (see [39, 2.4.18]). It is now the moment to introduce some analytic structures. A Jordan algebra M endowed with a complete norm satisfying a ◦b ⩽ab, a, b ∈Mis called a Jordan Banach algebra. A JB-algebra is a real Jordan Banach algebra Nwhose norm satisfies the following two geometric axioms: (i)a2 =a2, for all a ∈N; (ii)a2 ⩽a2+b2, for all a, b ∈N, (see [39, Definition 3.1.4]). The Jordan mathematical model closest to C∗-algebras is given by the class of JB∗- algebras. A JB∗-algebra is a complex Jordan Banach algebra Mtogether with an algebra involution a → a∗, whose norm satisfies the following generalization of the GelfandNaimark axiom: Ua(a∗)=a3,for every a∈M. Both of the just introduced Jordan structures are intrinsically related thanks to a result due to J. D. M. Wright proving that every JB-algebra corresponds to the selfadjoint part of a (unique) JB∗-algebra (see [69]). If a C∗-algebra Ais equipped with its original norm and involution and the Jordan product given by a ◦b =1 2(ab +ba), then the resulting structure is a JB∗-algebra. Jordan ∗-subalgebras of C∗-algebras are called JC∗-algebras, and their symmetric parts are known as JC-algebras. The class of JB∗-algebras is strictly bigger than the collection J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 573 of all associative C∗-algebras since, for example, the exceptional Jordan algebra H3(O)is a purely exceptional JB-algebra, that is, there is no nonzero homomorphism from H3(O) into a JC-algebra (cf. [39, §7.2]). From a purely algebraic point of view, a complex Jordan triple system is a complex linear space Eequipped with a triple product {x, y, z}which is bilinear and symmetric in x, zand conjugate linear in yand satisfies the following ternary Jordan identity: L(x, y){a, b, c}={L(x, y)a, b, c}−{a, L(y,x)b, c}+{a, b, L(x, y)c},(3) for all x, y, a, b, c ∈E, where L(x, y) :E→Eis the linear mapping given by L(x, y)z= {x, y, z}. The analytic structures known as JB∗-triples, whose origins go back to the theory of holomorphic functions on infinite dimensional complex Banach spaces [49], are defined as those complex Jordan triple systems Ewhich are Banach spaces satisfying the next “geometric” axioms: (a)For each x ∈E, the operator L(x, x)is hermitian with non-negative spectrum; (b){x, x, x} =x3for all x ∈E. The triple product of each JB∗-triple Eis a non-expansive mapping, that is, {a, b, c} ⩽abc,(4) for all a, b, c ∈E(cf. [37, Corollary 3]). JBW∗-triples (respectively, JBW∗-algebras) are defined as those JB∗-triples (respectively, JB∗-algebras) which are also dual Banach spaces. The bidual of every JB∗-triple is a JBW∗-triple (see [24]). It is further known that each JBW∗-triple admits a unique (isometric) predual and its product is separately weak∗continuous [11](see also [23, Theorems 5.7.20 and 5.7.38]). Each C∗-algebra Acarries a natural structure of JB∗-triple with respect to the triple product given by {a, b, c}=1 2(ab∗c+cb∗a).(5) The same triple product equips the space B(H, K), of all bounded linear operators between two complex Hilbert spaces, with structure of JB∗-triple. In particular, there exist infinite-dimensional complex Hilbert spaces which are JB∗-triples. For each element ain a JB∗-triple E, the symbol Q(a) will denote the conjugate linear operator on Edefined by Q(a)(x) ={a, x, a}. Every JB∗-algebra Mis a JB∗-triple with triple product {a, b, c}=(a◦b∗)◦c+(c◦b∗)◦a−(a◦c)◦b∗.(6) 574 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 It follows that Q(a)(x) =Ua(x∗)for all a, x ∈M. We refer to [39,22]and [23]for the basic background on JB∗-triples and JB∗-algebras. An element e ∈Eis called a tripotent if {e, e, e} =e. When a C∗-algebra Ais regarded as a JB∗-triple with the triple product in (5), it is known that the tripotents in Aare precisely the partial isometries in A. In the same way that each partial isometry in a C∗-algebra Ainduces a Peirce decomposition, each tripotent ein a JB∗-triple Eproduces a Peirce decomposition of Ein the form E=E2(e) ⊕E1(e) ⊕E0(e), where Ei(e)is the i 2 eigenspace of the operator L(e, e), i =0, 1, 2. This decomposition satisfies the following Peirce rules: {E2(e),E 0(e),E}={E0(e),E 2(e),E}=0 and {Ei(e),E j(e),E k(e)}⊆Ei−j+k(e), when i −j+k∈{0, 1, 2}and is zero otherwise. The Peirce k-projection, Pk(e), is the natural projection of Eonto Ek(e). Peirce projections are non-expansive (cf. [33, Corollary 1.2]) and they can be expressed in the following terms: P2(e)=Q(e)2,P 1(e)=2(L(e, e)−Q(e)2), and P0(e)=IdE−2L(e, e)+Q(e)2. It is known that the Peirce-2 subspace E2(e)is a JB∗-algebra with unit e, Jordan product x ◦ey:= {x, e, y}and involution x∗e:= {e, x, e}, respectively. It is worth to note that a linear bijection between JB∗-triples is an isometry if and only if it is a triple isomorphism (cf. [49, Proposition 5.5]). Consequently, the triple product in E2(e) is uniquely given by {x, y, z}=(x◦ey∗e)◦ez+(z◦ey∗e)◦ex−(x◦ez)◦ey∗e, for all x, y, z∈E2(e). A subspace Bof a JB∗-triple Eis a JB∗-subtriple of Eif {B, B, B} ⊆B. A JB∗- subtriple Iof Eis called an inner ideal of Eif {I, E, I} ⊆I. A subspace Iof a C∗-algebra Ais an inner ideal if IAI ⊆I. Every hereditary σ-unital C∗-subalgebra of a C∗-algebra is an inner ideal. A complete study on inner ideals of JB∗-triples is available in [28,29] and the references therein. It follows from Peirce rules that for each tripotent ein a JB∗-triple E, the Peirce-2 subspace E2(e)is an inner ideal. Let Ebe a JB∗-triple. The JB∗-subtriple, Ea, of Egenerated by a single element ais identified, via the Gelfand theory, with the commutative C∗-algebra J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 575 C0(Ωa)={f:Ω a→Ccontinuous with f(0) = 0 if 0 ∈Ωa}, for a unique compact set Ωacontained in [0, a], such that a ∈Ωaand 0 cannot be isolated in Ωa; and under this identification acorresponds to the continuous function given by the embedding of Ωainto C(cf. [49, Corollary 1.15] and [50, Lemma 3.2]). A consequence of this representation affirms that every element in a JB∗-triple admits a cubic root and a (2n −1)th-root (n ∈N) belonging to the JB∗-subtriple that it generates. The sequence (a[1 2n−1])of all (2n −1)th-roots of aconverges in the weak∗(and also in the strong∗) topology of E∗∗ to a tripotent in E∗∗, denoted by rE∗∗ (a), and called the range tripotent of a. The tripotent rE∗∗ (a)is the smallest tripotent e ∈E∗∗ satisfying that ais positive in the JBW∗-algebra E∗∗ 2(e). It is also known that, if a =1, the sequence (a[2n−1]), of all odd-powers of a, converges in the weak∗-and strong∗-topology of E∗∗ to a tripotent (called the support tripotent of a, u(a)in E∗∗, which satisfies u(a) ⩽a ⩽rE∗∗ (a)in E∗∗ 2(rE∗∗ (a)) (compare [27, Lemma 3.3]; beware that in [30], r(a) is called the support tripotent of a). In case that ais a positive element in a JB∗-algebra M, the support and the range tripotents of ain M∗∗ are projections, called the support and range projections of ain M∗∗. For each element ain a JB∗-triple E(in which we generally do not have a cone of positive elements), the symbol E(a) will stand for the norm-closure of {a, E, a}= Q(a)(E)in E. It was proved by L. J. Bunce, C.-H. Chu and B. Zalar that E(a)is precisely the norm-closed inner ideal of Egenerated by a. Clearly, Ea⊂E(a). It is further shown in the just quoted reference that E(a)is a JB∗-subalgebra of the JBW∗- algebra E(a)∗∗ =E(a)w∗ =E∗∗ 2(rE∗∗ (a)) and contains aas a positive element, where rE∗∗ (a)is the range tripotent of ain E∗∗ (cf. [16, Proposition 2.1]). The reader will need some basic knowledge on the strong∗-topology of a JB∗-triple. If we are given a norm-one functional ϕin the predual, W∗, of a JBW∗-triple W, and a norm-one element zin Wwith ϕ(z) =1, the mapping (x, y)→ ϕ{x, y, z} defines a positive sesquilinear form on W. Moreover, the mapping does not depend on the chosen z, that is, if w∈Wsatisfies ϕ(w) =1, we have ϕ {x, y, z}=ϕ {x, y, w}, for all x, y∈W(see [9, Proposition 1.2]). The mapping x →xϕ:= (ϕ{x, x, z})1 2, defines a prehilbertian seminorm on W. The strong∗-topology (denoted by S∗(W, W∗)) is the topology on Wgenerated by the family of all semi-norms  · ϕwith ϕrunning in the unit sphere of the predual of W(cf. [10]). For the purposes of this note we recall that the triple product of every JBW∗-triple Wis jointly strong∗continuous on bounded sets. The first proof of this result appeared in [60], however the difficulties affecting Grothendieck’s inequalities in [9]also impacted the original proof and an alternative argument can be found in [57, Theorem 9]. The recent proof of the Barton-Friedman conjecture on Grothendieck’s inequalities for JB∗-triples in [38] reinstates the validity of the original proof. 582 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 Remark 2.8. The characterization provided in Proposition 2.7 is not valid without the hypothesis of finiteness. Consider, for example, the Hilbert space H=2with orthonormal basis {ξn:n ∈N}and A =B(H). Take a partial isometry vsuch that 1 −vv∗=ξ1⊗ξ1is a rank-one projection and 1 −v∗v=ξ1⊗ξ1+ξ2⊗ξ2 has rank 2. If for J={v}⊥=A0(v) ⊥A(v) =A2(v) there were a partial isometry esatisfying that (ξ1⊗ξ1)A(ξ1⊗ξ1+ξ2⊗ξ2) =A0(v) =J⊆A2(e)and (1 −ξ1⊗ξ1)A(1 −ξ1⊗ξ1−ξ2⊗ξ2) =A2(v) ⊆A0(e)we would have ξ1⊗ξ1≤ee∗, ξ1⊗ξ1+ξ2⊗ξ2≤e∗e, 1 −ξ1⊗ξ1≤1 −ee∗and (1 −ξ1⊗ξ1−ξ2⊗ξ2) ≤1 −e∗e. Therefore ξ1⊗ξ1=ee∗and ξ1⊗ξ1+ξ2⊗ξ2≤e∗e, which is impossible. Remark 2.9. The partial isometry eappearing in the statements of Proposition 2.5 need not be unique. Actually if eis a partial isometry satisfying the desired conclusion, then the partial isometry λe satisfies the same property for all λin the unit sphere of C. The partial isometry eappearing in Proposition 2.5(a) induces a local order in the C∗-algebra (A2(e), •e, ∗e)and we actually obtain a strengthened version of the statement. Proposition 2.10. Let Abe a C∗-algebra. Then Ais a weakly Rickart C∗-algebra if, and only if, given x ∈Aand an inner ideal J⊆Awith I=A(x) ⊥J, there exists a partial isometry ein Asuch that I⊆A2(e), e∗e =RP(x), ee∗=LP(x), xis a positive element in the C∗-algebra (A2(e), •e, ∗e), A(x)is a C∗-subalgebra of the latter C∗-algebra and J⊆A0(e). Proof. It suffices to prove the extra properties in the “only if” implication. Suppose Ais a weakly Rickart C∗-algebra. We shall assume that Ais non-unital, and its unitization A1=A ⊕C1is a Rickart C∗-algebra [12, Theorem 5.1] (see also [63, Lemma 3.6]). Fix x ∈A. Another essential contribution by P. Ara and D. Goldstein (see [3, Corollary 3.5], [35, Corollary 7.4]) assures the existence of a polar decomposition for x, that is, there exists a partial isometry e ∈A1such that x =e|x|, ee∗=LP(x)and e∗e =RP(x) (cf. also [12, Proposition 21.3]). If we write ein the form e =e1+λ1with λ ∈C, e1∈A, we infer from the fact e1e∗ 1+λe∗ 1+λe1+|λ|21 =ee∗=LP(x) ∈Athat e =e1∈A, that is, weakly Rickart C∗-algebras satisfy the existence of polar decompositions. Let I=A(x)and let Jbe an inner ideal orthogonal to I. By considering the partial isometry ein the polar decomposition of x, we can easily check that xis a positive element in the C∗-algebra (A2(e), •e, ∗e), namely, ee∗(e|x|1 2)e∗e =e|x|1 2=(e|x|1 2)∗e, (e|x|1 2) •e(e|x|1 2) =e|x| =x, and hence xis positive in (A2(e), •e, ∗e). Finally, given y∈Jthe conditions x ⊥y, ee∗=LP(x)and e∗e =RP(x)imply that y⊥e, and therefore J⊆A0(e).  Corollary 2.11. Let Abe a C∗-algebra. Then Ais a weakly Rickart C∗-algebra if, and only if, given x ∈Athere exists a partial isometry ein Asuch that A(x) ⊆A2(e), xis a positive element in the C∗-algebra (A2(e), •e, ∗e), and A0(e) ={x}⊥. J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 583 Proof. (⇒)By applying Proposition 2.10 to I=A(x)and J={x}⊥we find a partial isometry e ∈Asatisfying that I⊆A2(e), e∗e =RP(x), ee∗=LP(x), xis a positive element in the C∗-algebra (A2(e), •e, ∗e), A(x)is a C∗-subalgebra of (A2(e), •e, ∗e), and {x}⊥=J⊆A0(e). We shall show that {x}⊥=A0(e). To this end take a ∈A0(e). The identities xa∗= xRP(x)(1 −e∗e)a∗=x(e∗e)(1 −e∗e)a∗=0, and a∗x =a∗(1 −ee∗)LP(x)x =a∗(1 − ee∗)(ee∗)x =0, show that a ∈{x}⊥. (⇐)This is a clear consequence of Proposition 2.10, since for each x ∈Aand each inner ideal J⊆Awith I=A(x) ⊥J, by taking the partial isometry egiven by the hypothesis we have J⊂{x}⊥=A0(e)and A(x) ⊆A2(e).  We have seen in the proof of Proposition 2.10 that, as a consequence of the result by P. Ara and D. Goldstein [3,35], weakly Rickart C∗-algebras satisfy polar decomposition. It is well known that the partial isometry appearing in the polar decomposition of an element ais uniquely determined by |a|(cf. [12, Propositions 21.1 and 21.3]). We shall conclude this section by showing that the properties of the partial isometry ein Corollary 2.11 provide a characterization of the partial isometry in the polar decomposition. Corollary 2.12. Let xbe an element in a weakly Rickart C∗-algebra A. Suppose eis a partial isometry in A. Then the following are equivalent: (a)eis the partial isometry in the polar decomposition of x; (b)xis a positive element in the C∗-algebra (A2(e), •e, ∗e), and A0(e) ={x}⊥. Proof. The implication (a) ⇒(b)has been proved in the proof of Corollary 2.11. (b) ⇒(a)Since eis a partial isometry, the elements ee∗and e∗eare projections in A. It is known that ee∗Aee∗and e∗eAe∗eare Rickart C∗-algebras (cf. [12, Proposition 5.6]). Since the mapping z→ ze∗(respectively, z→ e∗z) is a C∗-isomorphism from (A2(e), •e, ∗e)onto ee∗Aee∗(respectively, e∗eAe∗e), we derive that (A2(e), •e, ∗e)is a Rickart C∗-algebra. We shall next show that the left and right projections of xin A2(e)both coincide with e. Since xis positive in A2(e), we have RPA2(e)(x) =LPA2(e)(x) =q. Clearly q⩽ein A2(e). If q<e, the partial isometry (projection in A2(e)) e −qis orthogonal to qin A2(e) and also in A, because orthogonality in Acan be given in terms of the triple product {a, b, c} =1 2(ab∗c +cb∗a)and A2(e)is closed for this triple product (see section 4for additional details). When the triple product is computed with respect to the C∗-product of A2(e)and with respect to the one in Awe have x={q,x,q}A2(e)=q•ex∗e•eq=qe∗ex∗ee∗q=qx∗q={q,x,q}. It follows that xbelongs to A2(q), which combined with the fact e −q⊥q, implies that x ⊥e −q. It follows from the hypotheses that e −q∈A0(e). Therefore e −q= e •e(e −q) =ee∗(e −q) =0, leading to a contradiction. 584 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 Since xis positive in A2(e), RPA2(e)(x) =LPA2(e)(x) =ein this C∗-algebra, and the mapping z→ e∗zis a C∗-isomorphism from (A2(e), •e, ∗e)onto e∗eAe∗e, we deduce that e∗xis a positive element in Awith ee∗=LPA2(e)(x)e∗=LP(e∗x). Similarly, e∗e =e∗RPA2(e)(x) =RP(e∗x)(have in mind that the left and right projections of xe∗ and e∗xdo not change when computed in Aor in ee∗Aee∗or e∗eAe∗e, respectively [12, Proposition 5.6]). Furthermore, since ((e∗x)∗(e∗x))n=(x∗ee∗x)n=(x∗x)n,for all natural n, it can be deduced, via functional calculus, that |x| =e∗x. It clearly follows from the hypotheses that x =ee∗x =e|x|. We have seen above that RP(e∗x) =e∗eand ee∗=LP(e∗x). Therefore eis the partial isometry in the polar decomposition of xby uniqueness.  3. Jordan counterparts of Rickart and Baer ∗-algebras in terms of projections Sh. A. Ayupov and F. N. Arzikulov developed a deep study on the notions of Rickart and Baer ∗-rings in the setting of real Jordan algebras in the papers [7,8,4,5]. Before entering into details, we introduce the required nomenclature. Let Mbe a Jordan algebra. According to the standard notation (see [7,58]), the (outer) quadratic annihilator of a subset S⊂Mis the set Ann(S)=S⊥q:= {a∈M:Ua(S)={0}}.(8) The inner quadratic annihilator of Sis formed by the elements in the intersection of all kernels of all U-maps associated with elements in Sdefined by ⊥qS:= {a∈M:Us(a) = 0 for all s∈S}.(9) Let us denote M2:= {a2:a ∈M}for the set of all elements in Mwhich are the square of another element (do not confuse with the set of all elements of the form a ◦bwith a, b ∈M). Clearly, each idempotent in Mis inside M2. We consider the following two statements: (R1) For each element a ∈M2there exists an idempotent e ∈M(i.e. e2=e) such that {a}⊥q=Ue(M); (R2) For each element x ∈Mthere exists an idempotent e ∈Msuch that ⊥q{x} ∩M2= Ue(M) ∩M2. In any Jordan algebra M, (R1) implies (R2) and both properties are equivalent when Mis unital and lacks of nilpotent elements (cf. [7, Theorems 1.6 and 1.7]). According to [7,8], a Jordan algebra Msatisfying condition (R1) (respectively, (R2)) is called a J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 585 Rickart Jordan algebra (respectively, an inner Rickart Jordan algebra). That is, each Rickart Jordan algebra is an inner Rickart Jordan algebra. It should be noted here that in [8] inner Rickart Jordan algebras are called weak Rickart Jordan algebras, however since the term weak Rickart algebra is employed in the associative setting with another meaning (for example, for an uncountable set Γthe commutative C∗-algebra ∞,c(Γ) of all countably supported elements of the commutative von Neumann algebra ∞,c(Γ) is weak Rickart but not an inner Jordan Rickart algebra see, for example, [12]), here we shall employ the term mentioned above. The notion of (inner) Rickart is essentially addressed to real JB-algebras. For example, the exceptional JB-algebra H3(O)is a Rickart Jordan algebra (cf. [7, Proposition 3.4]). Moreover, for each associative Rickart ∗-algebra A, its self-adjoint part Asa is a Jordan algebra satisfying (R1) and (R2) (cf. [7, Proposition 1.1]). Reciprocally, if Ais an associative ∗-algebra with proper involution and Asa is a Rickart Jordan algebra, then Ais a Rickart ∗-algebra in the usual sense ([7, Proposition 1.3]). Every Rickart Jordan algebra possesses a unit element and lacks of nilpotent elements, it is further known that the set of idempotents of a Rickart Jordan algebra is a lattice, which is not, in general, complete (see [7, Lemma 1.4, Proposition 1.10]). There exist examples of inner Rickart Jordan algebras without unit element (cf. [8, Remark 1 in page 32]). However the properties gathered in the next lemma hold: Lemma 3.1 ([8, Lemma 2.3]). Let Mbe an inner Rickart Jordan algebra. Then the following statements hold: (a)There exists an element 12in Msatisfying a ◦12=afor every a ∈M2; (b)M2contains no non-trivial nilpotent elements. The element 12given in the above statement (a)is a unit for those elements in M2. If Mis generated by square elements (i.e., every element is a finite linear combination of elements in M2), then the element 12actually is a unit in M. Corollary 3.2 ([7, Theorems 1.6 and 1.7]). Suppose Mis a Jordan algebra linearly generated by M2and containing no non-trivial nilpotent elements. Then Mis a Rickart Jordan algebra if and only if it is an inner Rickart Jordan algebra. The lacking of associativity in Jordan algebras is somehow compensated with the celebrated Macdonald’s theorem asserting that if Gis a multiplication operator in two variables x, ywith G(a, b) =0for all a, bin all special Jordan algebras, then G =0in all Jordan algebras, equivalently, any polynomial identity in three variables, with degree at most 1in the third variable, and which holds in all special Jordan algebras, holds in all Jordan algebras (cf. [39, Theorem 2.4.13]). The following identities, which hold true for any Jordan algebra M, can be directly deduced from Macdonald’s theorem: 2TalUam,an=2Uam,anTal=Uam+l,an+Uam,an+l,(10) 586 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 Un a=Uan,(11) for every natural numbers l, m, n(see [39, Lemma 2.4.21]). In the set of all idempotents in a Jordan algebra Mwe can consider a partial order defined by e ⩽fif e ◦f=e. The following equivalences can be easily checked by applying (10)and (11): e⩽f⇔e∈Uf(M)⇔Ue(M)⊆Uf(M).(12) A Jordan algebra Mis called a Baer Jordan algebra if it satisfies the following property: For each subset S⊂M2there exists an idempotent e ∈Msuch that S⊥q=Ue(M). We say that Mis an inner Baer Jordan algebra if for each subset S⊂Mthere exists an idempotent e ∈Msuch that ⊥qS∩M2=Ue(M) ∩M2. Let us observe that in [7,8,4–6] inner Baer Jordan algebras are called weak Baer Jordan algebras, which is a term not completely compatible with the notation in the associative setting. Each Baer Jordan algebra is an inner Baer Jordan algebra [7, Theorem 2.6] or [8, Proposition 3.1]. If Mis a Jordan algebra containing no nilpotent elements, then Mis an inner Baer Jordan algebra if and only if it is a Baer Jordan algebra [7, Theorem 2.6]. As we have seen in the comments after Lemma 3.1, if a Jordan algebra Mis linearly generated by elements in M2and Mis an inner Baer Jordan algebra, then Mis unital. A C∗-algebra is a Baer C∗-algebra if and only if Asa is a Baer Jordan algebra (cf. [7, Propositions 2.1 and 2.3] or [8]). To conclude our tour through the algebraic Jordan alter-egos of Rickart and Baer algebras, we appeal to a couple of results also proved by Sh. A. Ayupov and F. N. Arzikulov, where they establish that a Jordan algebra Mis a Baer Jordan algebra if, and only if, it is a Rickart Jordan algebra and the set of all idempotents in Mis a complete lattice (see [7, Theorem 2.7]); moreover, Mis an inner Baer Jordan algebra if, and only if, it is an inner Rickart Jordan algebra and the set of all idempotents of Mis a complete lattice (cf. [8, Theorem 3.5]). Following [7,8,4,5], and in coherence with the terminology of C∗-algebras, (inner) Rickart JB-algebras and (inner) Baer JB-algebras or AJBW-algebras are defined as those JB-algebras which are (inner) Rickart and (inner) Baer Jordan algebras, respectively. We shall also deal with the complex structures. A JB∗-algebra Mwill be called a Rickart JB∗-algebra (respectively, a Baer JB∗-algebra or an AJBW∗-algebra) if its self-adjoint part, Msa, is a Rickart JB-algebra (respectively, a Baer JB-algebra or an AJBW-algebra). That is, Mis a Rickart JB∗-algebra if and only if for each a ∈M+there exists a projection p ∈Msuch that {a}⊥q∩Msa =Up(M)∩Msa =Qp(M)∩Msa; which by Corollary 3.2 is equivalent to prove that for each x ∈Msa there exists a projection p ∈Msuch that J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 587 ⊥q{x}∩M+=Up(M)∩M+=Q(p)(M)∩M+. A similar restatement can be applied to the definition of Baer JB∗-algebras. A JBW∗- algebra (respectively, a JBW-algebra) is a JB∗-algebra (respectively, a JB-algebra) which is a dual Banach space. It is known that a JB∗-algebra Mis a JBW∗-algebra if, and only if, Msa is a JBW-algebra (cf., for example, [53, Corollary 2.12]). Two elements a, bin a Jordan algebra Aare said to operator commute if a◦(b◦x)=(a◦x)◦b for every x ∈A. By the mentioned Macdonald’s theorem or by the Shirshov-Cohn theorem [39, Theorem 2.4.14], it can be easily checked that operator commutativity of a couple of elements in a Jordan algebra of self-adjoint operators can be equivalently verified in any Jordan subalgebra containing these elements (cf. [67, Proposition 1]). A real Jordan algebra Nis called formally real if for every a1, ..., an∈Nthe condition n i=1 a2 i= 0 implies a1=... =an=0(see [39, §2.9]). Every JB-algebra is a formally real Jordan algebra. A Jordan subalgebra Bof Nis called strongly associative if the identity (x ◦y) ◦a =x ◦(y◦a)holds for all x, a ∈Band y∈N, equivalently, any pair of elements in Boperator commute as elements in N. A family Fof elements of Nis called compatible if the Jordan subalgebra J(F) generated by Fis strongly associative. The idea behind (weakly) Rickart and Baer C∗-algebras is to find a subclass of C∗- algebras, between general C∗-algebras and von Neumann algebras, in which every element can be approximated in norm by finite linear combinations of projections. In the setting of AJBW∗-algebras (i.e. Baer JB∗-algebras) this goal is achieved by the following theorem, in which Arzikulov established a Jordan version of the original result proved by Kaplansky for AW∗-algebras. Theorem 3.3 ([4, Theorem 2.1]). The following statements are equivalent for each JBalgebra N: (a)Nsatisfies the following properties: (1) Every subset of pairwise orthogonal projections in the partially ordered set of projections has a least upper bound in this set; (2) Every maximal strongly associative subalgebra of Nis generated by its projections (i.e., it coincides with the least closed subalgebra containing its projections); (b)Nis an AJBW-algebra; (c)Nis an inner AJBW-algebra. Let Mbe a JB∗-algebra. It is worth to notice that the JB∗-subalgebra generated by a single self-adjoint element in Mis strongly associative (cf. [22, Proposition 2.4.13 and Fact 3.3.34]). The set of all strongly associative subalgebras of Mcan be regarded as an inductive set when equipped with the order defined by inclusion. Therefore each 588 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 strongly associative JB∗-subalgebra of Mis contained in a maximal strongly associative JB∗-subalgebra. It follows from Theorem 3.3 that every self-adjoint element in a AJBW∗-algebra Mcan be approximated by finite linear combinations of projections in M–actually the same conclusion holds for any element in M. We shall see later that our notion of weakly Rickart JB∗-algebra also enjoys this property. As in the case of C∗-algebras, a couple of projections p, qin a JB∗-algebra are called orthogonal if p ◦q=0. Both notions are perfectly compatible in the case of a C∗-algebra regarded with its associative structure or as a JB∗-algebra. One of the new contributions in this note is to explore the notions of weakly Rickart and SAW∗-algebras in the setting of JB∗-algebras. In order to develop our study, we shall follow a similar method to that introduced by Ayupov and Arzikulov focused on the selfadjoint part and the lattice of projections. In the setting of JB∗-algebras we cannot define properties in terms of the left or right multiplication operator by an element. We gather next some reinterpretations for latter purposes. Lemma 3.4. Let aand xbe non-zero positive elements in a C∗-algebra. Then the following statements are equivalent: (a)ax =x; (b)a ◦x =x; (c)Ua(x) =x. Clearly, the elements aand xcommute in case that any of the previous statements holds. Proof. (a) ⇒(b)and (c). This is clear because xa =(ax)∗=x∗=x, and thus a ◦x = 1 2(ax +xa) =x. Similarly, Ua(x) =axa =xa =x. (b) ⇒(a)We can clearly embed Ainside its unitization, and thus assume that Ais unital. Since (1 −a) ◦x =0with 1 −a ∈Asa and x ⩾0, [18, Lemma 4.1] implies that x ⊥(1 −a)in A(as JB∗-and as C∗-algebra), then (1 −a)x =0 =x(1 −a), which proves (a). (c) ⇒(a)If a ⩽1the proof is much easier. First, the inequality x =Ua(x) ⩽ a2xassures that a =1. We can deduce from a simple induction argument that Uan(x) =anxan=xfor all natural n. Now, by applying that the sequence (an)n converges in the strong∗topology of A∗∗ to the support projection, s(a), of a, together with the join strong∗continuity of the product of A∗∗ [65, Proposition 1.8.12], we obtain s(a)xs(a) =x. Finally, since a =s(a) +(1 −s(a))a =s(a) +a(1 −s(a)) in A∗∗, it follows that ax =s(a)x +a(1 −s(a))x =x. For the general case we assume that axa =x. Since the same identity holds in A∗∗, it is easy to check that aza =zfor every zin the C∗-subalgebra of Agenerated by x (and also in the von Neumann subalgebra of A∗∗ generated by x). Therefore, the identity a r(x) a =r(x)holds in A∗∗. It is easy to deduce from the above that J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 589 (r(x)ar(x)) (r(x)ar(x)) = r(x). Having in mind that r(x) a r(x)is a positive element with r(x) a r(x) ⩽a r(x)whose square is r(x), a simple application of the local Gelfand theory proves that r(x) a r(x) = r(x). Now by mixing the identities a r(x) a =r(x)and r(x) a r(x) =r(x)we get ar(x)=(ar(x)a)r(x)=r(x),and r(x)a=r(x)(ar(x)a)=r(x). Finally, it is easy to see that ax =ar(x)x =r(x)x =x =xr(x) =xr(x)a =xa. As in the associative setting of C∗-algebras, a JB∗-subalgebra Bof a JB∗-algebra M is said to be a hereditary JB∗-subalgebra of Mif whenever 0 ⩽a ⩽bwith a ∈Mand b ∈B, then a ∈B, equivalently, B+is a face of M+(cf. [26,15,1]). It is known that a hereditary C∗-subalgebra Bof a C∗-algebra Ais σ-unital if and only if it has the form B=xAx for some positive x ∈A. The same statement remains valid in the case of a JB∗-algebra M, where each σ-unital, hereditary JB∗-subalgebra is of the form Ux(M), for some positive x ∈M. Corollary 3.5. Let aand xbe positive elements in a JB∗-algebra M. Then the following statements are equivalent: (a)a ◦x =x; (b)Ua(x) =x; (c)a ◦z=zfor all zin the inner ideal of Mgenerated by x. Furthermore, if any of the previous statements holds the elements aand xoperator commute as elements of M, and a ◦r(x) =r(x), where r(x)denotes the range projection of xin M∗∗. Proof. By Macdonald’s theorem (see also the Shirshov-Cohn theorem in [39]or [69, Corollary 2.2]), there exists a C∗-algebra Acontaining the JB∗-subalgebra of Mgenerated by aand xas JB∗-subalgebra. Lemma 3.4 proves that (a)is equivalent to (b)in A, and hence in M. Since ax =xa =xin A, [67, Proposition 1] assures that aand x operator commute in M. The implication (c) ⇒(a)is clear because x ∈M(x). To see the implication (a) ⇒(c), we recall that (a) implies that aand xoperator commute in Mand ax =xa =xin A (cf. Lemma 3.4). Then {a, x, z}=(a◦x)◦z+(z◦x)◦a−(a◦z)◦x=x◦z(x∈M), and thus, by the Jordan identity, we get 590 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 UaUx(y)={a, {x, y∗,x},a}=−{y∗,x,{a, x, a}} +2{{y∗,x,a},x,a} =−{y∗,x,x}+2(x◦y∗)◦x =−(x◦y∗)◦x−x2◦y∗+(x◦y∗)◦x+2(x◦y∗)◦x ={x, y∗,x}=Ux(y), for all y∈M. This shows that Ua(z) =zfor every z∈M(x). Now take z∈M(x) positive, then, by the equivalence (a) ⇔(b), Ua(z) =zgives a ◦z=z. Since, each z∈M(x)writes as a linear combination of four positive elements in M(x), we have a ◦z=z. The weak versions of Rickart and Baer Jordan algebras in the classical sense considered in Berberian’s book [12]have not been considered yet. The reader should be warned that, in order to work in the Jordan setting, the left and right multiplication operations do not make too much sense in a Jordan algebra. Definition 3.6. Let Nbe a JB-algebra. We shall say that Nis a weakly Rickart JB-algebra if for each element a ∈N+there exists a projection p ∈Nsuch that p ◦a =a, and for each z∈Nwith Uz(a) =0we have p ◦z=0. Nis called a weakly inner Rickart JB-algebra if for each element x ∈Nthere exists a projection p ∈Nsuch that p ◦x =x, and for each z∈N+with Ux(z) =0we have p ◦z=0. A JB∗-algebra Mwill be called weakly Rickart or weakly inner Rickart if its selfadjoint part satisfies the same property. Remark 3.7. Let Nbe a weakly Rickart JB-algebra. Then, for each a ∈N+, the projection pin Definition 3.6 is unique. This projection will be called the range projection of ain N(RPN(a) =RP(a)in short). Indeed, suppose that there exist projections p, pin Nsuch that p◦a=p◦a=a, and for any z∈Nwith Uz(a) =0we have p ◦z=p◦z=0. It follows from the original assumptions that (p −p) ◦a =0, and since a ⩾0, we deduce from (13)that p −p⊥a. It then follows that U(p−p)(a) ={p −p, a, p −p} =0. By applying the assumptions we get p ◦(p −p) =0 =p◦(p−p), which implies that p =p ◦p=p. It can be seen that RPN(a)is the smallest projection in Nsuch that a =p ◦a(= Up(a)). Namely, if qis any projection in Nsuch that q◦a =a, then (RPN(a) −q) ◦a =0, and thus RPN(a) ◦(RPN(a) −q) =0, therefore RPN(a) ◦q=RPN(a), which is equivalent to say that RPN(a) ⩽q. J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 591 Lemma 3.8. Let Nbe a JB-algebra. Then Nis weakly Rickart and unital if, and only if, it is a Rickart JB-algebra if, and only if, it is weakly inner Rickart and unital. Proof. Suppose Nis a unital weakly Rickart JB-algebra with unit 1. Let us fix a ∈N+. By assumptions there exists a projection pin Nsuch that a ◦p =aand for each z∈N with Uz(a) =0we have p ◦z=0. Given x ∈{a}⊥qwe have Ux(a) =0, and thus p ◦x =0, in particular (1−p) ◦x =x. We have shown that {a}⊥q⊆U1−p(N) =N2(1−p) =N0(p). Conversely, if x ∈U1−p(N) =N2(1−p) =N0(p), since p ◦a =a, we deduce that a ∈N2(p), and consequently, Ux(a) =0, by Peirce arithmetic. Suppose now that Nis a unital weakly inner Rickart JB-algebra with unit 1. So, given x ∈Mthere exists a projection p ∈Nsuch that p ◦x =xand for each z∈N+ with Ux(z) =0we have p ◦z=0. For each z∈⊥q{x} ∩N2we have Ux(z) =0, and hence p ◦z=0. It follows that ⊥q{x} ∩N2⊆U1−p(N) ∩N2. Reciprocally, each z∈U1−p(N) ∩N2is positive and must be orthogonal to N2(p)by Peirce arithmetic, then z∈⊥q{x} ∩N2, because x ∈N2(p). To conclude the proof we observe that every Rickart JB-algebra is unital and weakly (inner) Rickart.  Proposition 3.9. Let pbe a projection in a weakly Rickart JB∗-algebra M. Then the Peirce-2 subspace M2(p)is a Rickart JB∗-algebra with unambiguous range projections of positive elements in M2(p). Proof. Let us fix a positive element a ∈M2(p). Clearly, ais positive in M. Let q=RP(a) denote the range projection of ain M. Since (p −q) ◦a =0, it follows from (13)that (p −q) ⊥a, and thus U(p−q)(a) =0. Applying now that q=RP(a)we get q◦(p −q) =0. Therefore, p ◦q=qand thus Up(q) =q, witnessing that q∈M2(p)and satisfies the properties of a range projection for ain M2(p). We have proved that M2(p)is a unital weakly Rickart JB∗-algebra, Lemma 3.8 gives the rest.  Let hand xbe two elements in a JB∗-algebra Mwith hpositive. We know from [18, Lemma 4.1] that x⊥hif, and only if, h◦x= 0. (13) The orthogonal annihilator of a subset Sin a JB∗-triple Eis defined as S⊥ E=S⊥:= {y∈E:y⊥x, ∀x∈S}. The next result with the basic properties of the orthogonal annihilator has been borrowed from [19, Lemma 3.1] and [30, Lemma 3.2]. Lemma 3.10 ([30, Lemma 3.2], [19, Lemma 3.1]). Let Sbe a nonempty subset of a JB∗-triple E. Then the following statements hold: 598 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 {a}⊥={a∗}⊥={a}⊥∗=(M0(e))∗=M0(e∗), witnessing that e∗satisfies the properties of the range tripotent of ain M, and by the uniqueness of this element e =e∗. That is, eis a self-adjoint tripotent in M, and thus, by the local Gelfand theory, e =p −q, where pand qare two orthogonal projections in M. It follows from the properties of the range tripotent e =p −qthat 0 ⩽ain M2(e). Since 0 ⩽−q⩽ein M2(e), the element −qis a projection in M2(e). Therefore, having in mind that, by Kaup’s theorem, the triple product on M2(e)is uniquely given by the restriction of the triple product of Mand by the JB∗-structure of M2(e), the element UM2(e) −q(a)={−q,a∗e,−q}={−q,a,−q}={q,a,q}=Uq(a) is positive in M2(e)(cf. [39, Proposition 3.3.6]), and in M2(−q). Since, M2(−q) =M2(q) with (M2(−q))sa =(M2(q))sa we deduce the existence of y∈(M2(−q))sa =(M2(q))sa ⊆ Msa such that Uq(a)=y◦−qy={y,−q,y}=−{y,q,y}=−Uy(q), which implies that Uq(a)is a negative element in M. On the other hand, since ais positive in Mand qis a projection, the element Uq(a) must be positive in M[39, Proposition 3.3.6], which combined with the previous conclusion leads to Uq(a) =0. It follows from the first statement in Lemma 3.11 that q∈{a}⊥q∩Msa ={a}⊥∩Msa, that is, q⊥a. The properties of the range tripotent imply that q∈M0(e) =M0(p −q), and thus q⊥(p −q), and so q=0. We have therefore shown that the range tripotent e =RM(a)of ain Mis a projection in this JB∗-algebra. It can be easily checked that e ◦a ={e, e, a} =aand for each z∈Msa with Uz(a) =0we have p ◦z=0(cf. Lemma 3.11), that is Mis a weakly Rickart JB∗-algebra.  An element uin a unital JB∗-algebra Mis called unitary if it is invertible with inverse u∗. In the setting of JB∗-triples, the word unitary is applied to those elements usuch that L(u, u)is the identity mapping. Clearly, every unitary uin a JB∗-triple Eis a tripotent with E2(u) =E–this is actually a characterization. There is no ambiguity in case that a unital JB∗-algebra Mis regarded as a JB∗-triple because both notions are equivalent [13, Proposition 4.3]. Our next result is a strengthened version of Proposition 4.3. We recall first that for each tripotent ein a JB∗-triple Eand each unitary complex number λ, the mapping Sλ(e)=λ2P2(e)+λP1(e)+P0(e) (16) is a triple automorphism on E[33, Lemma 1.1]. It can be easily deduced from this fact that the mapping J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 599 Rλ(e)=P2(e)+λP1(e)+λ2P0(e) (17) also is a triple automorphism on E. Proposition 4.4. Let Ebe a woR JB∗-triple. Then for each tripotent e ∈E, the Peirce-2 subspace E2(e)is a Rickart JB∗-algebra. Proof. Having in mind Proposition 4.2 and Lemma 3.8(a), it suffices to show that each positive element ain E2(e)admits a range tripotent in E2(e). Let v=RE(a)be the range tripotent of ain E. Let S−1=S−1(e) =P2(e) −P1(e) +P0(e)denote the triple automorphism on Egiven in (16). Let us observe that S−1(a) =abecause a ∈E2(e). Since ais positive in E2(v)with {a}⊥ E=E0(v), we deduce that a =S−1(a)is positive in E2(S−1(v)) with {a}⊥ E={S−1(a)}⊥ E=S−1{a}⊥ E=S−1(E0(v)) = E0(S−1(v)). That is, S−1(v)satisfies the properties of the range tripotent for a, and hence it follows from its uniqueness that v=S−1(v) =P2(e)(v) −P1(e)(v) +P0(e)(v). This equality proves that v=P2(e)(v) +P0(e)(v), where P2(e)(v)and P0(e)(v)are two orthogonal tripotents in E. If in the previous argument we replace S−1(e)with Ri(e), and we apply it to v= P2(e)(v) +P0(e)(v), we derive that v=Ri(e)(v) =P2(e)(v) −P0(e)(v), witnessing that v=P2(e)(v). Now, it can be easily seen that v=P2(e)(v) ∈E2(e)satisfies the properties of the range tripotent for ain E2(e)(and in E). This concludes the proof.  We can now establish the result which has motivated our study. We shall see that every woR JB∗-triple contains an abundant collection of tripotents. Theorem 4.5. Every weakly order Rickart JB∗-triple is generated by its tripotents. Proof. Let abe an element in a woR JB∗-triple E. Let e =RE(a)be the range tripotent of ain E. Proposition 3.9 assures that E2(e)is a Rickart JB∗-algebra. By construction, ais a positive element in E2(e), and hence Theorem 3.16 implies that acan be approximated in norm by finite linear combinations of projections in E2(e). The proof concludes by just observing that, since E2(e)is a JB∗-subtriple of E, every projection in E2(e)is a tripotent in E. 5. Von Neumann regularity Regular elements in the sense of von Neumann have been intensively studied in the associative setting of C∗-algebras (cf. [41,42,14]and [59, §3]) as well as in the wider setting of JB∗-triples (see [31,32,50,20,21]and [44]). 600 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 Motivated by the study conducted by Rickart on von Neumann regular elements in B∗ p-algebras (now called Rickart C∗-algebras) in [59, §3], we devote this section to explore von Neumann regular elements in woR JB∗-triples. An element ain a JB∗-triple Eis called von Neumann regular if and only if there exists b ∈Esuch that Q(a)b =a, Q(b)a =band [Q(a), Q(b)] := Q(a) Q(b) −Q(b) Q(a) =0 (cf. [50, Lemma 4.1] or [31,32,20]). The element b ∈Esatisfying the previous properties is unique and is called the generalized inverse of ain E(denoted by a†). However, there exist von Neumann regular elements a ∈E, for which we can find many elements cin E such that Q(a)c =a. Several useful characterizations of von Neumann regular elements in JB∗-triples can be found in [31,32,50,20]. For our purposes here, we recall that an element ain a JB∗- triple E, whose range tripotent in E∗∗ is denoted by rE∗∗ (a) =r(a), is von Neumann regular if, and only if, r(a) ∈Eand ais positive and invertible in the unital JB∗-algebra E2(r(a)), and in such a case a†is precisely the inverse of ain E2(r(a)) (cf. [20, §2, pages 191 and 192]). It is further known that in this case L(a, a†) =L(a†, a) =L(r(a), r(a)) (see [20, §2, page 192] and [51, Lemma 3.2]). The next lemma goes in the line of [43, Lemma 2.2] and [59, Theorem 3.2]. Lemma 5.1. Let ebe a tripotent in a JB∗-triple E. The following statements hold: (a)Every invertible element ain the unital JB∗-algebra E2(e)is von Neumann regular in Ewith rE∗∗ (a)being a unitary element in E2(e). (b)Suppose that xis an element in Ewith e −x <1. Then Q(e)(x)and P2(e)(x) are von Neumann regular elements whose range tripotents (i.e. r(Q(e)(x)) and r(P2(e)(x)), respectively) in E∗∗ belong to E2(e)and are unitaries in the latter JB∗- algebra. Moreover, r(Q(e)(x)) and r(P2(e)(x)) satisfy the properties of the range tripotent in a woR JB∗-triple for the elements Q(e)(x)and P2(e)(x), respectively. The latter conclusion holds for the range tripotent in E∗∗ of any invertible element a ∈E2(e). Proof. (a)The statement is essentially proved in [43, Remark 2.3]. Namely, if ais invertible in E2(e), the just quoted remark assures that the range tripotent r=rE∗∗ 2(e)(a)of a in the bidual of E2(e)is a unitary element in E2(e). It is clear that rmust be also the range tripotent of ain E∗∗ and belongs to E. It follows from the characterization of von Neumann regular elements from [20], seen before this lemma, that ais von Neumann regular in E. (b)Since e −x <1and Q(e)and P2(e)are non-expansive mappings fixing the element e, we get e −Q(e)(x), e −P2(e)(x) <1. Having in mind that E2(e)is a unital JB∗-algebra with unit eand Q(e)(x), P2(e)(x) ∈E2(e), we deduce that these two elements are invertible in E2(e). The first part of the statement now follows from (a). We shall only prove the last statement for P2(e)(x). To simplify the notation, let r=r(P2(e)(x)) ∈E2(e)denote the range tripotent of P2(e)(x). Clearly, P2(e)(x)is J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 601 positive in E2(r)(let us note that E2(r) =E2(e)as sets because ris a unitary in E2(e)). Finally, it follows from Lemma 3.2 in [19]that {P2(e)(x)}⊥=E0(r), which concludes the argument.  The next result is a triple version of [59, Theorem 3.3]. Proposition 5.2. Let Ebe a woR JB∗-triple. Suppose that ais a von Neumann regular element in E. Then the range tripotent of ain Eas woR JB∗-triple coincides with the range tripotent of ain E∗∗ (and in E), that is R(a) =rE∗∗ (a). Furthermore a†∈ E2(R(a)) is the inverse of ain E2(R(a)) and R(a†) =R(a). Proof. We know from Lemma 5.1(b)that the range tripotent r(a)satisfies the properties of the range tripotent of ain the definition of woR JB∗-triple. Then the uniqueness of R(a)(see Lemma 4.3(b)) implies that R(a) =r(a). It is known that r=r(a) =R(a)and a†both belong to the JB∗-subtriple of E generated by a(cf. [51, Lemma 3.2]), and hence a†∈E2(R(a)). Finally, we know from the properties of the generalized inverse that a†is the inverse of ain E2(r).  As we have seen in subsection 1.1, for each element ain a JB∗-triple E, its triple spectrum Ωa⊆[0, a]can be employed to identify the JB∗-subtriple, Ea, of Egenerated by awith the commutative C∗-algebra C0(Ωa), and under this identification acorresponds to the continuous function given by the embedding of Ωainto C(cf. [49, Corollary 1.15] and [50, Lemma 3.2]). The triple spectrum Ωadoes not change when computed with respect to any JB∗-subtriple Fof Econtaining the element a[50, Proposition 3.5(vi)]. It is further known that ais von Neumann regular if and only if 0 /∈Ωa(cf. [50, Lemma 4.1]). In particular if Fis a JB∗-subtriple of a JB∗-triple E, then an element a ∈Fis von Neumann regular in Fif and only if it is von Neumann regular in E. Furthermore, if a ∈Eis von Neumann regular, then a†and r(a)both belong to the JB∗-subtriple of Egenerated by a. Our next goal is a triple version of [59, Theorem 3.13] and a refinement of Theorem 4.5. Proposition 5.3. Let Ebe a woR JB∗-triple. Suppose ais an element in Ewhose range tripotent is R(a). Then for each ε >0there exists a tripotent eε∈Eand an element b in the JB∗-subtriple of Egenerated by asatisfying eε⩽R(a), {b, R(a), b} =a, {b, eε, b} is von Neumann regular and a −{b, eε, b} <ε. Proof. Proposition 4.4 assures that E2(R(a)) is a Rickart JB∗-algebra. By definition, ais positive in E2(R(a)). Let Cbe a maximal strongly associative JB∗-subalgebra of E2(R(a)) containing a. Lemma 3.15 implies that Cis a Rickart JB∗-algebra. Therefore Cis a commutative Rickart C∗-algebra whose product and involution will be denoted by ·and ∗, respectively –observe that ∗coincides with ∗R(a). 602 J.J. Garcés et al. / Journal of Algebra 609 (2022) 567–605 Given ε >0, having in mind that Cis a commutative C∗-algebra, Theorem 3.13 in [59] proves the existence of a projection eε∈Csatisfying eε⩽R(a), eε·a ={eε, a, eε} = P2(eε)(a)is von Neumann regular in Cand a −P2(eε)(a) <ε. As observed in [34, comments after Theorem 2.1], since ais a positive in E2(R(a)) (and in C), the JB∗-subtriple Eaof E2(R(a)) (and of C) generated by acoincides with the JB∗-subalgebra that agenerates. Therefore the square root of ain Clies in Ea. Let b ∈Eadenote the square root of ain C. By applying that Cis a commutative C∗-algebra, it can be deduced that {b, eε, b} =(b ·b) ·eε=a ·eεis von Neumann regular in C. Clearly, {b, R(a), b} =a. Finally, since Cis a JB∗-subtriple of E, the element eεis a tripotent in Ewith eε⩽R(a), {b, eε, b}is von Neumann regular in Eand a −{b, eε, b} <ε. We can now prove that every inner ideal in a woR JB∗-triple Econtains an abundant collection of von Neumann regular elements. Theorem 5.4. Let Ibe an inner ideal of a woR JB∗-triple E. Then the von Neumann regular elements of Iare dense in I. Each von Neumann regular element xin Iis contained in E2(R(x)) =I2(R(x)), where R(x) ∈Iand E2(R(x)) is a Rickart JB∗- algebra. Furthermore, if I={0}, then Icontains a non-zero tripotent, actually Icontains the generalized inverse and the range tripotent of each non-zero element in I. Proof. Let us fix a ∈I. Proposition 5.3 proves that we can approximate ain norm by von Neumann regular elements of the form {b, e, b}, where e ∈Eis a tripotent satisfying e ⩽R(a)and b ∈Ea. Having in mind that Iis an inner ideal we deduce that Ea⊆I, and {b, e, b} ∈I, which concludes the proof of the first statement. The second statement is a consequence of Propositions 5.2 and 4.4. Take now a ∈I\{0}. In this case Ea⊆E(a). By the conclusion in the first paragraph, we can approximate ain norm by a sequence (an)nof non-zero von Neumann regular elements in I. It follows from Proposition 5.2 that the range tripotent of each anin E, R(an), coincides with its range tripotent in E∗∗ and by the theory on von Neuman regular elements a† n, R(an) ∈Ean⊆E(an) ⊆I, which concludes the proof.  Acknowledgments We would like to express our gratitude to the anonymous referee for many constructive comments and suggestions to improve the final form of the paper. J. Garcés and A.M. Peralta partially supported by MCIN/AEI/10.13039/501100011033/FEDER, EU, project no. PGC2018-093332-B-I00 and Junta de Andalucía grants number A-FQM-242-UGR18 and FQM375. L. Li partially supported by NSF of China (12171251) and Tianjin Natural Science Foundation (Grant No. 19JCYBJC30200). A.M. 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