A R K I V F O R M A T E M A T I K Band 5 nr 16
C o m m u n i c a t e d 12 F e b r u a r y 1964 b y L. GXRDING a n d L . CARLESO~
Real Banach algebras By
L A R S I N G E L S T A MC O N T E N T S
I n t r o d u c t i o n . . . 2 3 9 C h a p t e r I . G e n e r a l t h e o r y o f r e a l n o r m e d a l g e b r a s . . . 2 4 0 1. D e f i n i t i o n s . . . 2 4 0 2. B a s i c t e c h n i q u e s . . . 2 4 1 3. C o m p l e x i f i c a t i o n a n d s p e c t r u m . . . 2 4 3 4. G e l f a n d t h e o r y . . . 2 4 5 C h a p t e r IX. R e a l i t y c o n d i t i o n s . . . 2 4 7 5. T h e m o d i f i e d e x p o n e n t i a l f u n c t i o n . . . 2 4 7 6. R e a l i t y c o n d i t i o n s . . . 2 4 8 7. S i g n i f i c a n c e o f t h e c o n d i t i o n s . . . 2 5 1 8. A d e c o m p o s i t i o n t h e o r e m . . . 2 5 2 C h a p t e r I I I . T h e q u a s i - r e g u l a r g r o u p . . . 2 5 4 9. S t r u c t u r e o f G~ . . . 2 5 4 10. C o m p o n e n t s o f G q . . . 2 5 5 l l . T h e r e g u l a r g r o u p o f A~ . . . 2 5 7 12. R e m a r k s o n a c o h o m o l o g y r e s u l t . . . 2 5 8 C h a p t e r I V . S t r i c t r e a l i t y a n d f u l l f u n c t i o n a l g e b r a s . . . 2 5 8 13. C o n d i t i o n s f o r s t r i c t r e a l i t y . . . 2 5 9 14. O r d e r - t h e o r e t i c f o r m u l a t i o n . . . 2 6 0 15. A b s t r a c t c h a r a c t e r i z a t i o n o f CoB(~) . . . 2 6 1 C h a p t e r V . l=~eal a l g e b r a s w i t h i n v o l u t i o n . . . 2 6 3 16. D e f i n i t i o n s a n d p r e l i m i n a r i e s . . . 2 6 3 17. R e p r e s e n t a t i o n o f s y m m e t r i c b * - a l g e b r a s . . . 2 6 5 18. C h a r a c t e r i z a t i o n o f r e a l C * - a l g e b r a s . . . 2 6 8
R e f e r e n c e s . 2 7 0
Introduction
Most of the general theory of Banach algebras has been concerned with algebras over the complex field. The reason for this is clear: the power of function theoretic methods and the Gelfand representation [11]. But the complex algebras can be regarded as a subclass of the real algebras and it is natural to ask what can be said a b o u t this larger class. I n several respects the extension of results that are known for
~7:3 2~9
L. INGELSTAM, Real Banach algebras
complex algebras is easy; m a n y techniques valid for complex algebras will work in the real case as well (el. see. 2). Also since any real algebra can be embedded in a complex (see. 3), some general results can be obtained in this way from the complex case. In quite a few areas, however, new approaches are needed or different ("typically real") phenomena occur.
In much of the hterature on Banach algebras incidental remarks on the real scalar case can be found. Among more systematic contributions we can mention [1], [19], [20] and in particular the monograph [24], in large portions of which the t h e o r y is presented simultaneously for real and complex scalars. The author's papers [15], [16], [17] deal with special problems for real normed algebras.
This article intends to contribute to the t h e o r y of real Banach algebras in three central areas: (1) the structure of the (quasi-) regular group (Ch. III), (2) abstract characterization of real function algebras (Ch. IV), (3) the relation between real B*- a n d C*-algcbras (Ch. V). The questions studied here and in [15] lead us to introduce and investigate a certain classification of real Banach algebras (Ch. II). An introduc- t o r y chapter (Ch. I) gives some of the standard material from the general theory, modified for the real scalar case.
For a more detailed survey of the contents the reader is referred to the short sum- maries which are found at the beginning of each chapter.
Chapter I. General t h e o r y o f real normed algebras
This chapter intends to give a brief survey of those parts of the general theory of real normed algebras t h a t will be used in later chapters. Following the definitions (sec. 1), some standard notions and techniques (regular and quasi-regular group, adjunction of identity, natural norms, the function v (x), quotient algebras, comple- tion) are described (sec. 2). The spectrum of an element in a real algebra is defined and its relation to the complexification is discussed in section 3. For the Frobenius-Mazur Theorem 3.6 on real normed division algebras we give a complete proof whose alge- braic part is self-contained and elementary. In the last part (sec. 4) of the chapter the real counterpart of the Gelfand representation theory [11] for commutative complex Banach algebras is presented.
All the material in the chapter is known in principle, although m a n y things, most notably those of sec. 4, are rarely given explicitly for the real case.
1. Definitions
We let A be an associative algebra over the real numbers (R) or the complex numbers (C). A is called a topological algebra if it is also a Hausdorff topological space such t h a t addition, multiplication and multiplication b y scalars (from A • A, A • A and R • A or C • A into A, respectively) are continuous functions.
A norm on A is a real-valued function x- llxll on A with the properties
IIx +yll Ilxll + Ilyll, I1 11 =l [llZll,
LIxll >o if .o
&RKIV FOR MATEMATIK. B d 5 n r 1 6 for all x, y EA and scalars ~. A norm defines a topology on A , making A a topological vector
space.Two
norms, I I II and III Ill, define the same topology if and only if there are two numbers c and C such t h a t0<c-< I1 11-<c
for all x. Two norms satisfying such a relation are called equivalent.
A topological algebra, the topology of which can be defined b y a norm will be called a normed algebra. (This terminology, though slightly unusual, is quite practical; when a certain norm is replaced b y an equivalent norm we can still speak of the same nor- med algebra.) A norm t h a t defines the topology for A is called admissible. A B a n a e h algebra is a complete normed algebra.
For a given norm I I
II on
A multiphcation is continuous, i.e. A is a normed algebra under I1" II, if and only if there exists a number K such t h a tIlxyll <KIIxll IlYlI"
We will see t h a t there are always admissible norms for which we can take K = 1 (Proposition 2.3).
If A is an algebra over C we can make A an algebra over R simply b y restricting scalar multiplication to R. Hence the class of complex algebras can be regarded as a subclass of the real algebra~; later on we will devote some effort to the characterization of this subclass, the real algebras of complex type (Ch. II).
2. Basic techniques
Throughout this section A is a normed algebra. I f A has an identity element e (ex = x e = x for all x) the set A together with algebra multiplication forms a semigroup ( A , . ) with neutral element e. The elements with two-sided inverse in ( A , . ) are called regular a n d form a group G, the regular group.
If there is no identity we can still do something equivalent. Let x o y = x + y - x y ; then (A, o) is a semigroup with neutral element 0. The elements with two-sided inverse in (A, o) are caned quasi-regular and form a group G q, the quasi.regular group.
G and G q are topological groups with the (metric) topology of A [24, p. 19].
Proposition 2.1. I / A has identity e, G q is homeomorphically isomorphic to G, under the m a p x--> e -- x.
We can also "adjoin an identity". Let A be real and A x = R 9 A, direct sum as real normed vector spaces. Multiplication is defined b y
(fi, +fi
and the topology for instance given through a norm
II( ,x)ll = + Ilxll.
Thus A 1 is a normed algebra with identity (1, 0) and we have
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L. INGELSTAM, Real Banach a~gebras
~roposition 2.2. Through x-->(O,x) A is embedded (homeomorphieaUy and isomor- phicaUy) as an ideal o/codimension 1 in A 1. A n element x E A is quasi.regular in A i / a n d only i / ( 1 , - x ) is regular in A 1.
A norm is called natural if
II~yll < I1~11 Ilyll ~or an ~,y (submultiplicative), llell = 1 ~ a has identity.
Proposition 2.3. Every normed algebra has an admissible natural norm.
Proo/. If A has identity, let
sun Ilxyll III 9 III = ~K II y I1
for some admissible norm l" 9 Then "III is natural and equivalent to I1" I1" If there is no identity p u t x =H (O,x) , the latter t a k e n in A r
Remark. Using this device, the left regular representation, Gelfand [11] proved t h a t a Banaeh space t h a t is also an algebra such t h a t multiplication with a n y fixed element is a continuous operation, is actually a topological (Banach) algebra.
Definition 2.4.
/or some natural norm I1" II-
1
v(x) = i n f II x~ II ~
n
Proposition 2.5.
1
~(x)= lim ]]xn [[ ~
n - - ) oo
/or any admissible norm II" II.
For proof and further properties of ~, see [24, p. 10].
We notice t h a t v(x) is a topological invariant, independent of which admissible norm was chosen to define it.
Proposition 2.6. I / A is Banach and v(x) < 1 then x is quasi.regular. I / A has identity and v(e - x ) < 1 then x is regular.
~ X n
Proo]. The series - L , n = l and e + ~0ffil ( e - x) n converge and are the desired inverses.
Consequences of this are, among others, t h a t 0 has a whole neighbourhood con- sisting of quasi-regular elements and t h a t G q
(or G)
is an open subset of A.Proposition 2.7. I / I is a closed two.sided ideal in a normed algebra A , A / I is also a normed algebra. A [ I is Banach i / A is Banach.
A R K I V F O R MATI~.MATIKo B d 5 n r 1 6
Proo/. F o r [ x ] e A I I and II 11 a norm for _4, take I1[~]11
=inf~~
tion t h a t this is a norm, t h a t multiplication is continuous and t h a t completeness is preserved, is routine.
For normed algebras we also have a standard procedure of completion.
Proposition 2.8. A n y normed algebra A can be embedded (isomorphicaUy and topo- logically) as a dense subalgebra o / a Banach algebra A', called the completion o] A.
X co
Proo/. Let A~ be the space of all Cauchy sequences ( ,)n=l, xnEA. With a norm ll(xn)H =sup]]x,H A~ becomes a normed algebra. N={(xn); xn-->0 } is a closed two- sided ideal. Then A ' = A ~ / N is the desired Banaeh algebra and the embedding
X---~(X n - - X r162 - )n=l + N . The conclusion follows from Proposition 2.7 and routine calcula- tions.
3. Complexification and spectrum
As was indicated in sec. 1 the notion of a real normed algebra is more general t h a n t h a t of a complex normed algebra. The complex theory is better known and in some respects more satisfactory and we describe a procedure b y which we can draw some results (but far from all) for real algebras in general from the complex theory. The real algebra A can be embedded in a larger ("twice as big") complex algebra.
Let A c = A 9 A, direct sum as real normed vector spaces, and define multiplication (a, b) (c, d) = (ac - bd, ad + bc)
and multiplication b y complex scalars
(~ + i f ) (a, b) = (eta-fib, ~tb A-fa).
Then A c is a complex algebra and a real normed algebra, an admissible norm is for instance
II(a,b)H = HuH A-HbH.
To show t h a t A c is a complex normed algebra we construct an admissible complex- homogeneous norm, following Kaplansky [20, p. 400],
[llx[ll =maxr lI(cosq~ + isinq~)xll.
A is embedded in Ac b y the real-algebra monomorphism x-->(x, 0). (A more detailed discussion [24, p. 8] shows that, given a natural norm on A, there exists an admissible natural norm on A c such t h a t the embedding is an isometry with respect to these
n o r m s . )
The spectrum a c (x) for an element x of a complex algebra A is defined as the set of complex numbers $ such t h a t $-1 x is not quasi-regular, together with 0 if x -1 does not exist or A lacks identity. If the corresponding definition, with real numbers, is used for real algebras the spectrum would frequently be e m p t y and give no informa- tion at all a b o u t the element. We therefore adopt the following definition, due to Kaplansky [20].
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L. INGELSTAM, Real Banach algebras
Definition 3.1. The spectrum o] an element in a real algebra A is the set o / c o m p l e x numbers
(rA(x) = {o:+ifl ~=0; (~2 +fl~)-~ ( x 2 - 2 ~ x ) n o t quasi-regular}
plus 0 i / x -~ does not exist or A does not have identity.
W e notice in p a r t i c u l a r t h a t for a real o~=O,o~EaA(x) if a n d o n l y if ~ - l x is n o t quasi-regular.
Proposition 3.2. The spectrum o / a n element o/ a real algebra is equal to the spectrum o / t h e corresponding element i n the complexi/ication
a , (x) = a~o ((x, 0)).
Proo/. 0 is easily checked o u t to belong t o neither or b o t h of t h e sets. F o r ~ +ifl ~ 0 t h e element x' = (0~ + ifl) -1 (x, O) = (0~ 2 +fl2)-1 (ax, - ~ x ) e G q if a n d o n l y if x ~ = (~2 +/~2)-1 (o~x, fix) E G ~, hence if a n d o n l y if x' 0 x" = (~2 +fl2)-1 (x ~ _ 2my, 0) e G q.
Given a c o m p l e x algebra A, " r e g a r d i n g " it as a real algebra m a y distort t h e n o t i o n of s p e c t r u m b u t in a non-essential way:
Proposition 3.3. I / A is a complex algebra
aA (x) = ~ (x) u ~ (x).
P r o p o s i t i o n 3.2 enables us to quote, directly f r o m t h e well-known t h e o r y of complex algebras, t h e following t w o i m p o r t a n t results:
Proposition 3.4. For a n y element x in a normed algebra A , a A(x) contains at least one number ~ with [ ~ I >~v(x). I n particular a A(x) is never empty.
Proposition 3.5. I n a Banach algebra A , a A (x) is a compact set anal m a x I~1 = ~(x)
~ r A ( x )
/or all x E A .
A division algebra is a n algebra with i d e n t i t y in which e v e r y non-zero element has a two-sided inverse.
T h e o r e m 3.6. A normed real division algebra A is isomorphic to the real numbers (R), the complex numbers ( C) or the quaternions (Q).
Proo/. F r o m Definition 3.1 a n d P r o p o s i t i o n 3.4 follows t h a t e v e r y element t h a t is n o t a scalar multiple of e satisfies a n irreducible q u a d r a t i c equation. S u c h a n algebra is called quadratic a n d we p r o v e t h a t e v e r y q u a d r a t i c real (associative) algebra is isomorphic to R, C or Q.
I r A has dimension 1 or 2 it is clearly isomorphic to R or C. A s s u m e therefore t h a t there are x, y E A so t h a t {e, x, y} is linearly independent. Since a n y x o which is n o t a multiple of e satisfies an e q u a t i o n (x 0 - ~ e ) 2 = - ~ 2 e , ~ :4:0, we c a n assume x ~ = y 2 = - e . W i t h a = x + y, b = x - y we h a v e ab + ba = 0 a n d
x y + yz = a ~ - x ~ - y~ = aea +~e, x y + y x = x ~ + y2 _ b ~ =fib + (~e
ARKIV FOR MATEMATIK. B d 5 n r 16 for scalars ~,fl,~,,5. Since {e, a, b} is also linearly independent ~ = f l = 0 and ? = & We now p u t e ~ = ( 2 - ? ) - t a , e ~ = ( 2 + ? ) - 8 9 b, ea=ele ~ and notice t h a t e,e~,e2,e3 satisfy exactly the multipheation rules of the four basis elements of Q. Let their linear span in A be Q0. We prove t h a t Q0 = A.
Assume t h a t there exists a z ~ Q0. As before we can take z 2 = - e . F r o m the quadratic law follows, for i = 1, 2, 3:
e~z + ze, = (et + z) 2 - e~ - - z 2 = ~ ( e , +z) +~,e, e,z + z e , = -- (e, --z) ~ + e~ + z ~ =#,(e, +z) +v,e.
The linear independence gives u~ =p~ = 0 and v~ =2~. Take u =r + ~.~2~ e~). Then e~ u +ue~ = 0 for i = 1, 2, 3 and ~ can be chosen so t h a t u 2 = - e . B u t the associative law is not satisfied; we compute uel e2u in two ways:
u( (ele2)u ) = u(eau ) = u( - ue3) = - u2ea = e.a,
(uel) (egu) = ( - e l u ) ( - u e 2 ) = e 1 u2e2 = - e l e 2 = -e3:
Hence Qo = A and the theorem is proved.
Related results are given in [16] a n d [17].
4. G e l f a n d r e p r e s e n t a t i o n
The remarkable result of Gelfand t h a t a complex eomr~utative semi-simple Banach algebra is isomorphic to an algebra of continuous functions generalizes to real alge- bras in general. Some modifications of a technical nature are needed.
Throughout this section we let A be a real normed algebra. A left ideal I is called m o d u l a r if there exists an element e~ such t h a t x - xez E I for all x E A. A direct conse- quence of Theorem 3.6 is t h a t if A is commutative a n d M is a closed modular maximal ideal A I M is isomorphic to R or C. The (Jacobson) radical R A is the intersection of all modular maximal left ideals. A is called s e m i - s i m p l e if RA = (0} and radical if RA = A . Given a real normed algebra A, let CA be the set of non-zero continuous real algebra homomorphisms of A into C. We will call CA the Gel/and space of A. The connection between CA and the set of closed modular maximal ideals, ~ , is slightly more com- plicated t h a n in the complex case and will now be described.
Given ~0 ECA we define ~ b y ~q(x) =~(x) (complex conjugate). This ~, as a function from CA into CA, is called the conjugate mapping. Since a n y ~ EtA maps A onto either R or C we can distinguish a "real" and a "complex" part of CA
r = {r o f ( A ) = R ) = {cp; ~q; = e l ) ,
The extreme eases when either r or CA c is e m p t y will be discussed somewhat in Chap- ter I I .
Two homomorphisms ~0,~p ECA are called equivalent if ~ =yJ or ~ =vyJ. I f C~ is the set of equivalence classes we have
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L . INGELSTAM, Real Banach algebras
Proposition 4.1. For a commutative real normed algebra A with Gel/and space CA and conjugate mapping 1:, the set C~ is in one-to-one correspondence with ~ , the set of closed modular maximal ideals.
Proof. We define the function T-+Ker~o,~0
EtA.
Since K e r ~ =KerT~0 a n d Ker~0 is a closed modular m a x i m a l ideal this defines a function g from Ca into ~/. F r o m Theorem 3.6 follows t h a t g is surjective. To prove t h a t it is also injective assume K e r ~0 = Kerry.Then ~0 and ~0 induce an a u t o m o r p h i s m a on R or C so t h a t qo = a o ~o. B u t for R, ~ = identity a n d for C, ~ = i d e n t i t y or complex conjugation. Hence ~ a n d ~ are equivalent and g injective.
We now proceed to define the Gelfand representation a n d state the results in two theorems.
Definition 4.2. For an element x o / a normed algebra A, the Gelfand /unction ~ is a complex-valued/unction on CA defined by
~(~) =~(x), ~ Eta-
Theorem 4.3. iT/A is a real normed algebra and the Gel/and space Ca is given the weakest topology in which all Gel/and functions 2~, x E A, are continuous, then
(i) CA is locally compact,
(ii) Ca is compact i / A has identity, (iii) ~ is a homeomorphism,
(iv) C~ is a closed subset of ~ ~.
Proof. A * d~notes the real normed dual of A. We e m b e d Ca in A* • A* in a one-to- one manner ~-->~' = (89 +T), ~ ( ~ - ~ ) . 1 An element (/,g)EA* x A* belongs to the image C~ if and only if (f,g) 4~0 a n d
/(xy) =/(x) f (y) - g(x) g (y), g(xy) =/(x) g (y) +/(y) g (x)
for all x, y E A . We topologize A* with the weak* topology a n d A* • r and Ca accordingly. This topology on Ca is the weakest in which the Gelfand functions are continuous. A n a t u r a l n o r m on A induces a n o r m on A* and it is easy to see t h a t C~ c S 1 • S1, where S 1 is the unit sphere in A. Since S 1 is weak* compact, the closure q~ is compact. B u t a (f,g)E q~ m u s t satisfy the multiplicative relation above a n d consequently either belong to C~ or be = 0 . Thus we have two cases:
I. 0 r q~; Ca is compact. I n particular if A has identity 0 cannot be a n accumulation point.
2. q~ =C~ U {0}; Ca is equal to a compact Hausdorff space minus one point, hence locally compact.
Since v corresponds to changing the sign of the second component in A* • A * (iii) is clear. A n y ~ E ~ m u s t satisfy v~ = ~ , hence ~ EC~ a n d CAR is closed.
Remark. Since T is a homeomorphism it is easy to verify t h a t ~ in a c o m m u t a t i v e A, given the topology of C~, is c o m p a c t or locally compact as Ca [7, w 10, No. 6, 10].
ARKIV FOR MATEMATIK. Bd 5 nr 16 Now we can give the Gelfand representation theorem for real commutative Banach algebras. We recall t h a t in a Banach algebra all the modular maximal ideals are closed [24, p. 43], in particular RA = f/M~mM. If gs is a locally compact Hausdorff space C0(~ ) denotes the Banach algebra of all complex continuous functions t h a t
" t e n d to 0 at infinity" (i.e. tend to 0 after the filter generated b y the complements of compact sets); the topology being defined b y the maximum norm. If ~ is compact C0(~ ) = C(~), the algebra of all continuous functions on ~.
Theorem 4.4. Let A be a real commutative Banach algebra and r its Gel/and space.
The algebra homomorphism h :x-->F: o / A into Co(~bn) has the properties (i) Ker h=RA,
(ii) an(X) =&(r (the range o/ &) /or all x, with the possible exception o/ 0 i/ A does not have identity,
(iii) max+E+Al&(~)l =v(x), (iv) h is continuous.
Proo/. F r o m the way the topology of CA is defined follows t h a t & E Co(~bA). From Proposition 4.1 follows t h a t K e r h = f/CECA K e r q = r MeTnM=R A and (i)is proved.
For given x and oc+ifl#O p u t Xo= (g2+f12)-i (x2_2ax). If oc+iflggan(X) then there exists a y so that x o + y = x o y , ~o+?)=~o?) and ~o(~)#1, &(q~)#o~+ifl for all % If, on the other hand, o:+iflEan(X) the set I = { y - y x o ; y E A ) is a proper modular ideal with x o = % B u t I is contained in some maximal modular ideal M with e M =e I = x 0.
According to Proposition 4.1 there exists ~E~bA so t h a t &0(~0)=&0(z~0)=1; hence takes the value ~ + ifl at ~0 or vq. If A has identity every ideal is modular, and x is regular if and only if it does not belong to any maximal ideal; in other words 0~an(X) if and only if &@) =t=0 for all ~ ECA. Now (ii) is proved. F r o m (ii) and Proposition 3.5 follows (iii). Since v(x) <~ IIx]] for a n y natural norm ]] "]1, h is continuous and the proof is finished.
Chapter II. Reality conditions
The main object of this chapter is to present a certain classification of real Banach algebras. I n see. 6 the class of real algebras t h a t can also be complex is characterized and four "reality conditions", R1-R4, are introduced. I t is shown (Theorems 6.5 and 6.8) t h a t these conditions stand for increasing degrees of "reality" and also that, when an identity is adjoined to an algebra, the reality properties are preserved (Lemma 6.9). The significance of the conditions in various parts of the t h e o r y is discussed in sec. 7. I t is finally shown that, under suitable finiteness assumptions, an algebra can be decomposed in a "complex" and a "real" part (Theorem 8.1). This result, like several notions and results of t h e c h a p t e r , is purely algebraic.
5. The modified exponential function
In a Banach algebra without identity we cannot define the usual exponential func- tion. I t is, however, always possible to define the function
~. xn
x --> ixp x = - n.T
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L, INGELSTAM, Real Banach algebras
w h i c h is d e f i n e d a n d c o n t i n u o u s for all x a n d m o r e o v e r satisfies i x p (x § y ) = i x p x o i x p y for c o m m u t i n g x a n d y. I f t h e a l g e b r a A is c o m m u t a t i v e i x p is a s e m i - g r o u p h o m o m o r p h i s m of (A, + ) i n t o (A, o). T h e f u n c t i o n
x2
n = l n
is d e f i n e d a n d c o n t i n u o u s for all x w i t h v(x) < 1 a n d satisfies i x p l(x) = x. I f t h e r e is a n i d e n t i t y e we c l e a r l y h a v e
e x p x = e - i x p x , log(e - x ) = l(x).
F o r f u t u r e reference we m a k e t w o t e c h n i c a l r e m a r k s c o n c e r n i n g t h e i x p f u n c t i o n . L e m m a 5.1. I / , / o r a topologically nilpotent element x i n a B a n a c h algebra A , i x p ~x is a bounded/unction o/ the real variable c~( - ~ < ~ < ~ ) then x = 0 .
Proo/. F o r a c o n t i n u o u s l i n e a r f u n c t i o n a l / , t h e f u n c t i o n
(Z n
~0 : ~ ( a ) = [ ( i x p ~ x ) = - ~ ~.. ](x ~)
72=1
c a n be e x t e n d e d t o a n e n t i r e f u n c t i o n F on t h e whole c o m p l e x p l a n e . Since /(x=) lI~ <~ K 1/~ ]1 x~ II 1/n w e h a v e t h a t
Izl
for e v e r y 5 > 0. H e n c e ~ is a t m o s t of o r d e r one, m i n i m u m t y p e , a n d since i t is b o u n d e d on t h e r e a l axis a P h r a g m d n - L i n d e l 6 f t h e o r e m [5, p. 84] tells t h a t i t m u s t b e b o u n d e d . H e n c e / ( x ) = 0 for a r b i t r a r y / a n d x = 0.
L e m m a 5.2. I / /or some x there exists a convergent sequence ~ n , n = l , 2 ... o/ real numbers such that i x p ~ x = 0 / o r all n then x = O.
Proo/. L e t / be a l i n e a r f u n c t i o n a l a n d ~, yJ as in t h e p r o o f of L e m m a 5.1. N o w yJ is a n a n a l y t i c f u n c t i o n w i t h a n o n - i s o l a t e d zero, hence ~ = 0 , / ( x ) = 0 for all / a n d x = 0 .
6. R e a l i t y conditions
A s we h a v e a l r e a d y p o i n t e d o u t (sec. 1) t h e c o m p l e x (normed) a l g e b r a s can be r e g a r d e d as a subclass of t h e r e a l (normed) algebras. N e x t we l o o k i n t o t h i s s i t u a t i o n in some d e t a i l .
Definition 6.1. A real (normed) algebra is said to be o / c o m p l e x type i / i t is possible to extend the scalar multiplication to complex scalars so that the algebra becomes a complex (normed) algebra.
A s o m e w h a t m o r e t e c h n i c a l d e s c r i p t i o n can be given:
ARKIV FOR MATEMATIK. B d 5 n r 1 6 Proposition 6.2. A real (normed) algebra A is o/complex type i / a n d only i/there exists a (continuous) linear operator J on A satis/ying
J(ab) = J a .b =a .Jb and such that - j 2 is the identity map.
Proo]. The "only if" part is trivial; a-->ia is a map with the desired properties.
Assume J given with the properties above. A complex scalar multiplication can then be defined
(~ +i~)a = ~a + ~ J a .
If A is normed, let II" II be an admissible (real) norm. Following Kaplansky [20] we can construct an equivalent, complex norm
[11 x I1[ = max~ II cos v 9 x + sin V. J x II.
Corollary 6.3. A real (normed) algebra A with identity e is o] complex type i / a n d only i] there exists an element ~ in the center o] A, satis/ying ~2= _ e.
Proo/. If such a j exists, take J:x-+]x. If A is of complex type, ] = J e with the J of Proposition 6.2 has the desired properties.
The question whether scalar multiplication can be extended to complex numbers or not is not trivial even if the compatibility with multiplication is disregarded, i.e.
for real vector spaces. If the dimension is finite complex scalars can be introduced if and only if it is even. A n y infinite-dimensional real space can be given a complex structure in the algebraic sense. There are, however, infinite-dimensional real Banach spaces on which no complex multiplication exists, making them complex Banach spaces; Dieudonn6 [10].
Our main concern in the sequel will be real algebras t h a t are not of complex type.
We introduce conditions, signifying different degrees of "reality" of an algebra A.
The first set is such t h a t it can be used for algebras with or without topology:
Definition 6.4. A real (normed) algebra A is said to be R1, o/real type, i/ A is not o/complex type.
R 2 i / A does not contain any subalgebra o/complex type with identity.
R 4, o/strictly real type, i/ - x 2 is quasi.regular/or every x.
I t is clear t h a t the two following conditions are equivalent to R~:
R~': A does not contain a n y subalgebra isomorphic to the complex numbers.
R'2": The equation x 8 § = 0 has no solution in A except 0.
Theorem 6.5. For a real algebra A we have
(a) R 4 ~ R2,
(b) R 4 ~ R 2 ~ R 1 i / A has identity.
Proo/. If A is not R 2 we have some ]c=~0 satisfying/c +k3 =0. If A is also R 4 there exists a y such t h a t
0 = y o
(--]~2)=y_k2_t_yk~.
249
L. INGELSTAM, R e a l B a n a c h algebras
Multiplying b y / c gives/c3_ y(/c + ]ca) = 0 a n d k = 0 against t h e assumption. This proves (a). I n (b) it is trivial t h a t R 2 ~ R 1 if A has identity.
F o r a B a n a c h algebra we h a v e alternative formulations a n d one additional condi- tion.
Definition 6.6. A real B a n a c h algebra A is said to be R~ i / i x p x = 0 i m p l i e s x = O,
R'3 i / i x p ~x is a b o u n d e d / u n c t i o n o/c~, - co < ~ < ~ , o n l y i / x = O, R'4 i/ (~A(x) is r e a l / o r every x E A .
Proposition 6.7. F o r a B a n a c h algebra A (a) R 2 ~ R'2, (b) R 4 ~:~ R'4.
Proo/. (a) Assume t h a t A is n o t R~, i x p x = 0 with x ~ : 0 . Then, according to L e m m a 5.2, i x p 2 -~ x ~ : 0 b u t i x p 2 -n+l x = 0 for some n > 0 . P u t y = 2 - ~ - l x a n d t a k e e 0 = l / 2 i x p 2 y a n d i o e o - e o i x p y . T h e n e~ = e 0 a n d i ~ = - % a n d so t h e algebra g e n e r a t e d b y e 0 a n d i 0 is isomorphic to the complex numbers, a n d A is n o t R 2. If A is n o t R 2 we h a v e an element/C 4 0 such t h a t / c +lca = 0 . T h e n ixp2~/c = - / c s i n 2 z /c2(1 - c o s 2 z ) = 0 a n d A is n o t R~.
(b) I t is well k n o w n t h a t R 4 a n d R~ are equivalent if A is c o m m u t a t i v e [24, p. 119].
F o r a given x let B be a maximal, c o m m u t a t i v e subalgebra containing x. T h e n quasi- inverses of elements in B are a l r e a d y in B; in particular spectra of elements do n o t change (except possibly for 0) when restricted to B. I f A is R4, B is also R4, hence aB (x) a n d ~A(X) are real. Thus R 4 ~ R~ a n d since R~ ~ R a is obvious f r o m Definition 3.1 the proof is complete.
F r o m n o w on we leave o u t t h e primes in t h e conditions a n d use the different formu- lations interchangeably.
Theorem 6.8. F o r a real B a n a c h algebra A (a) R 4 ~ R 3 ~ R2,
(b) R 4 ~ R a ~ R 2 ~ R 1 i / A has identity, (c) R 4 ~ R 1 i / A is not radical.
(d) A n y radical A is Ra.
Proo/. Assume t h a t A is Ra. The quotient algebra A / R A (RA is the radical) is semi.
simple a n d R~, hence c o m m u t a t i v e (Theorem 7.1). If, for z C A / R A , i x p ~ z is bounded this is true also for its Gelfand f u n c t i o n h(ixp ~z) = ixp as B u t since 2 is real s = 0 a n d z = 0 . Hence, if x E A a n d i x p ~ x is bounded, x E R A a n d r ( x ) = 0 . B u t then, according to L e m m a 5.1, x = 0 a n d A is R a. R 3 ~ R 2 is obvious since if i x p x = 0 , i x p ~ x is periodic in ~, hence bounded, a n d x = 0 . T h u s (a) is proved. T h e n (b) follows f r o m T h e o r e m 6.5. F o r (c) let A be strictly real a n d P a primitive ideal (see [18, Ch. I]). Since P is closed A l P is a primitive B a n a c h algebra. A n a r g u m e n t b y K a p l a n s k y [20, p. 405]
shows t h a t A l P is isomorphic to the real numbers. I f A is also of complex t y p e then, for e v e r y x, x 2 = - ( J x ) 2. T h e n x m u s t be m a p p e d into 0 of A / P a n d x E P for e v e r y P . B u t this implies t h a t every x is in t h e radical, c o n t r a r y to t h e assumption. (d), finally, is trivial since all elements in a radical algebra are quasi-regular.
250
ARKIV F()R M~TEMATIK. B d 5 n r 1 6 T h e o r e m s 6.5 a n d 6.8 a p p a r e n t l y give us r e a s o n to r e g a r d R 1 t h r o u g h R 4 as suc- c e s s i v e l y s t r o n g e r r e a l i t y c o n d i t i o n s . I t is e a s y t o see b y e x a m p l e s t h a t n o n e of t h e f o u r are e q u i v a l e n t (see [15, p. 30]).
W e also n o t i c e t h a t t h e r e a l i t y c o n d i t i o n s a r e p r e s e r v e d w h e n a n i d e n t i t y is a d j o i n e d . G i v e n A , A 1 is t h e a l g e b r a d e f i n e d in sec. 2.
L e m m a 6.9. For a real (normed) algebra A (a) A1 is always R1,
(b) A 1 is R 2 i] and only i/ A is R 2.
I / A is a Banach algebra
(c) A 1 is R 3 i/ and only i/ A is Ra, (d) A 1 is R 4 i / a n d only i/ A is Ra.
Proo]. (a) a n d (b) a r e t r i v i a l . F o r (c) we n o t i c e t h a t for ( ~ , x ) E A 1 i x p ~(~, x) = (1 - e x p a~, e x p a ~ . i x p ~x).
F o r t h i s t o be b o u n d e d i t is n e c e s s a r y t h a t ~ = 0 ; t h u s i t is b o u n d e d if a n d o n l y if
= 0 a n d i x p ~x is b o u n d e d in A .
I n (d), i t is clear t h a t if A 1 is Rr t h e n A is R 4. F o r t h e " i f " p a r t a t e c h n i q u e d u e t o Civin a n d Y o o d [9] is used. W e a s s u m e t h a t A is R 4 a n d p r o v e t h a t ~ , ( ( ~ , x ) e) is non- n e g a t i v e r e a l for a n y (~, x) E A 1. T a k e h - 2~x § x 2 a n d l e t B be a m a x i m a l c o m m u t a t i v e s u b a l g e b r a of A c o n t a i n i n g h. B is a c o m m u t a t i v e B a n a c h a l g e b r a and, t h a n k s t o t h e m a x i m a l i t y , a s ( x ) =0A(x) ( e x c e p t p o s s i b l y for 0) for x E B. I f B is r a d i c a l t h e conclu- s i o n is clear, h e n c e a s s u m e t h a t B h a s a n o n - e m p t y G e l f a n d s p a c e CB a n d a G e l f a n d r e p r e s e n t a t i o n x-~&. F o r e v e r y ~v E r t h e r e is a u E B such t h a t u(~) = 1. W i t h y = ~u
+ x u we h a v e y2 = ~2u2 + hu ~ C B a n d
0 < ~ ( q ) = ~2~(~) + ~(q) ~ ( ~ ) = ~ + ~@).
H e n c e a ~ ((0, h)) = aA (h) = aB (h) >~ - ~2 a n d f i n a l l y aA, ((~, x) ~) = aA~ ( ( ~ , h)) = ~2 + a~ (h) > 0 a n d t h e p r o o f is finished.
7. Significance o f the conditions
S t r i c t r e a l i t y , R4, was u s e d a l r e a d y in G e l f a n d ' s original p a p e r [11] in o r d e r t o m a k e sure t h a t t h e r e p r e s e n t a t i o n of a r e a l c o m m u t a t i v e B a n a c h a l g e b r a o n l y con- sists of r e a l - v a l u e d f u n c t i o n s . I t is clear f r o m s e c t i o n 4 t h a t , for a c o m m u t a t i v e r e a l B a n a c h a l g e b r a , R 4 a n d r =~b~ a r e e q u i v a l e n t c o n d i t i o n s , a n d t h a t , for a R 4 a l g e b r a , t h e G e l f a n d s p a c e CA c a n be i d e n t i f i e d w i t h t h e set of m o d u l a r m a x i m a l ideals. R 4 is in f a c t a v e r y s t r o n g c o n d i t i o n , w h i c h is seen for i n s t a n c e f r o m t h e r e m a r k a b l e r e s u l t t h a t a R 4 B a n a c h a l g e b r a is c o m m u t a t i v e m o d u l o i t s r a d i c a l :
T h e o r e m 7.1. ( K a p l a n s k y ) A strictly real semi-simple Banach algebra is commutative.
T h e p r o o f [20, p. 405] a m o u n t s t o s h o w i n g ( b y m e a n s of t h e d e n s i t y t h e o r e m ) t h a t a n y p r i m i t i v e R 4 B a n a c h a l g e b r a is i s o m o r p h i c t o t h e r e a l n u m b e r s . Since a s u b d i r e c t s u m of c o m m u t a t i v e r i n g s is c o m m u t a t i v e t h e conclusion follows.
L. INGELSTAM, Real Banach algebras
I n C h a p t e r I V s o m e c r i t e r i a for s t r i c t r e a l i t y a r e given. I n t h a t c o n t e x t t h e closely r e l a t e d p r o b l e m t o c h a r a c t e r i z e t h o s e r e a l a l g e b r a s whose G e l f a n d r e p r e s e n t a t i o n consists of all r e a l f u n c t i o n s in C0(r ) is discussed.
T h e R a c o n d i t i o n was i n t r o d u c e d b y t h e p r e s e n t a u t h o r in c o n n e c t i o n w i t h s t u d i e s of t h e g e o m e t r i c a l p r o p e r t i e s of t h e u n i t s p h e r e [15]. A p o i n t o n t h e b o u n d a r y of a c o n v e x set is c a l l e d a vertex if no s t r a i g h t line t h r o u g h t h e p o i n t is a t a n g e n t of t h e set.
A r e a l B a n a c h a l g e b r a w i t h i d e n t i t y e is s a i d t o h a v e t h e vertex property if e is a v e r t e x of e v e r y u n i t s p h e r e t h a t belongs t o a n a d m i s s i b l e n a t u r a l n o r m . W e h a v e [15, T h e o r e m 2]:
T h e o r e m 7.2. A real Banaeh algebra with identity has the vertex property i / a n d only i / i t is R 3.
I t w o u l d be i n t e r e s t i n g t o k n o w if t h e r e is a n a l g e b r a i c c o n d i t i o n e q u i v a l e n t t o R a (cf. P r o p o s i t i o n 6.7).
Since a n y a l g e b r a w i t h o u t n o n - z e r o i d c m p o t e n t s is Re, a n d so q u i t e a few c o m p l e x a l g e b r a s a r e in f a c t Re, t h i s c o n d i t i o n seems r a t h e r weak. F r o m C h a p t e r I I I , however, i t will be clear t h a t R e is t h e " c o r r e c t " r e a l i t y c o n d i t i o n for t h e q u a s i - r e g u l a r group;
in R e r e a l a l g e b r a s t h i s g r o u p h a s d i s t i n c t i v e l y d i f f e r e n t c o n n e c t i v i t y p r o p e r t i e s f r o m t h a t of c o m p l e x ( t y p e ) algebras.
8. A decomposition theorem
I n t h i s section, w h i c h is p u r e l y algebraic, we give a t h e o r e m to t h e effect t h a t a c o m m u t a t i v e r e a l a l g e b r a s a t i s f y i n g some s u i t a b l e finiteness c o n d i t i o n c a n be s p l i t in a d i r e c t s u m of a c o m p l e x t y p e a l g e b r a a n d a R2-algebra, t h u s e n a b h n g us t o a c e r t a i n e x t e n t to d i s c r i m i n a t e b e t w e e n t h e " c o m p l e x " a n d t h e " r e a l " p r o p e r t i e s of t h e algebra.
W e consider t h e two c o n d i t i o n s on a n a l g e b r a A : DCS = A n y d e s c e n d i n g c h a i n
A = A o ~ A I ~ . . . ~ A ~ A n + I ~ ...
of ideals of A , such t h a t An+l is d i r e c t s u m m a n d in An, h a s o n l y a f i n i t e n u m b e r of d i s t i n c t m e m b e r s .
F I = T h e c e n t e r of A c o n t a i n s o n l y a f i n i t e n u m b e r of i d e m p o t e n t s .
Clearly, DCS is i m p l i e d b y t h e m o r e f a m i l i a r A r t i n d e s c e n d i n g c h a i n c o n d i t i o n for ideals, a ]ortiori b y A b e i n g f i n i t e d i m e n s i o n a l . The F I c o n d i t i o n , on t h e o t h e r h a n d , is o f t e n s a t i s f i e d b y f u n c t i o n a l g e b r a s . Moreover, DCS i m p l i e s F I .
T h e o r e m 8.1. A commutative real algebra, satis/ying the F I (or the DCS) condition, is the direct sum o / a n algebra o/complex type with identity and a R 2 algebra.
Proo/. I f A is n o t R2, we h a v e e l e m e n t s e 1 a n d k such t h a t - k e = e 1 a n d e~ = e x.
W i t h A I = e l A a n d A e = { a - e l a ; a E A } we h a v e A = A 1 @ A e,
w h e r e A 1 is of c o m p l e x t y p e . I f A is R 2 we t a k e A 1 = {0}. T h e n we c a n s p l i t u p A e in t h e s a m e m a n n e r , A e = A 3 $ A4, t h e n A 4 a n d so on. The r e s u l t can be d e s c r i b e d b y t h e d i a g r a m
ARKIV FOR MATEMATIK. B d 5 n r 16
A --> A 2 --> A 4 --~ A G - * ...
t t ~
A 1 A a A ~ A 7
t t t t
0 0 0 0
where all the maps are algebra homomorphisms and 1. At] horizontal maps are onto and split.
2. Sequences containing only one horizontal map are exact.
3. All members of the middle row are of complex type with identity.
Since each non-zero A 1 , A 3 . . . contains a different idempotent, the F I condition guarantees t h a t only a finite number of them are == {0 } and the upper row becomes stationary from some element on. (The D C S tells directly that only a finite number of the A, A 2... can be different.) Hence, for some n, we have
A = A 1 @ A a 9 ... ~ A2n_ 1 9 A2~ ,
where it is impossible to split up A2n non-trivially, hence A2~ is R 2. Since a direct sum of complex type algebras with identity is again of complex type with identity, the conclusion follows.
Without a n y finiteness assumption of the type D C S or F I , the conclusion of Theo- rem 8.1 is no longer valid. To see this, let cR be the algebra of all sequences of complex numbers converging to real limits (with component-wise multiplication). Assume t h a t
cR =A~ 9 AR
with Ac of complex type with identity. Its imaginary unit/c = { ~ } satisfies/c + k 3 = 0 , hence we must have ~ = + i or 0, and consequently there is an N such t h a t ~ =0, n > ~ N . Hence ~n=0, n > ~ N for all {~} EAc. But A n is never R2: the elements with an arbitrary complex number in the N t h position and O's elsewhere belong to An, but they clearly form a subalgebra isomorphic to the complex numbers.
Without the commutativity condition the theorem does not hold; for a finite-di- mensional (i.e. D C S ) counterexample take the algebra of all 2 • 2 matrices with real entries. This algebra is obviously not of complex type, but it contains a subalgebra isomorphic to the complex numbers, namely matrices of the form
hence it is not R 2. B u t it is simple, thus indecomposable.
R e m a r k 1. Since the decomposition in Theorem 8.1 is in fact effected by an idempo- tent it is clear that, if A is a topological algebra, both summands are closed. I n parti- cular if A is a Banach algebra the summands are Banach algebras.
R e m a r k 2. There exists a purely ring-theoretic anMogue to Theorem 8.I. A ring (R, + , .) is said to be of complex type if there exists a group endomorphism J on 253
L. INGELSTAM, Real Banach algebras
(R~/+), satisfying J ( a b ) = J ( a ) . b = a . J ( b ) a n d such t h a t - J ~ is t h e identity. I t is called R~ if the equation x a + x = 0 has no solution x # 0 in R. I f the conditions D C S a n d F I are read with "ring" instead of " a l g e b r a " we have:
A commutative ring, satis/ying F I (or DCS), is the direct sum o / a ring o/complex type with identity and a R2 ring.
Chapter III. The quasi-regular group
I n this chapter the m a i n object of s t u d y is the group of quasi-regular elenl~nts of a c o m m u t a t i v e B a n a c h algebra. We first show t h a t for a "sufficiently real" (i.e. R2) algebra the principal component of the quasi-regular group is simply connected (Theo- rem 9.3). This depends on the extension of a result b y B l u m [4] to the general case of real algebras without identity.
A theorem b y Lorch [21] says t h a t in a complex c o m m u t a t i v e B a n a c h algebra with identity the (quasi-) regular group either is connected or has an infinite n u m b e r of components. This is obviously not true for real algebras in general b u t we obtain a satisfactory analogue for real c o m m u t a t i v e R2 algebras: the components of the quasi- regular group are a t least as m a n y as the idempotents of the algebra, equal in n u m b e r if the set of components is finite (Theorem 10.3). I t is also shown t h a t the Lorch result is true even without the assumption of identity. Finally (sec. 12) we m a k e some r e m a r k s on how a recent result on the cohomology of the m a x i m a l ideal space is related to the questions discussed in the chapter.
9. Structure o f Gg
The set of quasi-regular elements, which we have called G q, is a group under the circle operation (xoy = x + y - ~ ) a n d a topological group in the topology of the alge- bra (see. 2). I t s principal component (the m a x i m a l connected subset of G r containing 0) is called G~. The following lemma, which is well known for the case with identity [24, p. 14], is technically important.
L e m m a 9.1. I n a Banach algebra A , Gg is the subgroup generated by ixp (A). I / A is commutative x - + i x p x is a homomorphism o / ( A , +) onto (G~, o).
Proo/. Since ixp tx, 0 ~ t <. 1, is a continuous p a t h in G q between 0 a n d ixp x we h a v e ixp (A) c G~. I f G' is the group generated b y ixp (A) we show t h a t G' is open a n d closed in G q.
There exists a neighbourhood U of 0 consisting only of elements i x p a , a E A (see. 5). F o r a x E G' the set x o U is a nei_ghbourhood of x a n d belongs to G', hence x is an interior point a n d G' is open. I f x E G' t h e n (x o U) f) G' is not e m p t y , hence an a exists so t h a t x o i x p a = g E G ' a n d xEG'. (As by-products we get t h a t G~ is pathwise connected a n d open in Gq.)
F r o m here on we restrict our a t t e n t i o n to c o m m u t a t i v e algebras. L e t HI(Gg) (or only YI1) denote the f u n d a m e n t a l group of G~. The following t h e o r e m generalizes a result b y Blum [4] for complex algebras with identity. We t a k e P = (x; ixp x = 0 ), which is a subgroup of (A, + ).
Theorem 9.2. I n a commutative Banach algebra HI(G~) is isomorphic to P.
ARKIV F f R MATEMATIK. B d 5 nr 16
Prop/. P is the kernel of the epimorphism ixp :A-->G~, hence A l P is isomorphic to G~. We show t h a t P is discrete, which is the case if 0 is an isolated point of P. Assume t h a t x e P a n d I[xll < 1. Since i x p x = 0 the s p e c t r u m of x only consists of points 2zdN, N=O, _ 1 , • .... Since I l x l l < l we h a v e t h a t the spectrum of x is {0} a n d so x is topologically nilpotent. B u t ~ ( ~ ) = i x p a x is continuous a n d periodic in ~, hence bounded, a n d L e m m a 5.1. shows t h a t q0(~) = 0 a n d x = 0 . Since A is locally connected, simply connected a n d locally simply connected, P discrete a n d ixp an open m a p , a theorem b y Schreier [28] applies a n d shows t h a t the f u n d a m e n t a l group of (the to- rus) A l P is isomorphic to P.
Theorem 9.3. For a commutative Banach algebra A the/ollowing statements are equi- valent:
(a) ixp :A-->G~ is an isomorphism, (b) A is R,,
(c) G~ is simply connected, (d) G~ is torsion/ree.
Prop/. F r o m L e m m a 9.1, Proposition 6.7 a n d Theorem 9.2, respectively, follows t h a t {a), (b) a n d (c) are all equivalent t o P =(0}. We prove t h a t (b) a n d (d) are equiva- lent.
Assume t h a t A is not R 2. Then i x p x = 0 for some x # 0 . F r o m L c m m a 5.2 follows t h a t there exists an integer n >~ 2 such t h a t y~ = i x p n - i x #0. Then y~ EGg a n d y ~ 0 so G~ is not torsion free. N o w assume t h a t there is a z E G~ such t h a t z # 0 a n d z ~ = 0 . According to L e m m a 9.1 z = i x p u for some u, a n d so ixpNu=z~ and A is not R~, which completes the proof.
Thus we h a v e t h a t for "sufficiently real" Banach algebras G~ is simply connected.
F o r every non- R 2 algebra,.in particular every complex algebra with identity, how- ever, II 1 has a subgroup isomorphic to the additive group of integers. F o r complex algebras without i d e n t i t y it can v e r y well h a p p e n t h a t Gg is simply connected. F o r e v e r y radical B a n a c h algebra, for instance, we h a v e A = G q = G~.
We conclude this section with a technical r e m a r k on c o m m u t a t i v e complex t y p e algebras.
T h e o r e m 9.4. A commutative real Banach algebra A with identity e is o/complex type i / a n d only i/ - e belongs to the principal component o/the regular group.
Proo/. I f A is of complex type, exp (iqe), 0 ~<q ~<~t, is a p a t h in G between e a n d - e ; hence - e belongs to the principal component Ge. I f - e E Ge t h e n ( L e m m a 9.1) there is a u E A such t h a t exp u = - e. B u t then j = exp 89 is an i m a g i n a r y unit a n d A is of complex type.
t 0 . Components
of
G qI n this section we s t u d y the quasi-regular group G q of a c o m m u t a t i v e B a n a c h algebra, in particular the n u m b e r of components of G q. F o r complex B a n a c h algebras with i d e n t i t y we h a v e a result b y Lorch [21, Theorem 12]:
G q (and G) either is connected or has an infinite number o/components.
I t is i m m e d i a t e t h a t this does n o t hold for real algebras in general. F o r instance for the real n u m b e r s R, G q = R - ( 1 } a n d has two components. A c o u n t e r p a r t to Lorch's t h e o r e m will be given for real R2-algebras. As a p r e p a r a t i o n we prove two lemmas.
18:3
L. INGELSTAM, Real Banach algebras
L e m m a 10.1. Every element o/finite odd order in G q belongs to its principal component
Proof. LetZ~ ~ =0, n odd. The line segment
A@) = ( 1 - ~ ) r , 0 <T ~< 1, connects 0 and r. B u t with
B(T) = [(1-- T) n - ( - $ ) n ] - I ~ (1--T)k('T)n-k-lr'/~ n-1 k=l
we have A ( ~ ) o B ( ~ ) = 0 . Hence A@)EG q and rEGg. (This result obviously holds in a n y power associative real topological algebra, i.e. not necessarily commutative, associative or normed.)
L e m m a 10.2. Let A be an abelian topological group such that its principal component is torsion/ree. 1 / x and y are two elements o/finite order in A they belong to di//erent components unless x =y.
Proo/. Assume x and y are in the same component, A'. z-->x-lz maps A ' homeo- morphically onto the principal component Ae. Then x-lyEAe and x - l y is of finite order, hence x-ly = e and x = y.
F o r the formulation of the main theorem we introduce the set of idempotents i = { y ; y2 =~}. Gq/G~, the set of components of G q, is called K.
Theorem 10.3. I n a commutative R 2 Banach algebra holds (a) card I <. card K,
(b) card I =card K i / K is finite.
Proof. We first introduce the subset S of G q consisting of all elements of order 2, S = {x; x ~ = 2 x - x 2 = O } . Evidently x--->~x is a one-to-one map of S onto I , so card 1 = card S.
To every element x E S we associate the component of G q, Fx, in which it lies, defin- ing a function
/:x-+Fx
of S into K. B u t G8 is torsion free (Theorem 9.3) and L e m m a 10.2 shows t h a t / is injective, which proves (a).
Now assume t h a t K is finite, take an arbitrary element F E K and let k EF. The powers k ~ n = l , 2 , . . . , cannot all lie in different components. If k ~ and k ~162 are in the same component, k~176162176 and there exists u E A such t h a t i x p u = k ~ (Lemma 9.1). The element v = k o i x p ( - u / 1 ) satisfies v~ and v e F . L e m m a 10.1 together with the fact t h a t Gg is torsion free shows t h a t there are no elements in G q of odd, finite order. Hence l = 2 ~ and v ~ = 0 for some n ~>0. B u t in a R~ algebra, x ~ = 0 for some n > 1 implies t h a t already x~ To see this, assume t h a t x~ but x~ Then
k = 89 (2 x - 3 x ~ + x 3) = 89 (z ~ - x )
ARKIV FOR MATEMATIK. B d 5 n r 16
is 4=0, because otherwise x~176176 against the assumption. B u t k also satisfies k + k s = 0 (notice the formal identities k = 89 - x ) [ 1 - (1 - x ) 2 ] and (1 - x ) 4 = 1) a n d the algebra would not be R 2. Hence we have v~ y E S a n d F = F v so / is onto a n d the proof of (b) is complete.
The inequality in (a) above can be strict. An example (slightly modified after S.
K a k u t a n i ) is the following. A is the real R 2 algebra of all continuous complex func- tions on the unit circle I z l = 1 t h a t t a k e only real values a t z = 1. The e l e m e n t s / , , with /n(z) =zn, n = 0 , 1,2.., lie in different components of G, b u t the n u m b e r of idempo- tents is 2.
11. The regular group o f A 1
I n this section we investigate the connection between the properties of the quasi- regular group of A a n d t h a t of A 1 (A with adjoined identity, see. 2). Here we reserve the notation G and G 1 for the regular group of A~ a n d its principal component, while G q a n d G8 stand for the quasiregular group a n d its principal component in A.
F o r a given component F of G q we define the subsets of A r
F + = { ~ ( 1 , - g ) ; a > 0 , gEF}, F - = - F + Theorem 11.1. I[ A is a real Banach algebra then
(a) to every component F o/ G ~ in A correspond two components F + and F - o / G in A1,
(b) card G/GI=2 card Gq/G~,
(c) the/undamental groups o/ G~) and G 1 are isomorphic, Hi(G1) _~ 1L(aS).
Proo[. The m a p x--->(1, - x ) is a h o m e o m o r p h i s m of G q onto the subset (1, - G q) c G.
I f U is a m a x i m a l connected subset of G q then (1, - F ) o F + a n d F + is clearly m a x i m a l connected. F§ a n d U - are homeomorphic a n d lie in disjoint homeomorphic half- spaces, hence F - is also a component. Since for a n y component A of G either A or
- A m u s t contain a n element (1, - g ) , gEG q, (a) a n d (b) are proved.
F o r (c), we rely on Theorem 9.2. Since i x p ( ~ , x ) = 0 implies ~ = 0 we h a v e III(G1) _ {(~,x); ixp (~,x) = 0 } = {(0,x); i x p x = 0 } ~_ HI(Gg).
F o r an algebra of complex t y p e we can, as an alternative to the construction used above, adjoin a complex identity~ t a k e
A l e = C 9 A,
direct sum as n o r m e d vector spaces, with the algebra operations defined as in see. 2.
We use the notations G, G1, G q, Gg as above, a n d for a component F of G q we define the subset
r ~ {CO, - 9 ) ; C # 0 , g e r } of A w.
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L. INGELSTA~r Real Banach algebras
Theorem 11.2. I / A is a Banach algebra o/complex type and a complex identity i8 adjoined, then
(a) to every component F o[ G ~ in A corresponds a component F ~ o / G in A w, (b) card GIG 1 =card Gq/G~,
(c) the /undamental group o/ G 1 is isomorphic to the direct sum o/the fundamental group o/G~ with the integers Z,
YI I (G1) = YI 1 (G~) (t) Z.
Pro@ (a) and (b) follow as in Theorem 11.1. For (c) we notice t h a t i x p ( ~ , x ) = (1 - e x p , , exp~ .ixpx) = 0 if and only if ~ = 2 ~ i n , nEZ, and ixpx =0. Hence Theorem 9.2 gives
II~(G~) _~ {(~, x); ixp (~, x) = 0} = {(2~in, x); n EZ, ixp x = 0 } _ Z 9 II~(G~).
A simple consequence of Theorem 11.2 (b) is t h a t Lorch's theorem (see sec. 10) generahzes to the case without identity:
Theorem 11.3. I n a Banach algebra o/ complex type the quasi-regular group either is connected or has an in/inite number o/components.
12. Remarks on a cohomology result
A recent result due to Arens [2] and R o y d e n [25], [26] is that, for a complex com- mutative Banach algebra with identity, G/G1 (as a group) is isomorphic to HI(~/I,Z), the first ~ech cohomology group of the maximal ideal space ~ with the integers as coefficient group. We can make two remarks on this, due to the fact t h a t / o r any compact space ~ , HI(~,Z) is torsion/ree.
This is immediately clear when ~ is a finite simplicial complex. Then HI(~,Z) is isomorphic to the direct sum of the free (Betti) part of H I ( ~ , Z ) and the torsion part of Ho(~,Z ) [8, p. 127]. Since H o is always free HI(~,Z) is torsion free.
For ~ an arbitrary, compact space H~(~,Z) is the direct hmit of HI(g2o,Z), where
~o are finite simphcial complexes: the finite open coverings (or "nerves" of such coverings) of ~ (for terminology, see [14, p. 132]). Since a direct limit of torsion free groups is torsion free, the conclusion follows.
Hence, if HI(s is not trivial, it is infinite. Since we also know of real commuta- tive Banach algebras in which GIG 1 is finite but non-trivial, we can make the two remarks:
1. For real, commutative Banach algebras in general it is not true t h a t G/G~
~HX(~,Z) for any compact space ~ .
2. For complex, commutative Banach algebras with identity the result t h a t G/GI~HI(TII, Z) (Arens, Royden) implies t h a t GIG 1 has either one or an infinite number of elements (Lorch).
Chapter IV. Strict reality- and full function algebras
The first part (sec. 13) of this chapter contains criteria for a real Banach algebra to be strictly real (R4). One of the conditions (Theorem 13.1) is analytic in character, the other (Theorem 13.3) deals with geometric properties of the unit sphere and corn- 258
ARKIV FOR MATEMATIK, Bd 5 nr 16 plements previous results b y the a u t h o r [15]. These conditions can also be formulated in t e r m s of an ordering of the algebra (see. 14). I n section 15 we give criteria for a Banach algebra A to be homeomorphically (or isometrically) isomorphic to a C~(~), the algebra of all continuous real functions on a locally compact space t h a t tend to 0 at infinity. The k e y to these results is the observation (Theorem 15.2) t h a t if, in a strictly real algebra, t h e spectral radius v(x) is in a certain sense compatible with the topology, the algebra is isomorphic to C0(~bA). The results generahze a theorem b y Segal [29] t h a t has been used in his work on the foundations of q u a n t u m mechanics.
13. Conditions for strict reality
We recall the definition ixp x = - ~ n % ~ (n!)-i xn a n d s t a r t with an analytic criterion of strict reality.
Theorem 13.1. 1[ in a real Banaeh algebra i x p ( - ~ x 2 ) , ~ > ~ 0 , is a bounded/unction o / ~ / o r every x, then the algebra is strictly real.
Proo/. T a k e an a r b i t r a r y element x of the algebra A a n d let B be a maximal, com- m u t a t i v e subalgebra containing x. B is closed a n d the spectrum of x in B coincides with the spectrum of x in A (except possibly for 0). I f k is the image of x in the Gelfand representation of B, ixp ( - ~k~) is a bounded (complex function-valued) function of
~. Then Re &2>~0 a n d the spectrum of x is contained in the "double wedge" / I m p /
~<[Re~[ of the complex plane. Since this holds for every element it follows from the spectral m a p p i n g theorem (for powers) t h a t the spectrum of every element m u s t be real.
The condition t h a t i x p ( - gx 2) is bounded for all x, which is sufficient for strict reality, is not necessary, which will be shown b y examples. We first m a k e an observa- tion a b o u t "topologically v e r y n i l p o t e n t " elements.
L e m m a 13.2. I[ x is an element o/ a Banach algebra such that IIx"[l~l"=o(n-1), n - + ~ , and ixp ~x is bounded/or ~ 0 then x = O .
Proo/. F o r a n y continuous linear f u n c t i o n a l / , / ( i x p ~c) =~(~) is an analytic function on the real line t h a t can be extended to an entire function on the whole complex plane. As such, it is at most of order ~,1 m i n i m u m type, a n d since it is also bounded on the positive real line a Phragmdn-Lindel6f theorem tells t h a t it m u s t be constant (cf. L e m m a 5.1). H e n c e / ( x ) =0 for all ] a n d x = 0 .
Remark. Results of this t y p e are obtained b y L u m e r a n d Phillips [22]. T h e y also disprove the conjecture b y Bolmenblust a n d Karlin [6] t h a t , in a B a n a c h algebra with identity, no r a y e + ~ x , a~>0, xE RA, can be a t a n g e n t of a n a t u r a l unit sphere.
I t is true, however, t h a t no full straight line e + ~x, - o o < a < ~ , x topologically nilpotent, can be a t a n g e n t of a n a t u r a l unit sphere ([15], cf. also L e m m a 5.1 and Theorem 7.2).
I n view of L e m m a 13.2 it is sufficient to exhibit an element x # 0 in a Banach algebra satisfying HxS"H a/- =o(n-1). The closed subalgebra generated b y x will then be a radi- cal Banaeh algebra and automatically strictly real (Theorem 6.8, (d)) b u t ixp( - ~ x ~) m u s t be unbounded. A trivial example is t h e n a nilpotent element x, where Ilxa"ll = 0 259
