Polar decomposition in Rickart C*-algebras

Publicacions Matem`atiques, Vol 39 (1995), 5–21. POLAR DECOMPOSITION IN RICKART C∗-ALGEBRAS Dmitry Goldstein Abstract A new proof is obtained to the following fact: a Rickart C∗-algebra satisfies polar decomposition. Equivalently, matrix algebras over a Rickart C∗-algebra are also Rickart C∗-algebras. Introduction. In this paper we give new proof of the following result: all Rickart C∗-algebras satisfy polar decomposition. This fact was established in [2] by P. Ara and author by using a suitable factorization of the elements in the regular overring of a finite Rickart C∗-algebra. New proof also uses the construction of the regular overring, but in a different way. In particular, we don’t need the result of Goodearl, Lawrence and Handelman about algebras without one-dimensional representations. The regular ring of measurable operators of a finite AW∗-algebra was constructed by S. K. Berberian in [3]. Later Saito modified Berberian’s approach for general AW∗-algebras [11]. E. Christensen constructed and investigated a ∗-algebra of measurable operators, associated to MSC C∗- algebras [5]. Handelman found a regular extension Q(T) for a finite Rickart C∗- algebra T, using a technique of the module-homomorphisms on the essential countably generated ideals (instead of Berberian’s coordinated sequences) [9]. It was established ([1], [10]) that a finite Rickart C∗- algebra Tsatisfies polar decomposition iff the bounded elements of Q(t) belong to T. Goodearl, Handelman and Lawrence have proved that T satisfies polar decomosition in the case where Thas no one-dimensional representations (see [8]). P. Ara in [1], using his special construction, proved that left and right projections of element in a Rickart C∗-algebra are equivalent and the 6D. Goldstein polar decomposition problem in general Rickart C∗-algebras can be reduced to the finite case. In this work Ara also proved an equivalence of the following conditions for a Rickart C∗-algebra T: (i) Tsatisfies polar decomposition; (ii) The matrix algebras Mn(T) over Tare the Rickart C∗-algebras for all n; (iii) The partial isometries of Tare ℵ0-addable. Finally, by development of the methods of [1], [8], it was proved in [2] that conditions (i)-(iii) are always fulfiled in Rickart C∗-algebras. In [6], [7] was constructed a ∗-algebra of measurable operators for a finite Rickart C∗-algebra and were proved some algebraic properties of this ∗-algebra. We continue to develope this approach in order to solve the polar decomposition problem. 1. Preliminaries. A∗-algebra Ais Rickart, if for all x∈Athere exists a projection e∈A such that R(x)={a∈A|xa =0}is eA. Because of the involution, L(x)={a∈A|ax =0}=Tf for some projection f. We shall write e=RA(x), f=LA(x), 1 −e=RP(x), 1 −f=LP(x) and P(A) for the set of all projections of A. The projections eand fare equivalent (e∼f)ina∗-algebra Aif e=uu∗,f=u∗ufor some partial isometry u∈A.Ais finite if p∼1 implies p= 1. A Rickart C∗-algebra is a C∗-algebra that is also a Rickart ∗-algebra. We recall some properties of the Rickart C∗-algebras. Theorem 2.1. A Rickart C∗-algebra satisfies the following properties: (i) P(T)is ℵ0-complete lattice partially ordered by p≥qiff pq =q (see [4]). If in addition Tis finite then the lattice P(T)is ℵ0-continuous [9, Cor. 1.1]. (ii) LP(x)∼RP(x)for all x∈T[1, Th. 2.5]. (iii) For given sequences (en)and (fn)of ortogonal projections such that en∼fnfor all n∈N, we have nen∼nfn[1]. The partial isometries are ℵ0-addable in a Rickart C∗-algebra Tif for every sequence of partial isometries {wn}such that {wnw∗ n}and {w∗ nwn} are the sequences of ortogonal projections there exists a partial isometry wsuch that ww∗ nwn=wnw∗ nw=wn. Polar decomposition in Rickart C∗-algebras 7 2. Strongly dense domains. Through this paper Tdenotes (if the opposite is not specified) a finite Rickart C∗-algebra. A sequence of projections (en)⊂P(T) is a strongly dense domain (SDD) in case en↑1. Let e∈P(T), x∈T. We define x−1(e)= RA[(1 −e)x]. Proposition 2.1. Let (en)and (fn)are SDD, xn∈Tsuch that m≤nimplies xnem=xmem. Then a sequence (tn=x−1 n(fn)en) is a SDD. Proof: Let dn=x−1 n(fn). If m≤nthen (1 −en)xntm=(1−en)(1 −em)xnemtm=(1−en)(1 −em)xmtm=0, so that tm≤tn. Let p∈P(T). We show that there exists a number k such that tkp= 0. For that choose a number isuch that q=pei= 0. If xiq= 0, then q≤ti. Now let xiq= 0. There exists a∈Tsuch that h=xiqa is non-zero projection [4, par. 8]. Observe that xiq=xnqfor all n≥iand h=fkh= 0 for sufficiently large k.Fork≥iwe have (1 −fk)xkqah=(1−fk)xiqah=(1−fk)hh=0. Therefore g=LP(qah)≤dk. In addition, g≤q≤ei≤ek, hence g≤tk.Thusptk≥g=0. Corollary 2.2. If (en)and (fn)are SDD, then (enfn)is also SDD. Proof: Put in Proposition 2.1 xn= 1 for all n. 3. A ring of measurable operators. An essentially measurable operator (EMO) is a pair of sequences (xn,e n) with xn∈T,(en) an SDD, and such that m≤nimplies xnem=xmemand x∗ nem=x∗ mem. Two (EMO) (xn,e n) and (yn,f n) are equivalent, if there exists an SDD (gn) such that xngn=yngn, gnxn=gnynfor all n∈N. Clearly that this relation is indeed equivalence relation (By Corollary 2.2). If (xn,e n) is (EMO), [xn,e n] denotes its equivalence class. We call [xn,e n] a measurable operator (MO) and denote by S(T) the set of all (MO), and use the letters x,y,z,... for the elements of S(T). Now we define the algebraic operations on S(T). We put [xn,e n]+[yn,f n]=xn+yn,e nfn λ[xn,e n]=[λxn,e n] [xn,e n][yn,f n]=[xnyn,k n], [xn,e n]=[x∗ n,e n], 8D. Goldstein where kn=fny−1 n(en)en(x∗ n)−1(fn). Summarizing, Theorem 3.1. The set S(T)of all MO is a ∗-algebra. The mapping x→ [x, 1](x∈T)is a ∗-isomorphism of Tinto Q, and [1,1] is a unity element for S(T). We write x=[x, 1], for x∈T. The image of Tin S(T)isT. We recall the construction by Handelman of the ∗-regular ring associated to a finite Rickart C∗-algebra. Let Abe a unital ring. A right (left) ideal E⊆Ais essential if Ehas nontrivial intersection with any nonzero right (left) ideal of A. We say that Eis essential countably generated (ecg) right ideal if there exist a sequence {tn}n∈N⊆Asuch that tiA is essential in A. Similarly, we define left ecg ideal. It was proved in [9] that every ecg ideal of a finite Rickart C∗-algebra is generated by SDD. Let Tbe a finite Rickart C∗-algebra. Consider the following pairs of mappings [f,E;f1,E 1], where fis right T-module homomorphisms from essential countably generated right ideal E,f1is left T-module homomorphism from essential countably generated left ideal E1, and they are balanced by the following condition: e1f(e)=f1(e1)efor all e∈Eand all e1∈E1. Two pairs [f,E;f1,E 1] and [g,J;g1,G 1] are equivalent if f(x)=g(x) and f1(y)=g1(y) for all x∈EJand all y∈E1J1. Let Qbe the set of equivalence classes of just defined pairs. It was shown in [9] that Qis endowed with algebraic operations, and with respect to these operation Qbecomes a ∗-regular algebra. Define mapping from S(T)toQ.If[xn,e n] is MO then E=∞ n=1 enT (E1=∞ n=1 Ten) is an essential countably generated right(left) ideal in Tcorrespondently. Define a right T-module homomorphism f:f(ent)= xnent, where ent∈E. Obviously, f(entx)=f(ent)xfor all x∈T. Let ent=ems(m≤n). Then f(ems)=f(em)s=xmems=xnems=xnent=f(ent). Thus this definition is correct. Similarly, we define a left T-module homomorphism f1:E1→T,f(ten)=tenxn. Now let e∈E,e1∈E1, e=emt,e1=t1en.Ifm≤nthen e1f(e)=t1enf(emt)=t1enxmemt=t1enxnemt =f1(t1en)emt=f1(e1)e. By a similar argument e1f(e)=f1(e1)e. Therefore [f,E,f1,E 1]∈ Q. We shall denote just defined mapping by π. Then π([xn,e n]) = Polar decomposition in Rickart C∗-algebras 9 [f,E,f1,E 1]. Let [xn,e n]=[x n,e  n], π([x n,e  n]) = [f,E,f 1,E 1]. Choose an SDD (pn) such that xnpn=x npn,pnxn=pnx nfor all n∈N.Putqn=pnene n. Note that qn∈EE1EE 1.We have f(qn)=xnqn=xnpnqn=x nqn=f(qn). Thus f=fon ∞  n=1 qnT. In the same way we obtain f1=f 1on ∞  n=1 Tqn. Theorem 3.2. The mapping πis a ∗-isomorphism from S(T)onto Q. Proof: Let [f,E,f1,E 1]∈Q,E=∞ n=1 enT,E1=∞ n=1 Ten,(en)an SDD, f(ex)=f(e)x, f1(xe1)=xf1(e1) for all e∈E,e1∈E1,x∈T.Putf(en)=yn,f1(en)=zn. Obviously, ynen=yn,enzn=zn. Set xn=yn+zn−znen=yn+zn−enyn so that xnen=yn,enxn=znfor all n∈N. It is easy to see that [xn,e n] is MO. Set π(xn,e n])=[g,E,g1,E 1], where g(en)=xnen, g1(en)=enxn. Then g(en)=yn=f(en), g1(en)=zn=f1(en), hence [f,E,f1,E 1]=[g,E,g1,E 1]. Thus πis surjective. Now we show that the mapping πpreserves the algebraic operations. Let [xn,e n], [yn,k n]∈S(T). Put π([xn,e n]) = [f,E,f1,E 1],π([yn,k n]) = [g,J,g1,J 1], where E= ∞  n=1 enT, E1= ∞  n=1 Ten, J= ∞  n=1 knT, J1= ∞  n=1 Tkn. We have (see [9, Section 2]) [f,E,f1,E 1]+[g,J,g1,J 1]=f+g,E J, f1+g1,E 1J1, [xn,e n]+[yn,k n]=xn+yn,e nkn. 10 D. Goldstein Let pn=enknand π([xn+yn,e nkn]) = [r, L, r1,L 1]. We can regard that L= ∞  n=1 pnT, L1= ∞  n=1 Tpn, r(pn)=(xn+yn)pn,r 1(pn)=pn(xn+yn). Since (enkn)T=(enT)(knT) ( see [9]) it follows L=JE,L1= J1E1. In addition r(pn)=(xn+yn)pn=(f+g)(pn),r 1(pn)=pn(xn+yn)=(f1+g1)(pn). Consequently [r, L, r1,L 1]=f+g,E J, f1+g1,E 1J1. Further, [xn,e n][yn,k n]=[xnyn,t n], where tnis a suitable SDD. On the other hand, [f,E,f1,E 1][g,J,g1,J 1]=[fg,g−1E,g1f1,f−1 1J1]. We shall establish that ∞  n=1 tnTis an essential subideal in g−1E. By the definition, tn=hngn, where hn=en(x∗ n)−1(kn),g n=kny−1 n(en),g −1E={x∈J:g(x)∈E}. We have tn≤gn≤kn, therefore tn∈Jfor all n∈N. It remains to prove that g(tn)∈Efor all n. Really, g(tn)=g(gn)tn=g(gnkny−1 n(en))tn=g(kn)y−1 n(en)gntn =ynkny−1 n(en)gntn=yny−1 n(en)kngntn =enyny−1 n(en)kngntn=enyny−1 n(en)tn, therefore g(tn)∈E.Sotn∈g−1Efor all n. Similarly, we obtain ∞  n=1 Ttn⊂f−1J1.Thus [fg,g−1E,f1g1,f−1 1J1]=fg, ∞  n=1 tnT,g1f1, ∞  n=1 Ttn. It implies π([xn,e n][yn,k n]) = π([xn,e n])π([yn,k n]). Obviously, π(λ[xn,e n]) = λπ([xn,e n]). It is sthrightforward to check that π([xn,e n]∗)=π([xn,e n])∗. Polar decomposition in Rickart C∗-algebras 11 Corollary 3.3. S(T)is a Rickart ∗-algebra and ℵ0-continuous ring. Proof: It follows from Theorem 3.2 and [9, Th. 2.1]. 4. Some algebraic properties of S(T). Cayley transform. Lemma 4.1. If x=[xn,e n]∈Qand the xnare all invertible then x is invertible and x−1=[x−1 n,h n]for a suitable SDD (hn). Proof: Set fn=LP(xnen). We show that (fn) is a SDD. If m≤n then fn(xmem)=fnxnem=fnxnenem=xnenem=xmem,fm≤fn. Since xnis invertible then RP(xnen)=en. We have fn∼en[1, Th. 2.5]. As p∼qimplies 1 −p∼1−qfor the projections pand qin a finite Rickart C∗-algebra, so en+1 −en∼fn+1 −fn. Then by ℵ0-additivity, 1 = [sup n (en+1 −en)] e1∼sup n (fn+1 −fn)f1=f. Set yn=x−1 n.Ifm≤n, then ynfm=ymfm. Really, ynxmem=ynxnem=em=ymxmem, hence (yn−ym)xmem= 0 and (yn−ym)fm= 0. Similary on setting gn=LP(x∗ nen), we have that (yn) is a SDD and y∗ ngm=y∗ mgmwhen m≤n.Puthn=fngn, then y=[yn,h n] is MO, and xy =yx =1. Corollary 4.2. For any x∈S(T)an element 1+x∗xis invertible. Proof: It follows immediately from Lemma 4.1. Lemma 4.3. If x=x∗, one can write x=[xn,e n]with x∗ n=xn. Proof: If x=[yn,f n], then x=1/2(x∗+x)=[1/2(y∗ n+yn),f n]. Corollary 4.4. If x=x∗, then x+iis invertible. Proof: Let x=[xn,e n], x∗ n=xn; then x+i=[xn+i, en] and each xn+iis invertible. Theorem 4.5. The formulas u=(x−i)(x+i)−1,x=i(1 + u)(1 −u)−1 define mutually inverse one-one correspondences between the self-adjoint elements x∈Q, and the unitary elements ufor which 1−uis invertible. Proof: It follows from Corollary 4.4. We call this uthe Cayley transform of x. 12 D. Goldstein Lemma 4.6. Let x=[xn,e n]∈S(T)and xn−→ xin norm, then x=x. Proof: Evidently, xen−xnen=xen−xkenfor all k≥n. Then xen−xnen≤x−xkfor all k≥n,xen−xnen=0,xen=xnen. In just the same way, enx=enxn. Lemma 4.7. Let x=[xn,e n]∈S(T). Then xen=xnen. Proof: Obvious. 5. The bounded measurable operators. Let Tbe a finite Rickart C∗-algebra, Q=S(T) denotes a ∗-algebra of measurable operators of T. An element x=[xn,e n]∈Qis bounded, if supnxn≤∞. Let Bbe a set of all bounded MO. It is clear that Bis ∗-algebra. Since P(Q)⊂B hence Bis Rickart ∗-algebra. We define the mapping ·1:Bx→ x1= inf supn{xn|(xn,e n)∈x}∈R. The bounded elements of S(T) play a crucial role in the following discussion of the polar decomposition problem (or ℵ0-addability of the partial isometries, see Introduction) in a finite Rickart C∗-algebra. It is easy to see that the partial isometries of Bare ℵ0-addable (Corollary 7.3). On the other hand, it is well known that the algebras Band Tcoincide if Tis AW∗-algebra [3]. We shall prove a similar result for a general Rickart C∗-algebra. Theorem 5.1. The mapping ·1is a C∗-norm. Proof: Let x=[xn,e n]∈B. Clearly, x1≥0. If x1= 0 then for any ε≥0 there exists EMO (xn,e n)∈xsuch that supnxn≤ε. Let y=[yn,e n]∈B, supnyn=α. We can choose (x n,e  n)∈xwith x n≤ε/α for all n∈N. Therefore xnyn≤ε,xy1=0. Now assume that there exists x∈Bsuch that x= 0 and x1=0. Choose number nsuch that xen= 0. By Lemma 4.7 xen=xnen.As it was shown above xen1=xnen1= 0. Let a=xnen. By the definition of the norm ·1we have a1= inf supn{an|(an,k n)∈a}. For any (an,k n)∈athere exists an SDD (pn) such that apn=anpn. Note apn=anpn≤an. Choose bsuch that ba =eis a non-zero projection [4, par. 8]. Since P(T)isℵ0-continuous, there exists k∈N such that q=epk=0. Consequently 1=q≤epk=bapk≤apkb, Polar decomposition in Rickart C∗-algebras 13 hence apk≥1/b. It follows ak≥1/b, hence a1=0,a contradiction. Thus x1= 0 implies x=0. Obviously λx1=λx1for each x∈B. Further, let x,y∈B,x=[xn,e n], y=[yn,f n]. Then x+y1= inf sup n{cn|(cn,g n)∈x+y} ≤inf sup n{x n+y n|(x n,e  n)∈x(y n,f n)∈y} ≤inf sup n{x n+y|(x n,e  n)∈x,(y n,f n)∈y} =x1+y1. In just the same way, we get xy1≤x1y1. From previous property we have x∗x1≤x2 1. On the other hand, let (bn,q n)∈x∗x,(sn,k n)∈xfor a suitable SDD kn. Hence there exists SDD (pn) such that npn=s∗ nsnpn,p nbn=pns∗ nsn. Let tn=pnknqn. Then (tns∗ n)(sntn)=tnbntnand so tns∗ nsntn≤ bn. In addition, [tns∗ n,f n]=[s∗ n,k n] for suitable SDD (fn), therefore (tns∗ n,f n)∈x∗. Hence for any EMO (bn,q n)∈x∗xthere exists EMO (zn,f n)∈x(zn=sntn) such that zn2≤bn.Thus x2 1≤x∗x1. Corollary 5.2. The norms ·and ·1coincide on T. Proof: Let xbe a positive element of T. By the definition, we have x1= inf sup n{xn|(xn,e n)∈x}. Obviously x1≤x. Set (xn,e n)∈x. Then there exists SDD pnsuch that xnpn=xpnfor all n. Therefore xpn=xnpn≤xn. Choose a sequence of the positive numbers εnwith εn↑x. Set {x} =C(K), for some Hausdorff space K.PutUn={a∈K:x(a)>ε n}, bn(a)=   1 x(a),a∈U 0,otherwise. 20 D. Goldstein Corollary 7.4. All Rickart C∗-algebras satisfy polar decomposition. Proof: By [1, Th. 3.4], this assertion is reduced to a finite case. Now combine Corollary 7.3 and [1, Prop. 2.1] and the Corollary follows. Corollary 7.5. Let Tbe a Rickart C∗-algebra, then the matrix algebras Mn(T)over Tare also Rickart C∗-algebras for all n∈N. Proof: See [1, Th. 3.5]. 8. Axiom (PSR) in Q. Using Theorem 7.1 and the methods of [3], [11]or[6], we can describe the self-adjoint elements in Q. Theorem 8.1. Let x=x∗∈Q,u=u(u∈T) its Cayley transform. One can write x=[xn,e n]with xn,en∈{u},x∗ n=xn,xnen=xn, x2 n↑. Proof: See [3, Th. 4.2]. An element x∈Qis positive, written x≥0, if x=y∗yfor some y∈Q. Theorem 8.2. Let x=x∗∈B,u=uits Cayley transform. The following conditions are equivalent: a) x≥0; b) one can write x=[yn,f n]with yn≥0; c) the spectrum of ucontained in {eiΘ:−π≤Θ≤0}; d) one can write x=[xn,e n]with xn,en∈{u},xn≥0,xnen= xn. Proof: See [3, Th. 6.1]. Corollary 8.2. Qsatisfies axiom (PSR). Proof: See [3, Cor. 6.2]. Acknowledgments. The author express his gratitude to V. I. Chilin for stating the problem and for his concern about the work, and also to P. Ara and D. Handelman for valuable discussions. I am very grateful to Ben-Gurion University for the help in preparing this paper. Polar decomposition in Rickart C∗-algebras 21 References 1. P. Ara, Left and right projections are equivalent in Rickart C∗-algebras, J. Algebra 120 (1989), 433–448. 2. P. Ara and D. Goldstein, A solution to the matrix problem for Rickart C∗-algebras, Math. Nachr. 164 (1993), 259–270. 3. S. K. 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Saito, On the algebra of measurable operators for a general AW∗-algebra, Tohoku Math. Journ. 21 (1969), 249–270. Institute of Mathematics Hebrew University of Jerusalem ISRAEL Primera versi´o rebuda el 4 de febrer de 1993, darrera versi´o rebuda el 17 de febrer de 1995