A Characteristic Property of the Algebra C(Q)B

Siberian Mathematical Journal, Vol. 50, No. 1, pp. 77–85, 2009 c 2009 Karakhanyan M. I. and Khor kova T. A. Original Russian Text Copyright 

A CHARACTERISTIC PROPERTY OF THE ALGEBRA C(Ω)β M. I. Karakhanyan and T. A. Khor kova

UDC 513.83

Abstract: We study some properties of the algebras of continuous functions on a locally compact space whose topology is defined by the family of all multiplication operators (β-uniform algebras). We introduce the notion of a β-amenable algebra and show that a β-uniform algebra is β-amenable if and only if it coincides with the algebra of bounded functions on a locally compact space (an analog of M. V. She˘ınberg’s theorem for uniform algebras). Keywords: β-uniform algebra, cohomology, derivative, β-topology, amenability

Introduction Let Cb (Ω) be the algebra of all bounded continuous complex-valued functions on a locally compact space Ω with the uniform norm  · ∞ (f ∞ = supΩ |f |). Define the family of seminorms {Pg }g∈C0 (Ω) on Cb (Ω) as Pg (f ) = Tg f , where Tg : Cb (Ω) → Cb (Ω) is the multiplication operator Tg f = gf and C0 (Ω) is the subalgebra of all functions from Cb (Ω) vanishing at infinity. The topology on Cb (Ω), defined by this family of seminorms, is called the β-topology. The algebra Cb (Ω) in the β-topology is denoted by C(Ω)β (see [1, 2]). Thus, the β-topology on Cb (Ω) is the weakest topology in which all linear operators Tg , g ∈ C0 (Ω), are continuous and convergence of a net of functions {fi }i∈I of Cb (Ω) to f0 in the β-topology means that limI fi g − f0 g∞ = 0 for every g in C0 (Ω); i.e., the β-topology coincides with the strong operator topology under the standard isometric embedding of Cb (Ω) into the space of bounded operators C0 (Ω). A subalgebra closed in the β-topology A in C(Ω)β is called β-uniform if A contains constants and separates the points of Ω (i.e., for all x1 , x2 ∈ Ω, x1 = x2 , there exists f ∈ A such that f (x1 ) = f (x2 )). In this note we point out a number of properties of β-uniform algebras. We introduce the notion of a β-amenable uniform algebra and prove that a β-uniform algebra is amenable if and only if it coincides with C(Ω)β . An analogous result for uniform algebras was obtained by She˘ınberg (see [3]). § 1. The Algebra C(Ω)β The maximal ideal space of MΩ of Cb (Ω) can be represented as MΩ = F ∪ Ω, where F ∩ Ω = {∅}, F is the compact set that is the boundary of Ω in MΩ . Each function in Cb (Ω) is uniquely extendable to a function in C(MΩ ) of the same norm (C(MΩ ) is the Banach algebra of all continuous functions on MΩ with the uniform topology). For the sequel, we will need the following two simple assertions (see [1]): Lemma 1. (a) C(Ω)β is a β-complete locally convex algebra. (b) C0 (Ω) is everywhere dense in C(Ω)β . (c) The space of all β-continuous linear functionals on C(Ω)β is isomorphic to the space M (Ω) of all finite complex regular measures on Ω. Proof. (a) The definition of β-topology implies that every Cauchy net {fi }i∈I in the β-topology converges to some continuous function on Ω for every compact subset of the locally compact space Ω. Show that f0 ∈ C(Ω)β . Suppose the contrary. Then there exists {xn } in Ω such that |f0 (xn )| > n. Assume that g ∈ C0 (Ω) and g(xn ) =

f0 (xn ) √1 . n |f0 (xn )|

Since {fi } is β-Cauchy, {gfi } converges on Ω in the

Yerevan; Kazan . Translated from Sibirski˘ı Matematicheski˘ı Zhurnal, Vol. 50, No. 1, pp. 96–106, January–February, 2009. Original article submitted July 12, 2007. Revision submitted June 6, 2008. c 2009 Springer Science+Business Media, Inc. 0037-4466/09/5001–0077 

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uniform topology to a function bounded on Ω. On the other hand, √ lim(gfi )(xn ) = g(xn ) lim fi (xn ) ≥ n, n = 1, 2, . . . . i∈I

i∈I

We have a contradiction. Hence, f0 ∈ C(Ω)β . (b) The algebra C0 (Ω) includes a net {ei }i∈I that is an approximative unity for C0 (Ω), i.e., for every g ∈ C0 (Ω), the net {gei }i∈I converges to g uniformly on Ω. Given f ∈ C(Ω)β , the net {f ei }i∈I of C0 (Ω) β-converges to f since lim Tg f − Tg (f ei )∞ = lim gf − gei f ∞ ≤ f ∞ lim g − gei ∞ = 0 I

I

I

for every g ∈ C0 (Ω). (c) If φ is a β-continuous linear functional on C(Ω)β then φ is a continuous functional on the Banach algebra C(MΩ ). By the Riesz Theorem, there exists a finite regular Borel measure μ on MΩ representing φ; i.e.,  φ(f ) = fˆ dμ, MΩ

where fˆ is the Gelfand transform of f . Represent μ as the sum μ = μF + μΩ , where μF and μΩ are the restrictions of μ to F and Ω respectively. Show that μF = 0. Let {ei }i∈I be an approximative unity in C0 (Ω). Then the net {fi }i∈I , fi = 1 − ei , converges to the zero function in the β-topology of C(Ω)β . Therefore, the net of the functionals {fi φ}i∈I , (fi φ)(f ) = φ(fi f ), converges to the zero functional. Hence,     f f fˆ dμ 0 = lim(fi φ)(f ) = lim i f dμ + i f dμ = I

I

F

Ω

F

for all f in C(Ω)β . Consequently, μF = 0. Thus, to each β-continuous linear functional on C(Ω)β there corresponds some measure in M (Ω). The proof of the converse is trivial. The lemma is proven. Remark. Denote by C00 (Ω) the set of compactly-supported functions in C0 (Ω). If we define the β-topology on Cb (Ω) with the use of the operators {Tg : g ∈ C00 (Ω)} then Cb (Ω) is not β-complete and in this case the completion of Cb (Ω) coincides with the algebra of all continuous functions on Ω. Given an open set U in Ω such that the closure U of U in M (Ω) lies in Ω, denote by C0 (U ) the set of all functions in C(Ω) equal to zero on Ω\U . Lemma 2. (a) The uniform topology coincides with the β-topology on C0 (U ). (b) The linear space spanned by {C0 (Ui )}i∈I , where {Ui }i∈I is the family of all open sets in Ω such that Ui ⊂ U i ⊂ Ω, is β-dense in C(Ω)β . Proof. (a) Suppose that g ∈ C0 (Ω) is equal to 1 on U . Then Tg f = f for all f ∈ C0 (U ). Therefore, if a net {fi }i∈I β-converges in C0 (U ) then Tg fi = fi must converge to f0 uniformly. Since C0 (U ) is uniformly closed, f ∈ C0 (U ). (b) Obvious. The lemma is proven. § 2. β-Cohomology Let A be a β-uniform algebra on Ω. Since Cb (Ω) is complete in the β-topology, A is a closed subalgebra of Cb (Ω) in the  · ∞ -norm. Therefore, a β-complete uniform algebra A is also complete in the uniform norm. Denote by Ab the algebra A with the  · ∞ -norm. Let X be a Banach space and simultaneously a Banach Ab -module. We say that X is a β-complete Ab -bimodule if the fact that {fi }i∈I ⊂ A β-converges to f0 implies that the nets {fi x}i∈I and {xfi }i∈I 78

converge to f0 x and xf0 in the norm of the Banach space X for every x in X. The bimodule operation on X defines the bimodule operation on the dual X ∗ of X: (f ϕ)(x) = ϕ(xf ),

(ϕf )(x) = ϕ(f x)

for all f ∈ A , x ∈ X, and ϕ ∈ X ∗ . Call a linear functional ϕ ∈ X ∗ weakly∗ β-continuous if the fact that {fi }i∈I β-converges to f0 in A implies that the nets of the functionals {fi ϕ}i∈I and {ϕfi }i∈I converge to f0 ϕ and ϕf0 respectively in the weak∗ topology. If X is a β-complete Ab -bimodule then every linear functional ϕ ∈ X ∗ is weakly* β-continuous. Indeed, if a net {fi }i∈I β-converges to f0 then lim(fi ϕ)(x) = lim ϕ(xfi ) = ϕ(xf0 ) = (f0 ϕ)(x) I

I

for all x ∈ X. A continuous mapping D : Ab → X is called an X-derivative if D(f g) = f D(g) + D(f )g for all f and g in Ab . The mapping δx : Ab → X defined by the formula δx (f ) = [f, x] = f x − xf , x ∈ X, is called an inner derivative. Denote by Z 1 (A , X) the space of all continuous X-derivatives and denote by B 1 (A , X) the space of all inner derivatives. Define by H 1 (A , X) = Z 1 (A , X)/B 1 (A , X) the first cohomology group of Ab with coefficients in the Ab -bimodule X. The connection of the cohomology of topological algebras with the properties of the algebras can be found in [4, 5]. A derivative D : Ab → X is called β-continuous if the fact that a net {fi }i∈I in A converges to f0 in the β-topology implies that the net {D(fi )}i∈I converges to D(f0 ) in the norm of X. Now, let X be a β-complete Ab -bimodule. Then the inner derivative δx is β-continuous for every x. Denote by Zβ1 (A , X) the space of all β-continuous derivatives. Since every β-continuous derivative D : Ab → X is a continuous derivative from Ab into X; therefore, Zβ1 (A , X) is an abelian subgroup in Z 1 (A , X). Hence, Hβ1 (A , X) ⊂ H 1 (A , X) for a β-complete Ab -bimodule X. Applying the same arguments, we may define Zβ1 (A , X ∗ ), the abelian group of all derivatives D : A → X ∗ β-continuous in the weak* topology; i.e., if a net {fi }i∈I in A β-converges to f0 then the net of the linear functionals {D(fi )}i∈I converges to D(f0 ) in the weak* topology in X ∗ and Z 1 (A , X ∗ ), the abelian group of all derivatives D : Ab → X ∗ continuous in the weak* topology. Clearly, Zβ1 (A , X ∗ ) is a subgroup of Z 1 (A , X ∗ ). According to Johnson (see [6]), a Banach algebra Ab is called amenable if H 1 (A , X ∗ ) = Z 1 (A , X ∗ )/B 1 (A , X ∗ ) is trivial for every Ab -bimodule X, where B 1 (A , X ∗ ) is the abelian group of all inner derivatives δϕ (a) = aϕ − ϕa. Call an algebra A β-amenable if Hβ1 (A , X ∗ ) = Zβ1 (A , X ∗ )/B 1 (A , X ∗ ) is trivial for every β-complete Ab -bimodule X. Obviously, if A is an amenable algebra then A is βamenable; i.e., the quotient H 1 (A , X ∗ ) = 0 for every Ab -bimodule X implies that Hβ1 (A , X ∗ ) = 0 for every β-complete Ab -bimodule X. In this article we prove that β-amenability implies amenability for any β-uniform algebra. 79

§ 3. β-Complete Ab -Bimodules In this section we give two examples of β-complete Ab -bimodules which will be used in the sequel. Lemma 3. Let  μ ∈ M(Ω). Then there exist a measure ν in M (Ω) and a function g in C0 (Ω) such that μ = gν, i.e., f dμ = f g dν for every f in C0 (Ω). Proof. Without loss of generality, we may assume that μ is a positive probability measure. Let {Un }∞ n=1 be an increasing family of open sets such that the closure U n of Un is a compact subset in Un+1 and μ(Un ) > nk=1 21k . Using Urysohn’s Lemma, for each n, we may construct a positive function  n2 n2 n2 gn in C0 (Ω) such that gn ≡ 2n−1 on Un and gn ∞ = 2n−1 . The convergence of the series ∞ n−1 n=1 2  −1 implies that g = ∞ n=1 gn belongs to C0 (Ω). Show that ν = g μ also belongs to M (Ω). Indeed, since μ(Ω) = 1, we have n  1 1 μ(Ω \ Un ) = μ(Ω) − μ(Un ) < 1 − = n k 2 2 k=1

and μ(Un+1 \ Un )