ON SEMI-DISCRETE, SEMI-CONTINUOUS REAL W ∗-ALGEBRAS

REAL W

-ALGEBRAS

CORNELIU CONSTANTINESCU

We give a new proof of Theorem 6 in [1] stating that a semi-discrete, semi- continuos real W^∗-algebra (Definition 2) is isomorphic to a real W^∗-algebra of the formC(T,H) (the set of continuous H-valued functions onT), where T is a compact hyperstonian space andHis the field of quaternions.

AMS 2000 Subject Classification: 46L10, 46L35.

Key words: realW^∗-algebra.

1. INTRODUCTION

In general we use the notation and terminology of [2]. We give below a list of some notation used in this paper.

1. R(resp. H) denotes the field of real numbers (resp. quaternions). If T is a locally compact space, then C(T,R) (resp. C(T,H)) denotes the real C^∗-algebra of continuous R-valued (resp. H-valued) functions onT.

2. We put

ReE :={x∈E |x^∗ =x}, E^c :={x∈E |y∈E ⇒xy=yx}, PrE:={p∈ReE |p² =p}

for every C^∗-algebraE. If E is unital, then 1E denotes its unity.

3. A realC^∗-algebraEis called a GelfandC^∗-algebra ifE= ReE=E^c. 4. IfE is aW^∗-algebra, then ¨E denotes its predual. If (xi)i∈I is a family inE then we denote by

E¨

P

i∈I

xi the element ofE (if it exists) determined by the relation

a∈E¨ ⇒

* E¨

X

i∈I

xi, a +

=X

i∈I

hxi, ai.

Lemma 1. Let E be a C^∗-algebra such that ReE is commutative. Then (a) ReE⊂E^c;

(b)x∈E⇒x^∗x=xx^∗;

REV. ROUMAINE MATH. PURES APPL.,54(2009),2, 125–129

(c) p, q∈PrE, p∼q ⇒p=q;

(d)if E is not commutative, then there are x, y∈E with x^∗ =−x, y^∗ =−y, xy 6=yx.

Proof. (a)

Step1. x∈ReE,y∈E,y^∗ =−y⇒xy =yx.

Putz:=xy−yx.We have successively

z^∗ =y^∗x^∗−x^∗y^∗ =−yx+xy =z, z∈ReE,

x²y−xyx=x(xy−yx) =xz=zx= (xy−yx)x=xyx−yx², x²y+yx² = 2xyx.

Since y²∈ReE, we have xy²=y²x and

z² = (xy−yx)(xy−yx) =xyxy−xy²x−yx²y+yxyx=

= 1

2(x²y+yx²)y−x²y²−yx²y+1

2y(x²y+yx²) = 0.

Thus,z= 0 and xy =yx.

Step2. ReE⊂E^c

Let x∈ReE andy ∈E. Then (y−y^∗)^∗=−(y−y^∗) so, by Step 1, x(y−y^∗) = (y−y^∗)x.

Since y+y^∗ ∈ReE,

x(y+y^∗) = (y+y^∗)x, xy=yx, x∈E^c. (b) By (a), since x+x^∗ ∈ReE,

x²+xx^∗=x(x+x^∗) = (x+x^∗)x=x²+x^∗x, xx^∗ =x^∗x.

(c) This follows from (b).

(d) SinceE is not commutative, there areu, v∈E withuv 6=vu. Put x:= 1

2(u−u^∗), y:= 1

2(v−v^∗), a:= 1

2(u+u^∗), b:= 1

2(v+v^∗).

Then

x^∗ =−x, y^∗=−y, a, b∈ReE, x=u−a, y=v−b.

By (a),

xy−yx= (u−a)(v−b)−(v−b)(u−a) =

=uv−ub−av+ab−vu+va+bu−ba=uv−vu6= 0.

Definition 2 ([3], Definition 8.1.5). Let E be a real W^∗-algebra. A p∈PrEis calledsemi-abelianifp(ReE)pis commutative. Eis calledsemi- discrete if for every z ∈ PrE^c \ {0} there is a semi-abelian p ∈ PrE\ {0}

withp≤z. Eis calledsemi-continuousifp∈PrE, pabelian impliesp= 0.

Proposition 3. Let E be a real W^∗-algebra such that ReE is commu- tative.

(a) ReE is aW^∗-subalgebra of E and it is isomorphic to C(T,R), where T is the spectrum of ReE and a hyperstonian space.

(b) If E is not commutative then there are i, j ∈E and p ∈PrE\ {0}

with

i^∗=−i, j^∗=−j, i^∗i=j^∗j=p, ij =−ji.

Proof. (a). ReE is obviously the greatest Gelfand C^∗-subalgebra of E.

By [2], Corollary 4.4.4.12 b, it is a W^∗-subalgebra of E. The last assertion follows from [2], Corollary 4.4.1.10.

(b) By Lemma 1 (d), there arex, y∈E with x^∗=−x, y^∗ =−y, xy6=yx.

Put

a:=xy−yx, b:=xy+yx, c:=xyxy+yxyx.

Then

a^∗=−a6= 0, b, c∈ReE.

By Lemma 1 (a),

a^∗a= (yx−xy)(xy−yx) =yx²y−yxyx−xyxy+xy²x= 2x²y²−c, b²= (xy+yx)(xy+yx) =xyxy+xy²x+yx²y+yxyx=x²y²+c, so,

06=a^∗a= 4x^∗xy^∗y−b²≤4x^∗xy^∗y.

Using (a), we identify ReE with C(T,R) and find p ∈ PrE\ {0} and ε > 0 with

{p >0} ⊂ {a^∗a > ε} ∩ {x^∗x > ε}.

Define

f :T →R, t7→

( √ ₁

(x^∗x)(t) ifp(t) = 1 0 ifp(t) = 0 , g:T →R, t7→

( 2

q(x^∗x)(t)

(a^∗a)(t) ifp(t) = 1 0 ifp(t) = 0

,

h:T →R, t7→

( ₁

√

(a^∗ax^∗x)(t) ifp(t) = 1 0 ifp(t) = 0 , i:=f x, j:=bhx+gy.

Then f, g, h∈ReE,hx^∗x= ¹₂g, and, by Lemma 1 (a), i^∗=−i, j^∗ =−j, i^∗i=f²x^∗x=p,

j^∗j= (bhx^∗+gy^∗)(bhx+gy) =b²h²x^∗x+bhgx^∗y+gbhy^∗x+g²y^∗y=

=b²h²x^∗x−2b²h²x^∗x+ 4h²(x^∗x)²y^∗y =x^∗xh²(4x^∗xy^∗y−b²) =x^∗xh²a^∗a=p, ij+ji=f bhx²+f gxy+bhf x²+gf yx=

= 2bhf x²+f gb=−2bhf x^∗x+ 2f bhx^∗x= 0.

Theorem 4 ([1], Theorem 6). The statements below are equivalent for every real W^∗-algebra E.

(a)E is semi-discrete and semi-continuous.

(b)E is isomorphic toC(T,H), whereT is a compact hyperstonian space.

Proof. (a)⇒(b). Let (p_λ)λ∈Λbe a maximal orthogonal family in PrE\ {0}such that for every λ∈Λ there arei_λ, j_λ ∈E with

i^∗_λ =−i_λ, j_λ^∗=−j_λ, i^∗_λi_λ =j_λ^∗j_λ =p_λ, i_λj_λ=−j_λi_λ. Put (see [2], Proposition 5.6.3.20 b⇒a)

p:=

E¨

X

λ∈Λ

p_λ, i:=

E¨

X

λ∈Λ

i_λ, j:=

E¨

X

λ∈Λ

j_λ, k:=ij.

If p 6= 1_E then by Proposition 3 (b) there are p⁰ ∈ PrE \ {0} and i⁰, j⁰ ∈ E with

p⁰ ≤1_E−p, i^0∗=−i⁰, j^0∗=−j⁰, i^0∗i⁰=j^0∗j⁰ =p⁰, i⁰j⁰ =−j⁰i⁰ and this contradicts the maximality of (p_λ)λ∈Λ.

By [2], Proposition 5.6.3.20 c,

i^∗ =−i, j^∗=−j, i^∗i=j^∗j= 1_E, ij=−ji.

It follows that

k^∗ =−k, jk=−kj=i, ki=−ik=j, k^∗k= 1_E. Thus, we may identify C(T,H), where T is the spectrum of ReE, with

{a1_E +bi+cj+dk|a, b, c, d∈ReE}.

By Proposition 3 (a), T is a compact hyperstonian space.

Letx∈E. We have to show that x∈ C(T,H). Since ReE ⊂ C(T,R)⊂ C(T,H), we may assume x^∗ =−x. Put

a:= 1

2(ix+xi), b:= 1

2(jx+xj), c:=ai+bj+x, d:=ijc.

Then a, b∈ReE, c^∗ =−c, and we deduce (by Lemma 1 (a)) that ic+ci=ai²+bij+ix+ai²+bji+xi=−2a+ 2a= 0,

jc+cj =aji+bj²+jx+aij+bj²+xj=−2b+ 2b= 0, d^∗ =c^∗j^∗i^∗=−cji=ijc=d, d∈ReE,

d=aiji+bij²+ijx=aj−bi+kx, kd=−ai−bj−x, x=−ai−bj−kd∈ C(T,H).

(b)⇒ (a) is obvious.

REFERENCES

[1] Sh.A. Ayupov, A.A. Rakhimov and A.Kh. Abduvaitov,RealW^∗-algebras with Abelian Hermitian part.Math. Notes71(2002),3, 432–435.

[2] Corneliu Constantinescu,C^∗-Algebras. Elsevier, 2001.

[3] Bingren Li,Real Operator Algebras. World Scientific, 2003.

Received 9 November 2007 Bodenacherstr. 53

CH 8121 Benglen constant@math.ethz.ch