Banach Spaces of Continuous Functions
4.2 A characterization of real C ( K )-spaces
Proposition 4.1.4.If K is a totally disconnected compact Hausdorff space, then the collection of simple continuous functions (i.e., functionf of the form f =n
j=1ajχUj whereU1, . . . , Un are disjoint clopen sets) is dense inC(K).
Proof. This is an easy deduction from the Stone-Weierstrass theorem as the simple functions form a subalgebra ofC(K).
We conclude this section with another basic theorem from the classical theory, the Banach-Stone theorem, whose proof is proposed as an exercise (see Problem 4.2).
Theorem 4.1.5 (Banach-Stone). SupposeKandLare two compact Haus- dorff spaces such that C(K) and C(L) are isometrically isomorphic Banach spaces. Then K andLare homeomorphic.
The Banach-Stone theorem appears forK, Lmetrizable in Banach’s 1932 book [8]. In full generality it was proved by M. H. Stone in 1937. In fact, general topology was in its infancy in that period, and Banach was constrained by the imperfect state of development of nonmetrizable topology; thus, for example, Alaoglu’s theorem on the weak∗ compactness of the dual unit ball was not obtained till 1941 because it required Tychonoff’s theorem.
One needs to know that certain spaces such as∞andL∞(0,1) areC(K)- spaces in disguise. The standard derivation of such facts requires considering the complex versions of these spaces as commutative C∗-algebras (or B∗- algebras) and invoking the standard representation of such algebras asC(K)- spaces via the Gelfand transform ([32], pp. 242ff). Readers familiar with this approach can skip the next section, which is presented to remain within the category of real spaces.
4.2 A characterization of real C ( K )-spaces
The approach in this section allows us to avoid some relatively sophisticated ideas in Banach algebra theory and gives a direct proof that∞ andL∞[0,1]
are indeedC(K)-spaces.
Definition 4.2.1.Suppose A is a commutative real Banach algebra with identityesuch thate= 1. The state spaceofAis the set
S={ϕ∈ A∗:ϕ=ϕ(e) = 1}. An element ofS is called astate.
Remark 4.2.2.The set of statesS of a commutative real Banach algebraA with identity is nonempty by the Hahn-Banach theorem, and S is obviously weak∗compact.
76 4 Banach Spaces of Continuous Functions
A+ will denote the closure of the set of squares inA, that is, A+={a2:a∈ A}.
The following lemma states two properties ofA+ which are trivially veri- fied, and therefore we omit its proof.
Lemma 4.2.3.
(i) Ifx, y∈ A+ thenxy∈ A+.
(ii) Ifx∈ A+ andλ≥0 thenλx∈ A+. Proposition 4.2.4.
(i) Ifx∈ Ais such that x ≤1 thene+x∈ A+. (ii)A=A+− A+.
Proof. (i) Letx∈ Asuch that x<1. By writing (1 +t)1/2in its binomial series∞
n=1cntn(where, in fact,cn =1/2
n
), valid for scalarstwith|t|<1, we see that the series ∞
n=1cntn is absolutely convergent, therefore convergent to somey∈ A. By expanding out (1 +t)1/2(1 +t)1/2for a real variabletwhen
|t|<1 it is clear that
m+n=k
cmcn=
1 ifk= 0,1 0 ifk≥2.
We deduce that y2 =e+x.SinceA+ is closed we obtain thate+x∈ A+ if x ≤1.
(ii) follows immediately (using Lemma 4.2.3) since ifx ≤1 we can write x=12(e+x)−12(e−x).
We aim to show that a real Banach algebraAwith identity is aC(K)-space if it satisfies one additional condition, that is:
Theorem 4.2.5 ([1]). Let Abe a commutative real Banach algebra with an identityesuch thate= 1. ThenAis isometrically isomorphic to the algebra C(K) for some compact Hausdorff spaceK if and only if
a2−b2 ≤ a2+b2, a, b∈ A. (4.1) In our way to the proof of Theorem 4.2.5 we will need two preparatory Lemmas which rely on the following simple deductions from the hypothesis.
Equation (4.1) gives
x−y ≤ x+y, x, y∈ A+. (4.2) So, ifx, y∈ A+ we also have
x ≤ 12
x−y+x+y
≤ x+y. (4.3)
4.2 A characterization of realC(K)-spaces 77 Lemma 4.2.6.SupposeAsatisfies the condition (4.1). Thenϕ(x)≥0when- everϕ∈ S andx∈ A+.
Proof. Takex∈ A+ withx= 1. By Proposition 4.2.4,e−x∈ A+ and, by (4.3),
e−x ≤ (e−x) +x= 1.
Hence forϕ∈ S we have
1 =ϕ ≥ϕ(e−x) = 1−ϕ(x), and thusϕ(x)≥0.
Lemma 4.2.7.SupposeAsatisfies(4.1). LetKbe the set of all multiplicative states of A, i.e.,
K={ϕ∈ S:ϕ(xy) =ϕ(x)ϕ(y) for all x, y∈ A}.
Then K is a compact Hausdorff space in the weak∗ topology of A∗ which contains the set∂eS of extreme points of S (and in particular is nonempty).
Proof. It is trivial to show that K is a closed subset of the closed unit ball of A∗ and so is compact for the weak∗ topology. Suppose ϕ ∈ ∂eS. Since A=A+− A+ it suffices to show that ϕ(xy) =ϕ(x)ϕ(y) whenever x∈ A+
andy∈ A.
Let x ∈ A+ such that x ≤ 1 and y ∈ A with y ≤ 1. By Proposi- tion 4.2.4,e±y∈ A+. Therefore, by Lemma 4.2.6
ϕ(x(e±y))≥0, which implies
|ϕ(xy)| ≤ϕ(x).
Similarly,e−x∈ A+ by Proposition 4.2.4 and so
|ϕ((e−x)y)| ≤1−ϕ(x).
If ϕ(x) = 0 orϕ(x) = 1, using the previous inequalities it is immediate that ϕ(xy) =ϕ(x)ϕ(y).
If 0< ϕ(x)<1, we can define states onAby ψ1(y) =ϕ(x)−1ϕ(xy) and ψ2(y) = (1−ϕ(x))−1ϕ((e−x)y) and then write
ϕ=ϕ(x)ψ1+ (1−ϕ(x))ψ2.
By the fact that ϕ is an extreme point of S we must have ψ1 = ϕ and, therefore,
ϕ(xy) =ϕ(x)ϕ(y), x∈ A+, y∈ A.
78 4 Banach Spaces of Continuous Functions
Proof of Theorem 4.2.5.SupposeAsatisfies the condition (4.1). LetJ :A → C(K) be the natural map, given by
J x(ϕ) =ϕ(x).
Clearly, J is an algebra homomorphism, J(e) = 1 and J= 1. In order to prove thatJ is an isometry we need the following:
Claim.Supposex∈ Ais such thatJ xC(K)≤1. Then for any >0 there existstε>0 so that
e−tε(1 +)e−tεx<1.
If the Claim fails, there isx∈ AwithJ xC(K)≤1 so that for some >0 we have
e−t(1 +)e−tx ≥1, t≥0.
By the Hahn-Banach theorem (separating the set{e−t(1 +)e−tx: t≥0} from the open unit ball) we can find a linear functionalϕwithϕ= 1 and
ϕ(e−t(1 +)e−tx)≥1, t≥0.
In particularϕ∈ S andϕ((1 +)e+x)≤0.Hence|ϕ(x)| ≥1 +.But now by the Krein-Milman theorem and Lemma 4.2.7, we deduce thatJ xC(K)>1, a contradiction.
Thus, combining the Claim with Proposition 4.2.4 (i), we have that J xC(K)≤1 implies (1 +)e+x∈ A+ for all >0, soe+x∈ A+.
Applying the same reasoning to−xwe havee−x∈ A+.Hence, by (4.2), we obtain
x=12(e+x)−(e−x) ≤ 12(e+x) + (e−x)= 1.
ThusJ is an isometry.
FinallyJ is ontoC(K) by the Stone-Weierstrass theorem.
Remark 4.2.8.We only needed the full hypothesis (4.1) at the very last step.
Prior to that we only use the weaker hypothesis
a2 ≤ a2+b2, a, b∈ A. (4.4) The condition (4.4) implies (4.3), which was used in Lemmas 4.2.6 and 4.2.7.
However, this hypothesis only allows one to deduce that J xC(K) ≥ 12x and soAis only 2-isomorphic toC(K). That this is best possible is clear from the norm onC(K) given by
|||f|||=f+C(K)+f−C(K)
where f+ = max(f,0) and f− = max(−f,0). Under this norm C(K) is a commutative real Banach algebra satisfying equation (4.4) but not equation (4.1).
Let us observe that if we consider A = ∞ (with the multiplication of two sequences defined coordinate-wise), Theorem 4.2.5 yields thatA=C(K) (isometrically) for some compact Hausdorff space K. This set K is usually denoted by βN. We also note that if (Ω,Σ, µ) is any σ-finite measure space thenL∞(Ω, µ) is again aC(K)-space. In each case the isomorphismpreserves order(i.e., nonnegative functions are mapped to nonnegative functions) since squares are mapped to squares.