7

## Fundamental Theorems in W*-Algebras

### and the Kaplansky density theorem, ⅠI

### Akio Ikunishi

### Institute of Natural Sciences, Senshu University, 214-8580 Japan

### Abstract

### We shall glVC extraordinarily elementary proofs of that any W*-algebra has an

### identity and the Kaplansky density theorem･ We shall reconstruct Section 2 in [1].

### 1. SECOND DUALS OF C*-ALGEBRAS

### We can prove within the limits of the theory of C*-algebras and without using of

### rep-resentations on Hilbert spaces, that the second dual of a C*-algebra is a C*-algebra. By

### this fact, We imrnediately see the Kaplansky density theorem. Also, We glVe another proof

### of the Kaplansky density theorem in virtue of polar decomposition.

### Lemma 1. The second dual A** ofa Banach i-algebra A is a Banach algebra and A is a

### subalgebra of A**･ The involution ofA can be extended to the continuous linear mapping

### I - X* on A** by continuity with respect to ike a(A**, A*)-topology･ Furthermore, it holds

### that

(i) for any I ∈ A**, the linear mapping A** ∋ y L- Xy ∈ A** is c,(A**,A*)-Continuous;

(ii) for any y ∈ A, the linear mapping A** ∋ I Li Xy ∈ A** is J(A**,A*)-Continuous; (iii) foranyx･∈A** andy∈ A, X** -I and(xy)* -y*X*.

proof. Foranclcment I ∈ A**, definex* ∈ A** by (X*,p) - 7両with p ∈ A*. Obviously,

the linear mapping A** ∋ I - X* ∈ A** is J(A**,A*)-continuous andこr** - I.

For any y ∈ A and p ∈ A*, we can regard yp ∈ A* as a J(A**,A*)-continuous linear

form on A**･ For an element I ∈ A**, since the linear form A ∋ y ‥ (I,yp) is bounded,

there exists an element px ∈ A* such that (I,yp) - (y,px) for every y ∈ A. We have

I(y,px)L ≦ llyp旧匡= ≦ LIp‖llxl=Ly‖ foreveryx ∈ A**,y ∈ Aandp ∈ A* andso睡xll ≦

lLpHHx‖･ Regarding px ∈ A* as a J(A**,A*)-continuous linear form on A**, for each I and y in A**, we define a linear formこry On A* by (xy,p) - (y,px) with p ∈ A*. Since

I(xy,p)I ≦ lLp可=ly= ≦ LIpl=回目Ly‖ forevery I,y ∈ A**and p ∈ A*, xy isbounded on A*

β Bulletin of the Institute of Natural Sciences, Senshu University No.38

0-(A**,A*)-continuous and, for any y ∈ A, the mapping A** ∋ I i- xy ∈ A** is c,(A**7A*)-colltinuous. For any a,b ∈ A,I ∈ A** and p ∈ A*, it holds that

(xa)b - (u｣無A(ua))b - ull!.TEA(ua)b - u忠EAu(oJb) - I(ab)･

Similarly, we have (xy)I - I(yz) for cvcry I,y, I ∈ A**. Moreover, it holds the distributive

### law･ Since the conditions to the norm is clearly satisfied, A** is a Banach algebra and A is

a subalgebra ofA**. For any I ∈ A** and y ∈ 4 it follows that(xy)* - u莞:lEA('uJy)* - u莞:lEAy*u* - y*X*･

### □

Proposition 2. The second dual A** of a C上algebra is a Banach i-algebra with an idem-///I/･ I/IIIl/I･/･lIl･,/･･. ///, I/IIII//I,/I,･･///I,I/ ///.l‥ ′､ ､･IJ,Ir･l/Ill/ I,Ill////Il･Ju､ Jl.///I rL V,L･/ /,, I//,

### 0-(A**, A*)-topology and the involution is cT(A**, A*)-Continuous.

Proof･ Let -範be the Hilbert space associated with a state p ∈ S(A) and rip the canonical mapping of A illtO L9ep･ Since ll,rip(I)= ≦ l回I for every L･ ∈ 4 We can take the transposed

mapping t恥‥ L照一A* of恥SirlCe HtrIpH ≦ 1, let ttrIp be the bitranspose A** LjWJ*

-LjWp of町Since, for any I ∈ A** and i ∈ L#p*, (I,tnp(i)) - (ti侮(I),i), t毎is continuous with respect to the a-(考, L2g79)-topology and c,(A*, A**)-topology･

Since the unit ball AI OfA is (丁(A**,A*)-dense in the unit ball of A**, for any y ∈ A and any element二r Of the unit ball of A**, we have

ulx,,u∈Al I(uy,p)I ≦ p(uu*)1/2p(y*y)1/2 ≦ =np(y)1I･

### l(y,px)I-I(xy,p)l- 1im

Therefore there exists an element i ∈ L#p* such that px -冗(i) and llEH ≦ 1･ Hence tp･7: I Ll･ ∈ A**, llxLl ≦ 1) is included in the image of the unit ball ofL3eJ under tr]p･ Since the unit ball ofL9eJ is J(L9eJ, L9ep)-compact, the balanced convex set (px I I ∈ A**満目≦ 1) is relatively compact with respect to the (7(A*, A**)-topology･ For an element y ∈ A**, the linear form yp: A** ∋ I r- (xy,p) belongs to (A**)*･ Let g be afilter on A converging to y ∈ A** with respect to the 7-(A**7 A*)-topology; then the image of g under the mapping ･LL L- ,LLP COnVergeS uniformly to yp on the unit ball of A**. Since, for any u ∈ A,叩∈ A*

### is J(A**,A*)-Continuous, yp is J(A**,A*)-continuous on the unit ball of A**･ Hence, by

the Banach theorem, yp is o-(A**,A*)-continuous on A**, that is, yp ∈ A*. Therefore the mapping A** ∋ I ‥ (xy,p) is J(A**,A*)-Continuous･ By Jordandecomposition, the mapping A** ∋ I i--+ xy ∈ A** is a-(A**,A*)-Continuous. Hence, for anyこr,y ∈ A**, we have

### (xy)* - y*X*･ Consequently, A** is a Banach *-algebra.

### Let (el,)i be an approximate identity of A, Let 1 be a cluster point of (eL)i With respect

to the J(A**,A*)-topology; then, for any I ∈ 4 we have lこr - Xl - I. Therefore we have

### Fundamental Theorems in W*-Algebras and the Kaplansky denslty theorem, TT

9### Let A be a C*-algebra and p a state ofA. p IS Self-adjoint in A**. It holds that, for any

### elements I and y of the unit ball of A**,

lp(y*y) -p(X*X)I ≦ lp(y*(y-I))l + lp((y-I)*X)l

- lp(y*(y-I))I+lp(♂(y-I))I

≦2 sup l(pa)(y-I)ト

Ila‖≦1

Since the set (pa L a ∈ A**,‖all ≦ 1) is a J(A*,A**)-compact balanced convex set, the

### function I H p(X*X) is continuous on the unit ball of A** with respect to the

71(-M**7-4*)-topology･ Hence p IS positive on A**･ We can define the seminorms pp and p芸on A**‥

pp(I)-p(X*X)1/2 and p;(I)-p(a)1/2

### Wc call the topology defined by all pp (resp･, all pp and p;) the J-Strong tOpOlogy (resp･,

### the J-StrOng* topology)･ By Jordan decomposition, the (7-Strong tOPOlogy is finer than

### the J(A**,A*)-topology･ pp and p; are continuousOn the unit ball with respect to the

### T(A**, A*)-topology･ Hence, for a c,-strongly* continuous linear form 4, on A**, the

### inter-section of ker4, and the unit ball is 71(A**,A*)-closed and s0 0-(A**,A*)-closed. By the

### Banach theorem, ker4, is c,(A**, A*)-closed, and hence 4, is o-(A**, A*)-Continuous, that is,

4, ∈ A*. Therefore the JIStrOng tOPOlogy and J-StrOng* topology are compatible with the

### duality (A**, A*)･ Hence, the unit ball of A** is J-Strongly closed and the unit ball of A is

### cTIStrOngly* dense in the unit ball of A**.

### Notice that A** has an identity.

### Lemma 3. Lei A be a C*-algebra and S(A) the state space of AI Then it holds that, for

### any seljladjoint element I of A**,

l回l- sup lp(I)I･ p∈S(A)

### Proof. Let I be a self-adjoint element of A** and 6 an arbitrary positive real number;

then there exists an element p ofA* such that p(I) ≧ =xH -6 and lLpll ≦ 1･ We havep(I) - 2~1(p+p*)(I). put4, - 2-1(p+p*) an°let ¢ - 4,+-中一beaJordandecomposition

### of 4); then we have

p(I) -4,(I) ≦悼+(I)l + 14,-(I)L

≦(日中+ll+l悼IH) sup lp(I)I-lL4,日 sup lp(I)I

qD∈S(A) p∈S(A)

### ≦ sup lp(I)l≦=緋

p∈S(A)10 Bulletin of the Institute ofNatural Sciences, Senshu Universlty No.38

### Lemma 4. Let A be a C*-algebra. Then, for any self-adjoint element I ofA**, we have

Hx2日-瞳‖2 and so 1回L - r(I), where r(I) denotes the spectral radius ofx･

Proof. By Lemma 3 and the Cauchy-Schwarz inequality, for any self-adjoint eleIIlentこr ∈

### A**, wc have

‖xH ≦ sup p(.T2)1/2≦植2111/2,

9∈S(A) and so Hx2日- H.7;‖2･ Therefore we have

r(I) - 1im lLx2nLl2ーn - llxH. n→()〇

### □

Let B be a commutative Banach *-subalgebra of A** and the rllapPlng B ∋ X一念∈ Co(0) the Gelfand representation of B･ Ifx and y are self-adjoint elements of B and金-夢,

### then we have I - y･ For, since the spectral radius in B coincides with the spectral radius

### in A**, we have

l匡-yH -r(I-y)-supli(LJ)-a(LJ)I -0

LJ∈il

and soこr-y.

### Lemma 5. LetA be a Banach algebra with an identity and assume thatA is the dual space

### of a normed space E and the multiplication in A is separately continuous with respect to the

o-(AE)-topology･ Let B and入be a subset ofA and a complex rmmber, respectively, such

that supy∈B H(^1 -y)1日< +∞･ Ifx belongs to the T(4E)-closure ofB, then we have 入¢ Sp(X･) and (入1 - I)-1 - 1imy→X,y∈B(Å1 - y)～1 with respect to the J(A,E)-topolo.qy･

### Proof. Let g be an ultrafilter on B which converges to I with respect to the

### T(4E)-topology. Since the image of an ultrafilter under a function is an ultrafi1ter base, there is

a limit a - limy,17(入1 - y)AI with respect to the 0-(A,E)-topology･ For any p ∈ E alld I ∈ 4 pz is in E and the mapplng A ∋ I L- PZ ∈ E is continuous with respect to the J(A,E)-topology and c,(E,A)-topology, so that (pz L llzH ≦ 1) is compact･ Hence we have limy,.7(入1 - y)~1((入1 - y) - (入1 - I)) - 0 with respect to the a-(A,E)-topology. Therefore we obtain 1 a(入1I) 0 and so a (入1X)1. Hence we have (入1I)JA1

-1imyーX,y｡B(入1 - y)~1 ロ

Lemma 6. LetA be a C*-algebra. ThenwehaveSp(X*X) ⊂ R+forevery elements ∈ A**･

Proof. Since the function I ‥ X*x is 0--strongly* continuous on the unit ball of A**, for any

### I in the unit ball of A**, X*x belongs to J-StrOng* closure of the positive portion of A and so

### belongs to the closure of the positive portion of A with respect to the 7-(A**, A*)-topology･

For any入¢ R十andpositive element y ∈ A, wc have LI(入1-y)~111 ≦ d(A,R+)Jl < +∞･

### Fundamental Theorems in W*-Algebras and the Kaplansky denslty theorem, II

llIf Sp(I) ⊂ R and B is a Banach subalgebra of A** containing I and 1, then we have SpB(I) - Sp(I)･ For, if SpB(I) ¢ R, then there is a number A ∈ SpB(I) such that d(A,R) - sup〃∈S｡B(I)d(FLフR) > 0･入is a boundary point of SpB(I) and so is a boundary point of Sp(I). Hence we have A ∈ Sp(I) ⊂ R, which is a contradiction. Therefore SpB(I) is included in R and so coincides with Sp(∬)･

### Let L99 and Al* denote the unit ball of the second dual A** of a C*-algebra A and the

set of all二r*x withこr ∈ A**, respectively.Lemma 7･ Let A be a C*-algebra and I a self-adjoint element ofA** with Sp(3;) ⊂ R+･

### Then there exists a unique self-adjoint element y ofA**, denoted by xl/2, such thai I - y2

and Sp(y) ⊂ R+･ Therefore Al* Coincides with the set of all self-adjoint elements I such that Sp(I) ⊂ R.I Furthermore, the function Al* n Lプラx r- xl/2 ∈ Al* n Lプis J-Strongly continuous.

Proof･ Let I be a self-adjoint element in A** with Sp(I) ⊂ R+ and llx= ≦ 1･ There exists a sequence (pn)n of polynomials with real coe侃cients such that

### lim sup lpn(i)-tl/21 -0.

n→∝'o<t<1### Let B be the commutative Banach *-subalgebra of A** generated by I and 1, and the

mapping B ∋ y一歩∈ C(f7) the Gelfand representation･ ByLemma 4, it holds that

Hpn(I) -pm(I)= - suplpn(金(LJ)) -Pm(金(LJ))I ≦ sup lpn(i) -pm(i)I.

LJ∈f7 0_<t_< 1

### Hence the sequence (pn(I))n is a Cauchy sequence and so converges in norm to some

self-adjoint element y. Since 9(LJ) - limnー∞Pn(i(LJ)) - i(LJ)1/2, we have y^2 - 92 -金and so y2 - I and Sp(y) - 9(0) ⊂ R+･ Ifz is a self-adjoint elemellt, I - Z2 and Sp(I) ⊂ R+, theII I COmmuteS With二r. Hence there is a commutative Banach *-subalgebra C containlng I,I and 1. Since y ∈ CフCOnSidering the Gelfand representation ofC, we have ∂ -金1/2 -乏### andsoy-2:.

Al* n Lブヨx r-i pn(I) ∈ A** is c,-strongly continuous and

sup JIpn(Xトxl/2日- sup suplpn(i(LJ)上意(LJ)1/2I ≦ sup Jpn(i)-tl/2J.

X∈Al*∩,y x∈Al*∩亡5P LJ∈0 0<t<1

### Since the limit of a sequence of continuous functions with respect to the topology of uniform

convergcncc is continuous, the function Al* n LSP ∋ I r- xl/2 ∈ Al* n LSP is J-Stronglycontinuous. □

12 Bulletin of the Institute of Natural Sciences, Senshu University No.38

### X∈Aand入∈C,

llx+^1日2 ≦ liminfHx+入eLLl2

i

≦ liminf(lLL･-XeLll + HxeL, +入eLH)2

i

- 1iminf llxeL +入e/LH2 - 1iminf H(xeL +入eL)*(xeL +入eL)ll

I, i

-liminfHe,,(I+^1)*(I+Al)eLl[ ≦ Ll(I+Al)*(I+^1)ll･

L

### The subalgebra A + Cl of A** is therefore a C*-algebra.

### Theorem 8. The second dual of a C*-algebra is a C*-algebra.

Proof･ Let A be a C*-algebra･ For any I ∈ Al*, we have Sp(1 +I) ⊂ [1,+∞) and so Sp((1+I)~1) ⊂ (0,1]. Hence, by Lemma4, we have H(1+I)~1日- r((1+I)~1) ≦ 1･ Therefore the function Al* ∋ I L---i (1 + I)~1 ∈ A** is J-Strongly continuous･ Since the function L90 ∋ I L- X*X ∈ Al* n L90 is o'-strongly* Continuous, for a positive natural number n, the function Lプヨx L- (1 +n(X*X)1/2)ll ∈ A** is J-StrOngly* continuous, in virtue of Lemma 7. Therefore the function Lアヨx r- I(n-ll + (X*X)1/2)-1 ∈ A** is continuous

### with respect to the J-StrOng* topology and o1-Strong tOPOlogy･ By considering spectrum Or

the above remark, for any I ∈ A, we have lLx(n~11 + (X*X)1/2)~111 ≦ 1. Hence we obtainllx(n~11 + (X*X)1/2)~1日≦ 1 for every I ∈ LSP. Since

### x - I(n-ll + (X*X)1/2)~1(X*X)1/2 - n-Ix(n~11 + (X*X)1/2)ll,

we have

順一X(n111 + (.7:*X)1/2)~1(X*X)1/2日≦ n~1.

### Therefore it follows that

HxH - 1im lLx(n-ll + (X*X)1/2)~1(X*X)1/2日

n→OC)

≦ lI(X*X)1/2日- llx*xHl/2,

so that l匿可l - lLxH2. consequently, A** is a C*-algebra･

2. IDENTITIES IN Ⅳ*-ALGEBRAS

### In ll], We used the projection of the second dual of a W*-algebra of norm one, however,

### in the followlng, We do not need the second dual of a W*-algebra.

### Theorem 9. Any W*-algebra has an identity.

### Proof. Let LM be a W*-algebra and (eL)i be its approximate identity･ There exists a cluster

### Fundamental Theorems in W*-Algebras and the Kaplansky denslty theorem, II

### 13

### obtain p(1) - 1 - 1im,,p(eL). Since the state space of LM is algebraically total in the dual

### space of LM, 1 is a limit of (el,)I, With respect to the J(-M, -4g*)-topology･ Therefore, for any

.7; ∈ LM and p ∈ Lノ材*, it holds that

(1X,p) - (1,叩) -lim(e,/,秤) - 1im(ELK,P) - (I,P),

i i

so that l二r - I. Similarly, we haveこrl - I, and hence, 1 is an identity ofLM. □

### Lemma 10. Let u4g be a W*-algebra andE the projection of the second dualLM** onto LAW

### of norrTWne. Then E is positive and so self-adjoint･

Proof. LAW** is a C*-algebra. Sinceフfor any state p ofL4, Poe(1) - 1, Poe is positive and

so E is positive. Therefore E is trivially self-adjoint. □

### Theorem ll. The involution in a W*-algebra LM is J-Weakly continuous.

Proof. Let E be the canonical projection of Lノ材** onto LM of norm one; then, by Lemma 107

### there is a commutative diagram as follows:

LAW** lnVOlution〉 LAW**

### El lE

### 葛閤iiiiiiiiii夢闇

1nVOlution

Since the involution in LM** is c,(LM**, LM*)-Continuousフthe involution in -修is J-Weakly continuous. □

### 3. THE KAPLANSKY DENSITY THEOREM

### By Thcorcm 8, wc immediately see the Kaplansky density theorem. Also, Wc can show

### the Kaplansky density theorem in virtue of polar decomposition･

Proposition 12･ Let u4W and L〟 be two W*-algebrtLS and ◎ a o1-Weakly continuous

求-//,,Ill,,I//,･I･I,//I､JJH,./ ･〝 /II/,, ･ I. 'r/l･ (,小(･〝日､ ,T-Il.,,I/.Lil/ I/,,-/ ,IIl･I //,, ,II,// I〃lII ,,/一小(･〝)

･･･///, /,I- Ir///I //J･ /IIl,I!/-)./-/I/, IIIl// I"lil ,･I･〝 I/I/I/,I･小･

Proof. Let i be the canonical mapping of LAW onto LAW/ ker◎; then there exists a *-isomor-phismせof u4g/ker◎ intoレ〟 such that ◎ -せ｡j･ Since -AW/ker◎ is a C*-algebra,せis

### an isometry. The image of the open unit ball of u4g under i coincides with the open unit

ball of LM/ker◎. Hence the image of the open unit ball of LAW under ◎ Coincides with the open unit ball of ◎(LM)･ Since the closed unit ball of -/身is (7-Weakly compact, the image of the closed unit ball of u4g under ◎ is J-Weakly compact and so coincides with the closed unit ball of ◎(JW). Therefore ◎(LAW) is J-Weakly closed･ □14 Bulletin of the Tnstitute of Natural Sciences, Senshu University No.38

Lemma 13･ Let LAW and V be as above andA a a-(LM,V)-dense 求-subalgebra ofL/材. Then

### A is J-Weakly dense in LM.

### Proof･ The self-adjoint portion As of A is dense in u4Ws with respect to the

7-(LM,V)-topology. For any complex number入¢ R and self-adjoint element y, we have

ll(入1-y)-111 ≦d(A,R)Ll < +∞,

where d(A,R) denotes the distance between入and R. By Lemma 5, we have (入1 1X)1 -1imy→X,y∈As(入1 I y｢1 for every I ∈ LAWs. Hence, (入1 - X)ー1 belongs to the J-Weak closure ofA+Cl andalsodoes入1I. Sincex (入11X･)(A(入1X･)~111) and入(入1I)~11 -1imy→X,y∈As(入1 - y)Aly, I belongs to the J-Weak closure of A. Therefore A is J-Weakly dense inー虜′. □

Theorem 14 (Kaplansky)･ Let LM and V be as above and A a 辛-subalgebra ofLM which,

/･､汀(.〟.1十'),//､=〝.〟. 'r//･I) ///I IIIl// I･･I//･･I.I /.､ T(.〟..〟.ト′/I//､=/I //1' Ill/I/ (,'II/ '･/●.〟.

### Proof. We may assume, without loss of generality, that A is a C*-algebra.Let id denote

the identity Inapping of A into LM and ◎ the transpose mapping of the restriction tidILM.; then ◎ is a continuous *-homomorphism of A** equipped with the J(A**, A*)-topology into LAW equipped with the cT-Weak topology. We regard ◎ as an extension of id. By Theorem 8, A** is a W*-algebra･ By Proposition 12, ◎(A**) is J-Weakly closed and the image of the unit ball of A** under ◎ coincides with the unit ball of ◎(A**)･ Since ◎(A**) is c,(LM,V)-dcnsc in LM, ◎(A**) coincides with LAW, in virtue of Lemma 13. Since the unit ball of A is

### cT(A**,A*)-dense in the unit ball of A**, the unit ball of A is cTIWeakly dense in the unit

ball of -/財and so T(LAW,LM.)-dense in the unit ball of LM. □

Proposition 15 (Polar Decomposition)･ Letレ冴be a W*-algebra. For any element I of

･〝. /// ,I/､/,II/,,/ I"/I/ ,,//, IJ,I/I//,I/ /ヾ…I/,//･I/ ,. III p〝 ､′′./I //川/.,･一車いI//,/ ,･,.

### -β(回)･

Proof･ Put vn - I(n~11十回)-1 for each positive natural number n; then we have llvnH ≦ 1. Since lvnl - (n~11 +回)I1回, (lvnl)n is increasing and so c,-strongly convergent. Since s(LEI)lvnl - lvnL, we have s(IxL)limn→∞ Lvnl - 1imn→∞ lvnl. Since Lxl一剛vnl - n~1L7'nl, We have lxl 回limn→∞卜由･ Hencewe have s(Lxl)(11imn→∞毎l) 0 and so limγけ∞ lvnl -S(回)･ Since vニvm - 1vnHvml, we have (vn -vm)*(vn -vm) - (LvnトIvT,もl)2. Hence (7'n)n

### is a Cauchy sequence with respect to the o1-Strong tOPOlogy. Since the unit ball of LM is

### complete with respect to the o1-Strong tOpOlogy, (vn)n converges J-Strongly to some clement

v∈亡M･ Sincex-un回-n~1vn, weobtainx-申l. Since抗)- limこr*vn- lim lxHvnl -回S(lxL)-回,

n→ 0く) n → ○く)

### Fundamental Theorems in W*-Algebras and the KapZansky denslty theorem, II

### 15

### Let w be a partial isometry in LM such that I - wtxL and w*W - S(lxI); then we have

### v- 1im I(n-ll+回)~1- 1imwlxL(n~11十回)Jl-ws(回)-W.

n一斗∝) n→〔X〕

ii

### Another proof of the Kaplansky density theorem. Let g be afilter on A converging to

a partial isometry v ∈ LM with respect to the 7-(LAW,V)-topology. It holds that, for any

x∈レ/♂,

### (1+xx*)J1- (1+vv*)ll - (1+xx*)Jl(vv*-xx*)(1+vv*)-1

### - ((1+xx*)llx(XIV)* + (1+xx*)-1(I-V)V*)(1+vv*)~1

Since H2(1 +xT*)~1xH ≦ 1, we have (1 +vv*)ll - 1imx,17(1 +xx*)~11 Since limx,17(1 +### xx*)-1(I-V) - 0, we obtain ,U - 2(1+vv*)~1V - limx,172(1+xx*)~lx･ Hence v belongs to

the J-Weak closureオ百才of AnLSP. In particular, any projection in LM belongs t0万百タ.Hence, by spectral decomposition, any positive element in ｡プbelongs to A n L577. Therefore, by polar decomposition, we obtain L90 ⊂ A n ｡ア.