REAL OPERATOR ALGEBRAS
RE4M 
oNaWfioR ALGEBRAS
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REAL OPERATOR ALGEBRAS
Chinese Academy of Sciences, P.R.Chinc
World Scientific New Jersey • London • Singapore • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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ISBN 9812383808
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Preface
The theory of operator algebras is generally considered over the field of complex numbers and in the complex Hilbert spaces. So it is a natural and interesting problem: How is in the field of real numbers? Which results are still true in the real case? Which results don't hold in the real case? And which results are needed to change some respects and forms? Up to now, the theory of operator algebras over the field of real numbers seems not to be introduced systematically and sufficiently. The material in [17, 26] seems few and not enough. Moreover, we can also find some respects of the real case in some papers (see References). Similarly to the complex case, a real operator algebra, precisely speaking, is a * algebra consisting of bounded (real) linear operators on a real Hilbert space, i.e., a * subalgebra of B(H), where B(H) is the collection of all bounded (real) linear operators on a real Hilbert space H, and * is the adjoint of operators. Since it is an infinite dimensional object (generally, H is infinite dimensional), so for studying it we must ask that it is closed under some topology. Similarly to the complex case, we find that the closures of real operator algebras with respect to usual locally convex linear topologies in B(H) are just two classes: weak closure and uniform closure. Hence, we need mainly to study the weakly closed real operator algebras (real Von Neumann algebras or real W*algebras) and the uniformly closed real operator algebras (real C*algebras). The aim of this book is to set up the fundamentals of real operator algebras and to give a systematic discussion for real operator algebras. There are the systematic treatments on complex Banach algebras and complex operator algebras in [1] and [26] respectively. In some sense, this book is a real analogue of [1] and [26] on Banach algebras and operator algebras. Moreover, we shall put other results on real operator algebras into this book
VI
Real Operator Algebras
as more as possible, and give a uniform treatment. Since our treatment is from beginning (real Banach and Hilbert spaces, real Banach algebras, real Banach * algebras, real C*algebras and W*algebras, and etc.), and some basic facts are given , we can get some results on real operator algebras easily. However, for the aim of systematic discussion, many results in this book seem to be trivial (i.e., to be a simple movement from the complex case). In this case, we shall just give the statements, and not give their proofs. Readers can find the similar proofs from [1] and [26]. Generally, there are two methods for proving the results on real operator algebras: To move and to change the proofs of the complex case into the real case; and first to go to the complexification and then to go back to the real case. Sometimes just one method is available and another method is not available. Sometimes we must use these two methods simultaneously. In this book, we shall describe the differences between the complex case and the real case. Moreover, since A = (B, —) (real C*algebra, see Chapter 5) and M = (N, —) (real W*algebra, see Chapter 6), we shall stress the bar "" operation throughout this book. This book is also an introduction for real operator algebras, written in a selfcontained manner. For reading i t , you just have the general knowledge of Banach algebras and operator algebras, e.g., [1] and [26]. Moreover, we shall use the results in [1] and [26] freely, and not give their proofs. This book consists of 10 Chapters. Chapter 1 is the preliminaries. Section 1.1 discusses the complexifications of real Banach spaces and real Hilbert spaces. In particular, we let f + ir}\\ — £ — i77(V£, rj), i.e., the bar "" operation is an isometry. Then we can get Propositions 1.1.4 and 1.1.5, and it is important for this book. Section 1.2 is the spectral decomposition in real Hilbert spaces. For (real) normal operators, we need use spectral pair. And for (real) selfadjoint operators, the spectral decomposition theorem is the same as in the complex case. Chapter 2 contains the complexifications of real Banach algebras, spectrum, divisible real Banach algebras, radical, Arens products, abelian case, and etc. The complexification of a real Banach algebra can be chosen to be a complex Banach algebra, and the bar "" operation is still isometric. The spectrum of an element must be denned in the complexification, and it is symmetric under the complex conjugation. Proposition 2.4.6 gives a basic fact (cr(x) n l = {0},Vx € R(A)), and we shall use it later. About Arens products, we have Proposition 2.6.4 and etc. Then the regularity of
Preface
vn
real C*algebras can be obtained easily in Chapter 5. Section 2.7 is the Gelfand theory for abelian real Banach algebras. In particular, we give a systematic discussion for the general case (with or without identity). Chapter 3 is real Banach * algebras. Lemma 3.1.3 gives a basic fact ([f(A)] D AK) About abelian case, Theorem 3.2.3 is similar to the complex case, but we must put the hermitian condition. Sections 3.3, 3.4 and 3.5 are GNS construction, * representations, and * radical. Section 3.6 discusses symmetric real Banach * algebras. In particular, the right form of Ptak's theory in the real case is given. Chapter 4 is the fundamentals of real Von Neumann algebras. It is a movement of [26, Chapter 1]. Of course, there are many differences between the complex case and the real case, for examples, [P{M)\ = MJI(C M generally), [U(M)] C M( generally, but \U(M)] = M), and etc. Proposition 4.3.3 (M c , = Mt+iM„) seems interesting and useful. Moreover, the important Von Neumann's double commutation theorem and Kaplansky's density theorem and etc. are still true in the real case. Chapter 5 is the fundamentals of real C*algebras. We use the complexification to define real C*algebras, and it is equivalent to the definition in [3, 17]. In section 5.3, although ntransitivity (n > 2) is not true in the real case generally, but we can still prove that 1transitivity holds and a topologically irreducible * representation is also algebraically irreducible for a real C*algebra. In section 5.5, we point out that any real C*algebra is regular, and the Arens product in its bidual is the multiplication of operators indeed. The uniqueness of * operation in any real C*algebra is obtained in section 5.6. Section 5.7 is the structure theorem of finite dimensional real C*algebras, and a method of proof in the theory of operator algebras is given . Section 5.8 is the enveloping real C*algebra of a hermitian real Banach * algebra, and it is a continuity of section 3.6. Chapter 6 is real Walgebras. It is the abstraction of real Von Neumann algebras, and similarly to Chapter 5 we use the complexification to define real Walgebras. There is also an equivalent definition similar to Sakai's theorem . Section 6.3 discusses abelian real W*algebras and theorem 6.3.6 seems very interesting . Unitaries and partial isometries in real W*algebras are studied in section 6.4. Gelfand Naimark conjecture is very famous, and it is a basic problem for the theory of C*algebras. Its real analogue is studied in Chapter 7. Similarly to complex case, we have an affirmative answer for GelfandNaimark conjecture in the real case. These results seem very interesting, and in particular, Theorem 7.2.4 is remarkable. Moreover, the technique of
VU1
Real Operator Algebras
"sin" and "cos" is introduced here. Of course, compare with the complex case, there are many open questions. Chapter 8 discusses the classification of real W*algebras. It is a real analogue of Von Neumann  Murray theory. About first classification (dimension theory), we can study it with the complexification. But for second classification, the real case is more complicated. We have semiabelian projections, semidiscrete real V7*algebras, semicontinuous real W*algebras and etc. Here, results are very few, and there are many open questions. Chapter 9 is a real analogue of reduction theory. The proofs of many results seem to be simple, and the important point is the systematic creation of concepts and notations. In particular, Theorem 9.4.2 seems very interesting compare with the complex case. Chapter 10 is an introduction for (AF) real C*algebras. An equivalent analytic definition is proved, and it is more complicated than the complex case since finite dimensional real C*algebras have complicated structure. Moreover, more results on (AF) real C*algebras (e.g., Bratteli diagrams, iftheory and etc.) can be found from References. Moreover, a real C*algebra (or a real von Neumann algebra) can be regarded as a complex C* algebra (or complex von Neumann algebra) with an appropriate conjugation (just look at the complexification) or an appropriate * antiautomorphism. In general, we may have two distinct real C* algebras (or real von Neumann algebras) whose complexifications are the same. An easy example is the real quaternion ring H and M2(R). They are all noncommutative 4dimensional real C*algebras (and real von Neumann algebras), but their complexification should be the unique 4dimensional complex C*algebra (and complex von Neumannn algebra) M2(C) (see Chapter 10). There is a number of research work of the * antiautomorphisms on complex von Neumann algebras or complex C*algebras by E. Stormer, P. J. Stacey, T. Giordano and etc. ([14, 5659], also see some references in H. Schroder's book [60]). In particular, E. Stormer proved the analogue of Connes' result that a type 7/jfactor is injective iff it is hyperfinite, for real von Neumannn algebras ([58]). A complex operator algebra can be expressed as the complexification of some real operator algebra if and only if it has a * antiautomorphism. It is still a (very interesting) open question if every complex operator algebra has a * antiautomorphism, i.e. it is an open question if every complex operator algebra is a complexification of some real one. Though some special cases are known (see Stormer's paper).
Preface
IX
Finally, the author acknowledges gratefully the supports of the NSF of China and his home Institute. And the author is also very grateful to Professor ZhongJin Ruan for his recommendation. Bingren Li
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Contents
Preface
v
1.
Real Banach and Hilbert Spaces
1
1.1 Complexification of real Banach and Hilbert spaces 1.2 Spectral decomposition theorem in real Hilbert spaces . . .
1 9
2.
3.
4.
Real Banach Algebras
15
2.1 Definition and complexification 2.2 Divisible real Banach algebras 2.3 The topological group of invertible elements and its principal component 2.4 Radical 2.5 Functional calculus 2.6 Arens products 2.7 Abelian real Banach algebras
15 20
Real Banach * Algebras
37
3.1 3.2 3.3 3.4
Some basic lemmas Abelian real Banach * algebras Positive linear functionals and GNS construction * Representations and topologically irreducible * representations 3.5 * Radical 3.6 Symmetric real Banach * algebras
37 40 42
Fundamentals of Real Von Neumann Algebras
59
xi
21 22 25 26 28
45 48 51
xii
Real Operator Algebras
4.1 4.2 4.3 4.4
5.
6.
Banach spaces of operators on a real Hilbert space Locally convex topologies in B{H) Von Neumann's double commutation theorem Kaplansky's density theorem, tensor product commutation theorem, and comparison of projections 4.5 Positive linear functionals 4.6 ^Finite real VN algebras
68 72 76
Fundamentals of Real C*Algebras
77
5.1 Definition and basic properties 5.2 Positive functionals and equivalent definition of real C*algebras 5.3 Pure real states, their left kernels, and irreducible * representations 5.4 Ideals, quotient algebras and extreme points 5.5 The bidual of a real C*algebra 5.6 The uniqueness of * operation 5.7 Finitedimensional real C*algebras 5.8 The enveloping real C*algebra of a hermitian real Banach * algebra 5.9 * Representations of abelian real C*algebras
77
Real Vf*Algebras 6.1 6.2 6.3 6.4
7.
8.
59 61 63
83 88 98 99 104 108 110 120 123
Definition and basic properties 123 Normal linear functionals and singular linear functionals . . 130 Abelian real W*algebras 132 Unitaries and partial isometries 141
GelfandNaimark Conjecture in the Real Case
147
7.1 Real C*equivalent algebras 7.2 The closed unit ball of a unital real C*algebra 7.3 GelfandNaimark conjecture in the real case
147 152 161
Classification of Real W*Algebras
167
8.1 8.2 8.3 8.4
167 172 175 176
Classification of real W*algebras Finite real W*algebras Properly infinite real W*algebras Semifinite real VF*algebras
Contents
9.
10.
xiii
8.5 Purely infinite (type III) real Walgebras 8.6 Properties on other classes of real VF*algebras 8.7 Real factors and tensor products
183 184 188
Real Reduction Theory
191
9.1 9.2 9.3 9.4
191 197 200 205
Real Real Real Real
measurable fields of Hilbert spaces measurable fields of operators measurable fields of VN algebras reduction theory
(AF) Real C*Algebras 10.1 Standard matrix unit 10.2 Technical lemmas 10.3 Definition and basic properties
211 211 218 228
Bibliography
233
Notation Index
237
Index
239
Chapter 1
Real Banach and Hilbert Spaces
1.1
Complexification of real Banach and Hilbert spaces
Let X be a real Banach space. Then Xc = X + iX becomes naturally a complex linear space. First we want to ask the question: is there a norm on Xc which makes Xc a (complex) Banach space and induces the original norm on X. The answer is affirmative, and there are infinitely many ways to do so. (1) Let
\\t + iv\\ = su P {(/(0 2 + f(v)2^ \f e x\ H/II = l},V£,v e X, where X* denotes the continuous dual of X. Clearly, (Xc,  • ) will satisfy our requirements, and
\\Z + iv\\ =
\\tivl
max(£, M) < U + iv\\ < (Uf
+ IMI 2 ) 1 ,
^,V
e X.
(2) For 1 < p < +oo, let  • \v be the £ p norm on Xc such that (Xc, \ • \p) is a real Banach space and X,iX are closed (real) linear subspaces of Xc,i.e.,
ie+*7ip = (iKiip + N r ) * ,
I
£j = £2 cos 0 — 7/2 sin 0,
T/2
= £2 sin 0 + 772 cos 0.
Then we need to show
ice;+«?i) + (e 2 +»v 2 )ii, < lei+Wi\ P + ie 2 +«? 2 i P But ((Xc, I • \p) is a real Banach space, so the last inequality is obvious. Definition 1.1.1. Let X be a real Banach space, and Xc = X+iX. {Xc,  • ) is called a complexification of X, if (Xc,  • ) is a complex Banach space,  \X is the original norm of X(i.e.,£+iO = f,Vf € X), and£ + ir7 = l£"/1l,Vf,77€X. In this case, we have max(£, r7) < f + J??,V£,?? € X. In fact, lltll < (lle + iv\\ + lie  iv\\) = lie + ^11 Similarly, ry < £ + ir,\\. Remark. The condition "f + ir)\\ = \\£  irj\\, Vf,77 € X " in the Definition 1.1.1 of a complexification is very important for our purpose (e.g., see Propositions 1.1.4, 1.1.5 and etc.). By the above discussion, we have the following. T h e o r e m 1.1.2. For any real Banach space, there is a unique (up to equivalece) complexification of it.
Real Banach and Hilbert Spaces
3
Definition 1.1.3. Let X be a real Banach space, and Xc be a complexification of X. Define bar operation— : Xc —> Xc and — : X* —> X* as follows:
T+*i = Ziri,
/(&) = /&)>
V£,r,€X,tceXc,feX*c. Clearly, the operations "" are conjugate linear, isometric, and —2 = id; X = {(ce Xc\(c = &}; and if / e X*c, then J = f ^ f(Q e l , V ( € X. Proposition 1.1.4. Let X be a real Banach space, and Xc be a complexification of X. (i)Iffe X* andj = / , then f\X G X*, and \\f\X\\ = /. (ii) For any g 6 X*, denote gc(£ + in) = g(£) + «#(??), V£, n G X. Then gc G X*,g~c = gc, and \\g\\ = \\gc\\ In particular, if f £ X* and f = f, then
U\x)c = f. (Hi) In the sense of (ii), X* can be isometrically embedded into X*,X* = {/ e X*\f = f}, and X* = X*+iX* is a complexification of X*. Moreover, f + ig = f — ig,Vf,g € X*. Proof (i) Since /(£) e R,V£ e X, it follows that f\X e X*. Clearly, \\f\X\\ < \\f\\. Now for any e > 0, we can find £, n € X such that £ + M J   < 1 ,
aDd\\f\\\l,I' € A} will be a normalized orthogonal basis of Kr, and (t*t)?ei = A;ej, (t*t)*ieii = \\iei', V/,/' e A. Thus t € T(Kr) and \\t\\itKr = 2]r,A; = 2ii,K. By Remark following Proposition 1.1.6, we have the following diagram: B(K) I tTK T{K)
B(K)r =» 4 Retrx T(K)r
c C
B{Kr) I tixr T(Kr).
For any a e B(K) and t € T(tf), tr/f r .(ai) = X],(ate;,e() r + = 2Y,i(atei,ei)r
References.
1.2
^2l,(atiev,iei>)r =
2RetiK(at).
[1], [26], [27], [30], [42], [43], [50], [51], [52], [55].
Spectral decomposition theorem in real Hilbert spaces
Definition 1.2.1. Let H be a real Hilbert space, {ei(), e 2 ()} is called a spectral pair over C, if for any Borel subset A of C, ei(A) and e 2 (A) e B(H) and satisfy the following: (i) ej() is countably additive in strong operator topology of B(H), j = 1,2. (ii) ei(A)* = ei(A),e 2 (A)* =  e 2 ( A ) , V Borel subset A c C.
10
Real Operator
Algebras
(iii) ei(A) = ei(A),e 2 (A) =  e 2 ( A ) , V Borel subset A C C, where "" is the complex conjugation, (iv) d ( A i n A 2 ) = e 1 (Ai)ei(A 2 )  e 2 (Ai)e 2 (A 2 ), e 2 (Ai n A 2 )  c a (Ai)ei(A 2 ) + ci(Ai)ea(A 2 ), V Borel subset A a , A 2 C C. (v) e i ( C )  1, el{cj>)  e a (0) = e 2 (C) = 0. Proposition 1.2.2. Let H be a real Hilbert space, and ej(A) G B(H), V Borel subset A ofC, j — 1,2. Then {ei(),e 2 ()} is a spectral pair overC, if and only if, in Hc = H+iH, e() = (ei() + «e2()) is a spectral measure ( [52]) over C and e(A) = e(A), V Borel subset A c t Moreover, i/{ei(),e 2 ()} is a spectral pair over C, then (1) ej(A) < 1, V Borel subset A C C, j = 1,2. (2) e i ( A ) = e i ( A ) 2  e 2 (A) 2 , e 2 (A) = e i ( A ) e 2 ( A ) + e 2 ( A ) e i ( A ) , V Borel subset A C C. (3) For any Borel partition {A fc l < k < m} ofC, {Afcl < k < m} C R and ( e f f with £ < 1, we have m
\\Y.\kei(Ak)t\\
• e() is countably additive in strong operator topology of B(HC). (ii) of Definition 1.2.1  e(A)* = e(A), V Borel subset A C C. (iii) of Definition 1.2.1 e(A) = e(A), V Borel subset A C C. (iv) of Definition 1.2.1 • e(Ai n A 2 ) = e(Ai)e(A 2 ), V Borel subsets A i , A 2 C C. (v) of Definition 1.2.1 e() = 0 and e(C) = 1. (iv) of Definition 1.2.1 = » (2) e(A) 2 = e(A), V Borel subset A C C. Therefore, {ei(),e 2 ()} is a spectral pair, if and only if, in Hc e() is a spectral measure over C and e(A) = e(A), V Borel subset A c C. Now let {ei(),e 2 ()} be a spectral pair over C. Then e() is a spectral measure. In particular, e(A) < 1, V Borel subset A C C.
Real Banach and Hilbert Spaces
11
From Definition 1.1.1 and Proposition 1.1.11, we have e.,(A) < e(A) < 1, Borel subset A C C,
j = 1,2.
Let {Afcl < k < m} be a Borel partition of C, {Afcl < k < m} c R, and {, € H with £ < 1. Then II E r = i Afcei(AfeKH2 +  E L i Afce2(Afc)e(2 = II E L i A fcei (A fc )e + i E L i A fe e 2 (A,)^ 2 = II Er=x A*e(Afc)£H2 = E L i A!le(A fc K 2 < maxi< fc < m Afc2 E r = i lle(A*;)£ll2 ^ maxi   a + *6p, 15
Va,b € A.
16
Real Operator
Algebras
On the other hand,  • p ~  • i on Ac (see section 1.1 and see Ac as a real Banach space). Hence there are constants Kp,K'p > 0 such that ^ ( N + I N ) < I  O + ^ I I P < ^ ( I H I + 6), Va, b € A. Further, a + i&^=sup{(ac6d) + i(ad+&c) p  c + id p < 1} < Kpsup{ac
 bd\\ + \\ad + bd\\ \ \\c + id\\p < 1}
< K'p(\\a\\ + 6)sup{c + d \\\c + id\\p < 1} < K'pKp\\\a\\
+ ll&ll) < KpK2\\a
+ ib\\p,
Va,& S A. Therefore,  • ' ~  • p on Ac, and (.Ac,  *  p ) is a (complex) Banach algebra. By c + id\\p = \\c — idp(Vc, d G .A), we also have o + » 6   ; =   o  t 6   ; ,
Va,beA.
Moreover, for any a € A \\a\\ = \\a\\p < \\a\\'p = sup{a(c+id) p  c + id p < 1} = c" 1 sup c+id  <j sup0 £R (accos0 — adsin0 p + acsin# + a d cos0 p )p < llall su Pc+id p A**), then mf af,fm = fa,Wf€A*. Definition 2.6.1. as follows:
The first and second Arens products in A** are defined (mn)(f) (mn)(f)
V/
=
=
m(nf),
=
n(fm),
€A*,m,n€A**. It is easy to see that A** will be a real (or complex) Banach algebra with first or second Arens product, and A is a real (or complex) Banach subalgebra oi A**.
Real Banach
27
Algebras
Definition 2.6.2. A real (or complex) Banach algebra A is said to be regular, if mn = mn, Vro, n G A**. Proposition 2.6.3. Let A be a real (or complex) Banach algebra. Then the following statements are equivalent: (1) A is regular. (2) For any fixed m 6 i**, the map x—+mx:A**^
A**
is o(A**, A*)continuous (clearly, the map x —> xm : A** —> A** is automatically a(A**, A*)continuous). (3) For any fixed n G A**, the map £ —> z • n : A** —+ A** is a(A*'*, A*)continuous (clearly, the map x —> n • x : A** —> A** is automatically a(A**, A*)continuous). Proof. By Definition 2.6.2, (1) ==> (2) is obvious. (2) = • (1). By assumption, for any / G A* and m € A**, (m)(f) will be a a{A**, A*)continuous linear functional on A**. Then by [26, Appendix], there is a linear map Tm : A* —> A* such that mn(f) = n(Tmf), Now (T r o /)(a) = a(Tmf) A, i.e.,Tmf = fm. Thus
=
(mo)(/)
VeA**. =
m(af)
{mn)(f) = n{fm) = (m • n ) ( / ) ,

(fm){a),Va
6
V/ e A*
or mn = m • n,Vm, n G A**, and A is regular. The proof of (1) (3) is similar to that of (1) (2).
Q.E.D.
Let A be a real Banach algebra, and Ac ~ A+iA be a complexification of A as Banach algebra. By Proposition 1.1.4, A* = A*+iA* and A** = A**+iA** are the complexifications of A* and A** as Banach spaces respectively. The "" operations in A* and A** are induced by the "" operations in Ac and A* respectively (see Definition 1.1.3). Moreover, we also have first and second Arens products in A**. Clearly, the first and second Arens products in A**, are the natural extensions of the first and
28
Real Operator Algebras
second Arens products in A** to A** respectively. Therefore, we have the following. Proposition 2.6.4. Let A be a real Banach algebra, and Ac = A+iA be a complexification of A (see Definition 2.1.2). Then A is regular, if and only if, Ac is regular. Moreover, in this case Arens product in A** is the natural extension of Arens product in A**. References.
2.7
[4], [30].
Abelian real Banach algebras
Let A be an abelian (commutative) real Banach algebra, Ac = A+iA be a complexification of A, and fic be the spectral space of Ac. Definition 2.7.1. fl — il(A) = {p\A\p £ fic} is called the spectral space of A. In other words, fi is the set of all nonzero complex valued multiplicative (real) linear functionals on A. The "" operation in Ac can be transferred onto il, i.e., we can define p~(a) = p(a),
Va e A,
or p(x) = p(x),
Vz € Ac,
Vp€$l Theorem 2.7.2. Let A be an abelian real Banach algebra, and ft be its spectral space. Then (1) p is continuous on A, and \\p\\ < 1 (notice p# A* generally), Vp € $1 Moreover, p = p p € A* p(Vp € ft) is a homeomorphism of fi with period 2. (4) o~(a) = {p(a)\p € fi} (when A has an identity), or a(a) — {0} U {p{a)\p € fi} (when A has no identity), andr(a) — sup{p(a) \p € fi},Va € A.
Real Banach
Algebras
29
(5) The Gelfand transformation a —> a() is a homomorphism from A into C0(fl,—) (as real algebras), where a(p) = p(a), Va G A, p G fi, C0(fJ) = {//complex valued continuous function on Cl, and vanishes at oo}, and C 0 ( f i ,  ) = { / € Co(fi)/(p) = 7(p), Vp G n } . Proof.
By the definition of ft, they are obvious.
Q.E.D.
Now we consider the relation between U and the set of all maximal regular ideals of A ( see Definition 2.4.1). First, we assume that A has an identity. If p G ft and p = p, then p € A*, and A = JiR, where J = { a € Ap(a) = 0} is a maximal ideal of A. Clearly, Ac = Jc+C, and 7C = J+iJ is also a maximal ideal of Ac. If p G fi and p^p~, then p() can take nonreal value on A. Since p(l) = 1, it follows that there is v £ A such that p(v) = i. Clearly, p(l + v2) = 0. Then A=
J+[l,v],
where J = {a G Ap(a) = 0}, [l,u] = {a + 0va,/3 G R}. Thus A/ J S C, and J is a maximal ideal of A (notice that C contains no any nontrivial ideal). Conversely, let J be a maximal ideal of A. Similarly to the complex case, we can show that J must be closed and A/J is divisible. By Theorem 2.2.2, we have A/ J S R o r C , i.e., A
J + R o r A = J+[l,v]
where 1 + v2 G J. If A = JiR, then Ac = J c 4C and J c = J+iJ is also a maximal ideal of Ac Further, there is a unique p — ~p £ fl such that J ={a€ A\p{a) = 0}. UA = J+[l,v], then Ac = ( J c + C ( l + iw))+C
= (J c +C(l  w))+C
30
Real Operator Algebras
Since v(l ± iv) = (=R')(1 ± iv) ± i(l + v2) and (1 + v2) G J, it follows that (J c iC(l + iv)) and (J c 4C(l — iv)) are two maximal ideals of Ac, and their intersection is J c = J+iJ. Thus Jc is not a maximal ideal of Ac, and there is p G fi such that p ^ p and J = {a e A\p(a) = 0} (indeed, p(a + a + (3v) = a + j3i,Va e J,a,(3 € R). Now let a € ft be such that J = {a G A 0 such that / ( o ) 2 < Kf(a*a),
Definition 3.3.7.
Va G A
Q.E.D.
Let A be a real Banach * algebra with identity. Denote
5(^4) = {p\p > 0, hermitian on A, and p(l) = 1}. S(A) is called the real state space of A. p € S(A) is also called a real state on A. P r o p o s i t i o n 3.3.8. Let A be a real Banach * algebra with identity. S(A) is a a (A*, A)compact convex subset of A*.
Then
Proof. By Proposition 3.3.2, 5(^4) C A*. Clearly, S(A) is convex and a(A*, A)~closed. Now it suffices to show that S(A) is bounded. Since * operation is continuous on AC/R{AC), where R{AC) is the radical of Ac, it follows that there is a constant K > 0 such that P I < K2\\a\\,
Va£Ac/R{Ac).
For any p e S(A),pc e S(AC) and pc\R(Ac) = 0 obviously. Thus, we can define pc € S(AC/R(AC)), i.e., pc(a) = pc(a),Va €ae AC/R(AC). Then p(a) 2 =  p c ( S )  2 < r s ( a * a ) S(A) ^ <j>. Proof. If S{A) ^ <j>, then V(A) ^ (j> by Proposition 3.3.8 and KreinMilmann theorem. Thus, there are nonzero * representations of A and nonzero topologically irreducible * representations of A. Conversely, let {n, H} be a nonzero * representation of A. By restricting 7r onto a subspace of H, we may assume that there is £ € H such that n(A)Z = H and £ = 1. Then
p() = M)€.0e5(A), and S(A) ^ .
Q.E.D.
Now let A be a real Banach * algebra without identity, and A = A+R. (1) Let p0(a + A) = A, Vo e A, A € M. Then po is a pure real state on A, and the * representation {TTO,HO,£O} induced by p 0 is an onedimensional topologically irreducible * representation of A. In fact, the left kernel IQ of po is L 0 = {( a + A)a e A, A € R, p0((a + A)*(a + A)) = 0} = A, and Ho = A/Lo = R,5ro(a + A) = Al, Vo £ A, A € R. (2) Clearly *S{A) > 1 («=» {po} S 5 ( 2 ) < = • {po} CJP(2)) «=> {/I/ > 0, hermitian on A, and /  A ^ 0} # 1 «=>• there is a nonzero * representation of A «=» there is a * representation n of A such that TT\A ^ 0,
Real Banach * Algebras
47
where #E means the cardinal number of any set E. In fact, if {n, H} is a nonzero * representation of A, then it can be extended to a * representation of A (defining 7r(l) = 1) . Further, let a e_A, and £ G H such that 7r(a)£ ^ 0 and £ = 1. Then p()  1 there is a nonzero topologically irreducible * representation of A • there is a topologically irreducible * representation n of A such that n\A ^ 0. In fact, if * S{A) > 1, then * V(A) > 1 and there is p € V(A)\{p0}. Let {IT, H, £} be the cyclic * representation of A induced by p. We claim that K\A ^ 0. Otherwise,
p(o) = ( 7 r ( o ) e , 0 = 0 ,
VaGA
and p = po This is a contradiction. Clearly, n is topologically irreducible for A. Conversely, if A has a nonzero topologically irreducible * representation, then * S(A) > 1 by (3). By the above discussion, we have the following. Proposition 3.4.5. Let A be a real Banach * algebra without identity, and A = A+R. Then the following statements are equivalent: (i) There is a nonzero * representation of A. (ii) There is a nonzero topologically irreducible * representation of A. (Hi) # S{A) > 1. (iv) £(A)\{0} ? 4,. References.
[1], [27], [30], [35], [36], [40], [41], [53].
48
3.5
Real Operator Algebras
* Radical
Definition 3.5.1.
Let A be a real Banach * algebra. We call
R* = R*(A) = n{ker7r7r
is a * representation of A}
the *radical of A. A has no nonzero * representation, iff R* — A; if R* = {0}, then A is said to be * semisimple. Since any * representation of A is continuous, R* is a closed * twosided ideal of A. Clearly, A/R* will be * semisimple, and R*(A) = R*(A) — n{ker7r7r is a * representation of A}. Moreover, if R — R(A) is the radical of A, then R is also a closed * twosided ideal of A by Theorem 2.4.4. Remark. We don't know that R(AC) = R(A)+iR(A) Section 2.4). But we have R*{Ac) =
(see the end of
R*{A)+iR*(A).
In fact, let a + ib € R*(AC), where a, b € A, and {n, H} be any * representation of A. Then {TTC,HC} is a * representation of Ac, where 7rc = Tr+iir, Hc = H+iH, and wc(a + ib) = ir(a) + in(b) = 0. Hence, R*(A)+iR*(A). n(a) = n(b) = 0, a,b& R*(A), i.e., R*(AC) C Conversely, let o € R*(A), and {a, K} be any * representation of Ac. Let H = Kr (Section 1.1), and n = a\A. Then {K, H} is a * representation of A. Hence, a(a) = 7r(a) = 0, and a G R*(AC), i.e., R*(A) C R*{AC), and R*{A)+iR*{A) c R*(AC). Proposition 3.5.2.
Let A be a hermitian real Banach * algebra. Then J R(A)Cp
1
(0)ci?*(A).
Consequently, if A is also * semisimple, then A is semisimple. Proof. If 7r is any * representation of A, then 7r(a) < p(a), Va € A. Thus, p^O) C kervr, and p _ 1 (0) C R*.
Real Banach * Algebras
49
Let a € R. Then a*a & R. Since A is hermitian , it follows from Proposition 2.4.6 that a(a*a) = a(a*a) n R = {0}. Therefore, p(a) = r(a*a)? = 0, i.e., R C p _ 1 (0). Proposition 3.5.3.
Q.E.D.
Let A be a real Banach * algebra.
(i) J S(A) =£ 1, if A has an identity.
*' (ii) If R* C A, then
R* — n{ker7r7r topologically irreducible * representation of A} = n{ker7r7r topologically irreducible * representation of A}.
Proof. (i) It follows from Propositions 3.4.4 and 3.4.5. (ii) By R* (A) = R* (A) we may assume that A has an identity. If a £ R*, then there is a * representation {n,H} of A such that 7r(a) ^ 0. Further, we may also assume that 7r(l) = 1. Take f € H such that 7r(a*)£ ^ 0 and lieil = 1. Then p() = e 5 ( 4 ) , and p(aa*) > 0. By KreinMilmann theorem and Proposition 3.3.8, S(A) is the a(A*, A)closme of CoV(A), where CoE is the convex hull of a subset E in a, linear space. Thus , there is p S V(A) such that p(aa*) > 0. Let {•7Tp,Hp} be the topologically irreducible * representation of A induced by p. Clearly, a* 0 Lp (the left kernel of p). Further, we claim that aa* & Lp. Otherwise, by Schwartz inequality 0 < p(aa*)2 < p(aa* • aa*) = 0. It is impossible. Now •Kp(a)a* = aa* ^ 0 in Hp, and a $ kevnp. Therefore, (R* c ) n {ker7r7r topologically irreducible * representation of A} C R*. Q.E.D. T h e o r e m 3.5.4.
Let A be a real Banach * algebra.
50
Real Operator Algebras
(i) If A has an identity, and R* £ A, then R* = n{ker7i>p G S(A)} = n{ker7r p p G V{A)}
= n{Lp\P e S(A)} = n{Lp\p e V(A)}. (ii) If A has no identity, and R* C A, then R* = n{ker 7 r / / G £(A)} = n{Lf\f
G £(A)}
= n{ker7T// G £ (A), and ker7r/ C L/} n{ker7T// > 0 and hermitian on A}.
Proof, (i) Clearly, R* C k e r ^ C Lp,Vp € S{A). Conversely, if a & R*, then there exists a * representation {n, H} of A such that 7r(a) ^ 0. This shows that there is £ G H such that n(a)£ ± 0. Without loss of generality, we may assume f = 1 and n(A)£ = H. Then p() = (*•(•)£,£) € 5(A), and p(a*a) = 7r(a)£2 > 0, i.e., a & Lp. Since CoV{A)° = 5(A), it follows that n{Lp\p G V{A)} = n{Lp\p € 5(A)}. Moreover, Lp D ker7rp D n{ker7rCTa G V{A)} D R*,Vp G V{A). Therefore, we have the conclusion (i). (ii) By (i), we have R* = R*(A) = n{ker7r> G S(A)} = n { L > G 5(A)}. Let p0(a + A) = A,Va G A, A G R. Then po € 5(A),ker7r 0 = A,L0 = A, where LQ is the left kernel of po and {7P0, HQ} is the * representation of A induced by po Thus R* = n{Lp\p G 5(A)} n{(LpnA)pG5(A)\{p0}} = n{L// G
£(A)\{0}}
= n{Lf\fe£(A)}.
Real Banach * Algebras
51
Noticing that every * representation of A is a direct sum of some cyclic * representations of A and a zero * representation of A, we have R* = n{ker7r{7r, H, £} is a cyclic * representation oi A}. For a cyclic * representation {n,H, £} of A, let /(•) = (7r()£, £). Then / 6 E(A), and aekevn
Tr(a)n(b)£ = 0, f{b*a*ab) = 0, • afc € Lf,
V6 € A V6 6 A
V6 € A a G ker 717,
i.e., ker7r = ker7r/. Moreover, in this case f(a*a) = vr(a)^2 = 0 if a € ker7r, i.e., ker7r = ker717 C Lf. Therefore, we have R* = n{ker7r// € £(A), and ker717 C i / } = n{ker7r//e5(^)} = n{ker7T// > 0 and hermitian on
A}. Q.E.D.
Remark. If A has an identity, generally R* C n{p _ 1 (0)p € 5(^4)} since P\AK = 0,Vp € 0, if a* = a and IT (a) C R+. Denote the subset of all positive elements in A by A+, and o > b, if a* — o, b* — b € A and (a  b) e A+. Denote p{a) = r(a*a)?, Va € A. Theorem 3.6.2. Let A be a real Banach * algebra, which is hermitian and skewhermitian. (1) Ifa€A and a(a) c R, then r(a)2 < r(a*a).
52
Real Operator Algebras
(2) If hi = a*a, h2 = h2 € A, then r(hi h2) < r(hi)r(h2). (3) A+ is a cone, i.e., if a,b> 0, then a + 6 > 0. (4) If h* = hi &A,i = 1,2, thenr(h1+h2) 1. Since a(a) C R, it follows that r(a) 0.
Thus , we have r(a)2 < r(a*a). Now we prove that 1 — 6 is invertible. Clearly, r(b*b) = r(bb*) < 1. Since A is hermitian, it follows that 1 — b*b > 0 and 1 — bb* > 0. By Lemma 3.1.2, there are u > 0, v > 0 such that 1  b*b = u2, 1 — bb* = v2. Noticing that (1 + 6*)(1 — b) = u(l + u~x{b* — 6)u _ 1 )u and A is skewhermitian, 1 — 6 has a left inverse. Similarly, (1  6)(1 + 6*) = v(l + v^b*  & > " > implies that 1 — 6 has a right inverse. Hence, 1 — 6 is invertible. (2) Since A is hermitian, we have a{hi h2) U {0} = a(ah2 a*) U {0} C R. Accoding to (1), r(hi h 2 ) < r(h2 /if h2)%  r{h\ h2)1* 0,v = (1 + 6 ) _ 1 6 > 0. Clearly, r ( l  u ) < 1, r(u) < 1, r(u) < 1. By Lemma 3.1.2, we can write u = w2,w > 0. From (2) , we have r(uv) < r(u)r(v) < 1. Then 1 — uv is invertible, and hence 1 + o + b is invertible. Since X + a + b = A(l + A  1 a + A _1 6) for any A > 0, it follows from the preceding paragraph that A + a + b is invertible, VA > 0. Therefore, a+b> 0. (4)(7). The proofs are similar to the complex case. Q.E.D. Remark. The inequality "r(a) 2 < r(a*a)" or 'V(a) < p(a)" is called the Ptak's inequality ([40, 41]). If B is a hermitian complex Banach * algebra, then this inquality is valid for each element in B. But in the real case, we must assume A € R (see the proof of (1)). Thus we don't know whether the Ptak's inequality is valid for each element in the real case. Theorem 3.6.3. Let A be a real Banach * algebra. Then the following statements are equivalent: (1) A is hermitian and skewhermitian; (2) A is hermitian and r(9^) < p(a), Vo € A; (3) A is hermitian and r(a) = p(a),V normal a G A; (4) A is hermitian and p() is subadditive on A. Proof. (1) => (2) = > (3) and (1) =>• (4) follows easily from Theorem 3.6.2. (4) => (2) is obvious. (3) ==^ (1). Consider a maximal abelian * subalgebra containing a. Now it is in the abelian case, and this is just Theorem 3.2.3. Q.E.D. Definition 3.6.4. a*a > 0, Va € A.
A real Banach * algebra A is said to be symmetric, if
Theorem 3.6.5. Let A be a real Banach * algebra. Then the following statements are equivalent: (1) A is symmetric; (2) A is hermitian and skewhermitian;
54
Real Operator
Algebras
(3) 1 + a*a is invertible in A, Va G A; (4) Rea(a*a) > 0, Va € A; (5)  A g cr(a*a), VA > 0, o e A Proo/. (1) => (2). It is obvious. (2) =$> (1). We may assume that A has an identity. Suppose that there is a € A such that 8 = inf{AA € 0), we may assume that 5 e (—1, —1/3). Put b = 2a(l +
a*a)\
Then 1  6*6 = (1  a*a)2(l + a*a)2 > 0, and a(b*b) C (  o o , l ] . Write 6 = h + k, where ft* = ft, fc* = fc. By Theorem 3.6.2, 1 + 66* = 2(ft2  k2) + (1  6*6) > 0, and • (3) are obvious. Q.E.D. Remark. It is natural to ask: whether A is symmetric = > A c is symmetric? In the abelian case, the answer is affirmative (see Theorem 3.2.3 and note the fact that Ac is hermitian • 0 such that r(u) < a, Vu G U{A). Proof. (2) = > (3) =>• (4) are obvious. (4) =4> (3). For u G f/(i4), un is also in [/(A) for each n e N. Then r(u) n = r(u n ) < a, and r(u) < a " , Vn. Therefore, r(u) < l.Vu G I7(i4). (3) => (2). It is a consequence of the fact that u € U(A) implies u~l € U{A). (1) ==>• (2). It is obvious from Theorems 3.6.5 and 3.6.3. (2) =$» (1). By Theorem 3.6.5. it suffices to prove that A is skewhermitian. Let k* = —k € A with r(k) < 1, let B b e a maximal abelian * subalgebra of A containing k, and let Q be the spectral space of B. As in the proof of lemma 3.1.3, we have u = h+k
€ U(A),
where h* = h and h G B. Let p € fi. Since r(u) = 1 = r(u*), we have p(h) + p{k) = p(u) = eia, p(J0p(A)=p(u*)=e*, where a, /3 € K. Since A is hermitian , p(h) is real. Then 0 < p{hf
< \{\p{h) + p(k)\ + \p(h)  p(k)\)2 < 1.
At the same time p(h)2  p(k)2 = p(h2  k2) = p(u*u) = 1. Therefore, p(k)2 = p(h)2  1 < 0, and p(k) G iR, Vp G fi. Further, by Lemma 3.1.1 and Theorem 2.7.2, a(k) C iR. Q.E.D. Remark. There are some other results about symmetric real Banach * algebras in Section 5.8. Proposition 3.6.7. Let A be a symmetric real Banach * algebra with identity, f be a hermitian linear functional on A, and / ( l ) = 1. Then the following statements are equivalent: (1) / ( a ) > 0, Va G A+; (2) f > 0 on A, i.e., f G S(A);
56
Real Operator Algebras
(3) f(h2) > 0, Vft* = ft G A; (4)\f(h)\• (1). If o > 0, then a + e > 0 for any e > 0. By Lemma 3.1.2, there is u > 0 such that a + e = u2 = u*u. Then / ( a ) + e = f(a + e) = /(u*u) > 0, Ve > 0. Thus ,/(a) > 0. (3) = > (4). Let ft* = /i € A and r(/i) < 1. Then r ( l  (1 ± ft)) < 1. By Lemma 3.1.2, there are u* = u and v* = v such that 1 + ft = u 2 and 1  ft = v2. Then / ( l ± ft) > 0 by (3), i.e., /(ft) < 1. (2) =>• (5). It is obvious from Proposition 3.3.2. Now it suffices to show (4) = > (2). For any ft* = ft G A, we have a(ft)c[a,/3] since A is hermitian, where a = mincr(ft),/? = maxa(ft). Let p = \{a + /3),5 = §(/?  a). Then A  p < 5,VA € a{h), i.e., r(h  p) < S. By (4), \f(h)p\
=
\f(hp)\mina(a*a)
> 0,
Va G A Q.E.D.
Lemma 3.6.8. Let A be a symmetric real Banach * algebra with identity, and ft* = ft G A. Then for any A G o(h), there is p G S(A) such that p(h) = A. In particular, there is p G <S(A) such that \p(h)\=r(h).
Proof. Clearly, A — AH+AK For ft* = ft G A and A G cr(ft), define a (real) linear functional p on [1, ft] = {a + /3h\a,f3 € K} as follows: p{a + /3ft) =
57
Real Banach * Algebras
a + /3A, V a , / ? e Let
Then p(a) > 0,Va € [1, /i] n A+,p(l) = 1 and p{h) = A. E is a linear subspace of AH, and 1, h £ E;
(E,PE)
PE is a linear functional on E, PE{1) = 1, P£;(a) > 0 , V a 6 £ 0 , 4 + , andp E [l,/i] = p
and {E,pE) < (F,PF) if E C F and ( P F   E ) = PE By Zorn lemma, £ admits a maximal element (E,PE). We claim that .E = AH In fact, suppose that there is a £ AH\E. Let F = SjRa. Since 1 € .E,  r ( a ) l < a < r(o)l, and A+ is a cone (see Theorem 3.6.2), we can define PF on F such that (pp\E) = PE, and sup{pjs(6)6e E,
b 0, then a > d/X,pF(a) > pE{d/X), i.e., 0 < pF(d + Xa) = pE(d) + XpF(a). If A < 0, then a < —d/X, PF(O) < PE{—d/X), i.e., 0 < pF(d + Xa) = ps(d) + XpF(a). Therefore, (F,pF) € £ and (E,pE) < (F,pF). This is a contradiction. Further, let p\An — PE and P\AK = 0. Then p € S(A), and p(/i) = A. Q.E.D. Notes. In the complex case, S.Shirali and F.W.M. Ford proved that a Banach * algebra B is symmetric if and only if B is hermitian. Using their method, we can also get Theorem 3.6.5. in the real case. V. Ptak set up an important inequality, and from this he simply proved ShiraliFord theorem in the complex case. Ptak's theory is considered in the real case by J. Vukman and this section. References.
[1], [27], [30], [35], [36], [40], [41], [47], [53].
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Chapter 4
Fundamentals of Real Von Neumann Algebras 4.1
Banach spaces of operators on a real Hilbert space
Definition 4.1.1. Let H be a real Hilbert space. We shall denote by F(H), C(H) and B(H) the sets of all linear operators of finite rank, all compact linear operators, and all bounded linear operators on H respectively.  •  will be the operator norm on H; The identity operator on H is denoted by Iff, or 1 if no confusion arises; and Hc = H+iH is the Hilbert complexification of H (see Section 1.1). Moreover, F(HC),C(HC) and B(HC) are the corresponding operator spaces on Hc. Then B(HC) =
B(H)+iB{H)
is a complexification oiB(H) in operator norm  •  (see Proposition 1.1.11). Clearly, F(HC) = F(H)+iF(H) as linear spaces. Proposition 4.1.2.
Let H be a real Hilbert space. Then C{H) =
(F(H),\\\\);
C(H) is a closed * twosided of B(H); Banach space if dim H = +oo; and C{HC) = is a complexification ofC(H)
C(H) is not the dual space of any
C{H)+iC{H)
in  • .
Proof. By the prof in the complex case (see [26, Section 1.1]), and Proposition 1.1.11, it suffices to show that if (a + ib) € C{HC), then a, b e C(H), where a, b G B(H). 59
60
Real Operator Algebras
For any bounded sequence {£„} C H, clearly there is a subsequence {£nfc} such that {(a + ib)€nk} is convergent. Then {a£„fc} and {b£nk} must be convergent. Therefore, a,b G C(H). Q.E.D. Definition 4.1.3. Let H be a real Hilbert space. The set of all operators of trace class on H is denoted by T(H), i.e., a G T(H) if a € C(H) and X^n An < +00, where {An} is the set of all positive eigenvalues of (a*a)* (counting the multiplicity). For a G T(H), ai = ^Z n An is called the trace norm of a, and tr(a) = J2t{a£i, &) is called the trace of a, where {&} is any normalized orthogonal basis of H. Similarly to the complex case, tr() is welldefined on T(H). Similarly to the complex case (see [26, Section 1.1]), we have the following. Theorem 4.1.4. Let H be a real Hilbert space. (1) T(H) = (F(H),  • Hi)", and T{H) is a * twosided ideal of B(H); (2) C(H)* = T(H), i.e., for any f G C(H)* there exists a unique a G T(H) such that 11/11 = ai, andf{c) = tr(ac),Vc € C(H); conversely, for any a € T(H), tr(a) is a continuous linear functional on C(H) with norm ai; (3) T(H)* = B(H), i.e., for any F G T(H)* there exists a unique b € B(H) such that \\F\\ = ll&H, andF(a) = tr(ab),\/a G T(H); conversely, for any b G B(H), on T(H) with norm \\b\\.
tr(b) is a continuous linear functional
Remark. T{H) is called the predual of B{H), and also denoted by B(H), = T(H). Proposition 4.1.5. Let H be a real Hilbert space. T(H)+iT(H) is a complexification ofT(H) in  • i.
Then T(HC) —
Proof By C(H)* = T(H), C(HC)* = T(HC), Propositions 4.1.2 and 1.1.4, the conclusion is obvious. Q.E.D. References.
[26], [30], [50].
Fundamentals of Real Von Neumann Algebras
4.2
61
Locally convex topologies in B(H)
Let H be a real Hilbert space. Similarly to the complex case (see [26 , Section 1.2]), we introduce the following locally convex topologies in B(H) : (1) weak (operator) topology; (2) strong (operator) topology; (3) strong * (operator) topology; (4) crweak (operator) topology, and it is also equivalent to a(B(H), T(H))\ (5) astrong (operator) topology, and it is also equivalent to s(B(H),T(H)); (6) astrong * (operator) topology, and it is also equivalent to s*(B(H),T(H)); (7) Mackey topology
T(B(H),T{H));
(8) uniform (operator) topology. Similarly to the proofs in the complex case ([26 , Section 1.2]) and by the duality theory (see [26 , Appendix ]), we have the following. Proposition 4.2.1. Let H be a real Hilbert space, and f be a linear functional on B(H). (1) If f is a(B(H),T(H))continuous, then there exists a unique a e T(H) such that f{b) =
tr{ab),VbeB{H)
and there are {£n}> {%} C H with X)n(ll£nll2 + ll^nll2) < +°o such that
/(&) = $ > £ » , 1*0.
V6ej5(ff).
n
Moreover, if f > 0 (i.e., f(b*b) > 0,V6 6 B(H)), then we can take £ n = ?7„,Vn. (2) If f is weakly continuous or strongly continuous, then there exists a unique v € F(H) such that f(b) = tr(bv),
V6 e B{H)
and there are £i, • • •, £ n and r}\, • • •,rjn € B(H) such that 'n
Moreover, if f > 0, then v > 0 or we can take & — rji, 1 < i < n.
62
Real Operator Algebras
Theorem 4.2.2.
The relations between topologies (l)(8) top.3)
D
n top.8)
D
top.7)
D
top.2)
D
are as follows: top.l)
n
top.6)
D
where "D " means finer. Moreover, in any bounded ball ofB(H), top.5), top.3) ~ top.6).
top.5)
n , D
topA)
we have top.l) ~ top4), top.2) ~
Remark. We will see top.6) ~ top.7) in any bounded ball of B(H) (see the end of Section 4.3). Corollary 4.2.3. Let f be a linear functional on B(H). Then the following statements are equivalent: (1) f is cr(B(H),T(H))continuous; (2) f is s(B(H),T(H))continuous; (3) f is s*(B(H),T(H))continuous; (4) f is T(B(H),T(H))continuous; (5) f is weakly continuous on any bounded ball of B(H); (6) f is strongly continuous on any bounded ball of B(H); (7) f is strongly * continuous on any bounded ball of B(H). Proposition 4.2.4. Let K be a convex subset of B{H). (1) The following statements are equivalent: (i) K is a(B(H),T(H))closed; (ii) K is s(B{H),T(H))closed; (Hi) K is s*(B(H),T(H))closed; (iv) K is T(B(H),T(H))closed; (v) KnXS is weakly closed, VA > 0; (vi) K D \S is strongly closed, VA > 0; vii) K C\\S is strongly * closed, where S is the closed unit ball of B(H). (2) K is weakly closed, if and only if, K is strongly closed. Now we consider the problem of topological restriction from B(HC) B(H). Proposition 4.2.5. Let H be a real Hilbert space Then (top. j) in B{HC)\B{H)

top. j) in
B{H),
i.e., ai — > 0 with respect to top. j) in
ai — > 0 with respect to top. j) in
B(H)
B(HC),
to
Fundamentals
of Real Von Neumann
Algebras
63
where net {ai} C B(H), and j — 1,2, • • •, 8. Proof. It suffices to show that the conclusion holds for Mackey topology (top. 7)), and it is enough to notice the following fact. If E is a or(r(F c ),B(if c ))compact subset of T(HC), then Re E = {a € T(H)  there is b e T(H) such that (a + ib) € E}, Im E = {6 € T ( # )  there is a e TCtf) such that (a + ib) € £ } are a(T(H),B(H))compact subsets oiT(H), and Re E+i liaE — {(a+ib)\a e Re E, b 6 Im £ } is a ) ~ T(M,M.)
C
T(H))\M,
s(B{H),T(H))\M, s*(B(H),T(H))\M, T{B(H),T(H))\M.
Similarly to the complex case (see the end of [26 , Section 1.3]) , we still don't know if r(M,Mt) ~ T(B(H),T(H))\M1 Moreover, by Proposition 4.3.3 and similarly to the proof of Proposition 4.2.5, we have 0,Va G M+, and p(&) = 0,V6* =  6 e Af*. Clearly, if p > 0 on M, then we have p(a*) = p(a),Va G M, and the Schwartz inequality holds: p(6*a)2 < p(a*a)p(b*b),
Va,6 G M;
and pc > 0 on M c , where p c is the natural extension of p onto Mc — M+iM. Since p c G M* and /9C = p c (l) = p(l), it follows from Proposition 1.1.4 that p € M*, and pj = p(l). Moreover, if 0 on M c , then p > 0 on M, where p(o) = Re 0 on M. p is said supp(oj) = p ( s u p a ( ) i
i
for any bounded increasing net {a;} C M+\ p is said to be a real normal state, Up is normal and p = p(l) = 1; p is said to be completely additive, if
Ylp^
PC^PI) = i
i
for any orthogonal family {pi} of projections in M. Similarly to the complex case (see [26 , Theorem 1.8.6]) , we have the following. Theorem 4.5.3. Let M be a real VN algebra, and p > 0 on M. Then the following statements are equivalent: (1) p G M*, i.e., p is a(M, M*)continuous; (2) p is normal; (3) p is completely additive. In particular, if p is normal or completely additive on M, then pc is normal or completely additive on Mc. Now let M be a real VN algebra on a real Hilbert space H, and p be a
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real normal state on M. Then pc is a normal state on Mc. If L = {a 6 M\p(a*a) = 0} is the left kernel of p, then Lc = L+iL is the left kernel of pc. It is easy to see that (Mc/Lc,
(, ) c )  {MIL, (, ))+i{M/L,
(,))
where (,) and (,) c is the inner products in M/L and Mc/Lc and pc respectively. Then we can see that {^Pc = np+inp, HPc =
induced by p
Hp+iHp},
where {KP,HP} is the * representation of M induced by p, and is the * representation of Mc induced by pcLet Sn be the real normal state space on M, and {n — ®pes„np, H =
{^pc,HPc}
®PesnHp}
We claim that {IT, H} is faithful. In fact, if ir(a) — 0 for some a G M, then (n(a*a)lp, lp)  p(a*a) = 0,
Vp G <S„.
Thus, we have tr(a*af) = 0,Vt G T(H)+, and ti(a*ah) = 0,V/i* = /i G T(H). By the definition of trace, it is easy to see that tr(a*afc) = ti(aka*) = 0,
Vfc* = k G T ( F ) .
Therefore, we must have a* a = 0 and a = 0. Of course, {7r, i / } can be naturally extended to a faithful * representation of Mc. Thus, we have the following. Proposition 4.5.4. Let M be a real VN algebra, and Sn be the real normal state space on M. Then for each p € <Sn, the * representation {np,Hp} of M induced by p (GNS construction) is a — a continuous, itp(M) is a real VN algebra on Hp, and \\np\\ < 1. Moreover, {n = ©pe5n7Tp,i7 = ®PesnHp} is a a — a continuous, isometric, faithful * representation of M. Lemma 4.5.5. Let f G M*. Suppose there exists a G M+ with \\a\\ < 1 and ll/H = / ( a ) . Then / > 0.
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Algebras
Proof. By Proposition 1.1.4, / c  = / = f{a). Now from a result of complex case (see [26, Proposition 2.3.3]) , fc > 0 on Mc. Therefore, / > 0 on M. Another proof is direct. It is as follows. By 0 < a < 1, we have  1 < 2a  1 < 1. Since / ( l ) < / = f(a), it follows that / 1  & > 1/(1  6)1 = 1  A, and A > 0. Now for c* =  c G M, we must prove /(c) = 0. Let /(c) = fi(e K), and
11/11 = 1. Then A + M =  / ( C + A )  <   C + A = (C 2 + A 2 ) 5
and A2 + 2A/x + (i2 < A2 + c 2 ,
VA 6 R.
Therefore, /(c) = n = 0.
Q.E.D.
By Lemma 4.5.5 and similarly to the complex case (see [26, Theorem 1.9.3]), we have the following. Theorem 4.5.6. Let M be a real VN algebra, and
0 and unique v € M such that
i s a n abelian real C*algebra, where fi is a locally compact Hausdorff space, and "" is a homeomorphism of ft with period 2. In fact, it follows from Definition 5.1.1 immediately. Moreover, by Proposition 5.1.4 it is also the general form of abelian real C*algebras. (4) A = C{X,Y) = {/ e C{X)\f{Y) c R} is an abelian real C*algebra, where X is a compact Hausdorff space, and Y is a closed subset of X. Moreover, A Si C(Sl,), where fi = (X\Y) U F U (X\Y) (disjointed topological union ) such that y = y,Vy € Y, and "": x G first or second (X\Y) —> x e second or first (X\Y). References. 5.2
[3], [17], [24], [25], [26], [30].
Positive functionals and equivalent definition of real C—algebras
Definition 5.2.1. Let A be a real C*algebra. a € A is said to be positive, denoted by a > 0, if a* — a, and a{a) C R+. Denote A+ = {a € Aa > 0}. Proposition 5.2.2. Let A be a real C*algebra. (1) A+ = {Ac)+ fl A is a closed cone, and A+ (1 (—A+) = {0}. (2) If a £ A+, then there exists a unique a? € A+ such that (a^) 2 = a, and a3 g
CQ(O).
(3) If a e A, then a £ A+, if and only if, there is b € A such that a = b*b. (4) For any h* = h e A, there are unique h+ and /i_ € A+ such that h = h+ — h,
and h+ • h = 0.
Moreover, h+ and h € C^Qi). Consequently, AH = {a £ A\a* = a} = A+ — A+ = [A+] (the (real ) linear span). Proof. From the complexification and [26, Section 2.2], it is obvious. Q.E.D. Remark. For any (complex) C*algebra B, we have B = [B+] (the (complex) linear span).
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Definition 5.2.3. Let A be a real C*algebra. {d/} is called an approximate identity for A, if {di} is an increasing net in A+ such that 0 < di < 1, VI, and {\\adi  a\\ + \\dta  a\\) —> 0, VoeA Clearly, {d{\ is also an approximate identity for Ac. Proposition 5.2.4. identity.
Every real C*algebra A admits an approximate
Proof. Let {u; = di + ici} be an approximate identity for Ac, where di, ci € A,Vl. Clearly, {d{\ will be an increasing net in A+, and 0 < di < 1,VZ. Moreover, since Ac is a complexification of A, it follows from Definition 1.1.1 that \\adi — a\\ + \\dia — a\\ < \\aui — a\\ + \\uia — a\\ —> 0,
Va e A.
Therefore, {di} is an approximate identity for A.
Q.E.D.
Remark. Let A be a nonunital real C*algebra, and {di} be an approximate identity for A. Since {di} is also an approximate identity for Ac, it follows from the complex case (see [26, Proposition 2.4.4]) that the C*norm on (A+R) is as follows: \\a + A = limj \\adi + Xdi\\ = limj \\dia + Xdi\\ = sup{a6 + A6 \b£A,
\\b\\ < 1 } ,
Va € A, A € R. Definition 5.2.5. Let A be a real C*algebra. A (real) linear functional / on A is said to be positive, denoted by / > 0, if f\A+ > 0, and f\Ax = 0 (i.e., /(a*) = / ( a ) , Va G A ), where AK = {a e A\a* = a}. Let / > 0 on A. / i s called a real state on A, if / = 1. Denote the set of all real states on A by S(A). Proposition 5.2.6.
Let A be a real C*algebra.
Fundamentals
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Algebras
85
(1) If f > 0 on A, then we have the Schwartz inequality: \f(b*a)\2 < f(a*a) • f(b*b),
Vo, b G A.
(2) If f > 0 on A, then fc > 0 on Ac. Consequently, f G A*, and 11/11 — ll/dl = u m / ( ^ ) = lim/(^f), where {di} is an approximate identity for A. (3) Let f G A*. Suppose there exists a G A+ such that \\a\\ < 1 and
ll/H = /(a). Then / > 0 on A. (4) (5) r(h). (6) (7)
Let
on Ac. Then p = Re( 0 on A, and \\p\\ = \\ 0,Vp G S(A), then aeA+. Let A be nonunital, and p > 0 on A. Define p(a + A) = p(a) + Xno,
V o e A , A e R , where p,0 > \\p\\. Then p > 0 onA = A+R. (8) Let B be a real C* subalgebra of A. Then each real state on B can be extended to a real state on A. Proof.
From the complexification and Lemma 4.5.5, all are obvious. Q.E.D.
Proposition 5.2.7. Let Abe a real C*algebra, and S{A) be its real state space. (1) S(A) is a convex subset of A*. (2) (S(A),a(A*,A)) is compact, if and only if, A is unital. (3) If A is nonunital, then S(Aj
= Co(V(A)u{Q})a
= Q(A),
where V(A) = exS(A) (the subset of all extreme points of S{A)), Q(A) — {/ G A*\f > 0, ll/H < 1}, and """ means the a{A\ A)closure. Proof. (1) By Proposition 5.2.6(2), it is obvious. (2) Clearly, S(A) is cr(A*, A)compact if A is unital. Conversely, if A is nonunital, then Ac is also nonunital, and 0 G S(AC) (see [26, Proposition 2.5.5]). Thus, there is a net {ipi} C S(AC) such that (pi —> 0 in a(A*, Ac). Then pi = R e ( ^  A ) —> 0 in o{A*, A), and 0 G S(A)". Therefore, S(A) is not a{A*, A)compact.
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Real Operator Algebras
(3) Clearly, Q(A) is convex and a(A*, A)compact, and exQ(A) — V(A) U {0}. Thus, by KreinMilmann theorem, Q(A) = Co{V(A) U {0}) J . On the other hand, by the proof of (2), (V(A) U {0}) C S ( I f C Q(A). Clearly, S(A)
is convex. Therefore,
S(Aj
= Co(V(A)u{0}f
= Q(A). Q.E.D.
Remark.
By KreinMilmann theorem, S(A)
= CoV{A)
if A is uni
tal. Theorem 5.2.8. Let Abe a real C*algebra, f 6 A* is said to be hermitian, if f(a*) = / ( a ) , V o e A (i.e., f\AK = 0 ) . Let f € A* be hermitian. Then there are unique /+ and / _ > 0 on A such that / = / +  /  , and ll/H = / +  + /_.
Proof. Since / c is also hermitian on Ac, it follows from [26, Theorem 2.3.23] that there are unique 0 on Ac such that fc =
H
u= 0
Hp}.
peS(A)
Then {nu,Hu} is a faithful * representation of A, and A is isometrically * isomorphic to the uniformly closed * subalgebra TTU(A) of B{HU) on the real Hilbert space Hu. Consequently, A is * semisimple. Theorem 5.2.10. Let A be a real Banach * algebra. Then A is a real C*algebra, if and only if, A is hermitian, and \\a*a\\ = \\a\\2, Va G A. Proof. The necessity is obvious. Now let A be hermitian, and \\a*a\\ = a 2 ,Va e A. We may assume that A has an identity. In fact, if A has no identity, then (A+R) is also hermitian obviously. Further, let \\a + A = sup{a& + A6 \b € A, \\b\\ < 1}, Va € A, A G R. Similarly to the proof in the complex case (see [26, Proposition 2.1.2]), we also have (o + A)*(a + A) = a + A2, Va € A, A e R. For any k* = —k G A, since * operation is isometric, it follows that (etk)*  e~tk,
1 = (e tfe )V fe  = e tfc  2 ,
Vt € R. Thus , a(k) C iR, i.e., A is also skewhermitian. By Theorem 3.6.5, A is symmetric. By Lemma 3.6.8 and the assumption on A, A admits a faithful * representation {TT,H}, where K = iru,H = Hu (see Definition 3.3.9). Let
Nil = IkWII, VaeA
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Real Operator Algebras
Then  • i is a C*norm on A. For any h* = h G A, we have a(n(h)) C a(h) obviously. Thus, \\h\\i = r(n(h)) < r(h)  ft. On the other hand, 7r(fc)>K7r(fc)l p ,l p ) = p(ft)l, Vp G 5(A). By Lemma 3.6.8,
W i = lkWII>r(ft) = W. Thus, /ii = \\h\\,Vh* = heA.
Therefore,
oi = a*a12 = o*o2 = a,VaG A, and A is a real C*algebra from Proposition 5.1.2.
Q.E.D.
Corollary 5.2.11. Let A be a real Banach * algebra. Then the following statements are equivalent: (1) A is a real C* algebra; (2) A can be isometrically * isomorphic to a uniformly closed * subalgebra of B(H) on a real Hilbert space H; (3) A is hermitian, and \\a*a\\ = a 2 ,Va G A; (4) A is symetric, and \\a*a\\ = a 2 ,Va € A; (5) 1 + x*x is invertible in A,Vx e A, and \\a*a\\ = a 2 , Va G A. Remark. In Theorem 5.2.10, the hermitian condition of A is necessary. To see this, consider C as a real algebra, and let A = A and A* = A, VA € C. Then we get a counterexample. Notes. Theorem 5.2.10 is due to L.Ingelstam [21, 39]. Moreover, K. R. Goodearl ([17]), C.H.Chu et al. ([3]) define a real C*algebra as Corollary 5.2.11(5). References.
5.3
[3], [17], [21], [24], [25], [26], [30], [39].
Pure real states, their left kernels, and irreducible * representations
Definition 5.3.1. Let A be a real C*algebra, S(A) be its real state space, and V(A) = ex<S(A) (see Proposition 5.2.7). Each p G V(A) is called a pure real state on A.
Fundamentals
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89
Proposition 5.3.2. Let A = C(fl,  ) be a unital abelian real C* algebra, where Q is a compact Hausdorff space, and "" is a homeomorphism of tt with period 2. Then its real state space is S{A) — {p\p probability measure on Q, and p,o — = p}, and its pure real state space is
Proof. Let p € S{A). Then pc € S(AC), and there is a probability measure p on Cl such that
PM = f g(t)dn(t),Vg € C(fi). For any / G C(ft,  ) , we have pc(f) = p(f) G R. Thus
Jfdp = Jfdp = Jf(t)dp(t) = Jf(t)d»Ct) = Jf(twct), V/ G C ( f i ,  ) . Moreover, f fdp = f f(t)dp(t),\/f
G C(fi). Therefore,
p o — = p, and supp/Li = supp^i. If p G V(A), then supp^i = {t,t} for some t € fi. Since p o — = p, it follows that p = \(5t +