CONTRACTIBILITY OF THE MAXIMAL IDEAL SPACE OF ALGEBRAS OF MEASURES IN A HALF-SPACE. CDAM Research Report LSE-CDAM

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1 CONTRACTIBILITY OF THE MAXIMAL IDEAL SPACE OF ALGEBRAS OF MEASURES IN A HALF-SPACE AMOL SASANE Abstract. Let H [n] be the canonical half space in R n, that is, H [n] = {(t,..., t n) R n \ {} j, [t j and t = t 2 = = t j = ] t j > } {}. Let M(H [n] ) denote the Banach algebra of all complex Borel measures with support contained in H [n], with the usual addition and scalar multiplication, and with convolution, and the norm being the total variation of µ. It is shown that the maximal ideal space X(M(H [n] )) of M(H [n] ), equipped with the Gelfand topology, is contractible as a topological space. In particular, it follows that M(H [n] ) is a projective free ring. In fact, for all subalgebras R of M(H [n] ) that satisfy a certain mild condition, it is shown that the maximal ideal space X(R) of R is contractible. Several examples of such subalgebras are also given. CDAM Research Report LSE-CDAM Introduction The aim of this paper is to show that the maximal ideal space X(R) of some Banach subalgebras (possessing a certain mild property) of the convolution algebra M(H [n] ) of all complex Borel measures with support in the half space H [n], is contractible. It follows then that such Banach algebras are projective free rings. All the notation and precise definitions are explained below. In particular, our result can be viewed as a two-fold generalization: () of the result in [9], from the one dimensional case (of the half space [,+ ) of R) to the n-dimensional case (the half space H [n] of R n ). (2) of the result in [8], from the specific subalgebra of almost periodic measures of M(H [n] ) to all subalgebras of M(H [n] ) satisfying a certain condition. (The result in [8] was in turn a generalization of a one-dimensional result of A. Brudnyi [2] to the multidimensional setting.) Although the current result is a generalization of the result from the conference paper [9], it does not follow automatically. Definition.. Let H [n] R n be the canonical half space defined by H [n] = {(t,...,t n ) R n \ {} j, [t = t 2 = = t j =, t j ] t j > } {}. M(H [n] ) denotes the set of all complex Borel measures with support contained in H [n]. Then M(H [n] ) is a complex vector space with addition and scalar multiplication defined in the pointwise manner as usual. The space M(H [n] ) becomes a complex algebra if convolution 99 Mathematics Subject Classification. Primary 54C4; Secondary 46J, 32A38, 93D5. Key words and phrases. contractibility of the maximal ideal space, convolution algebra of measures, projective-free ring, coprime factorization.

2 2 AMOL SASANE of measures (denoted henceforth by ) is taken as the operation of multiplication in the algebra. With the norm of µ taken as the total variation of µ, M(H [n] ) is a Banach algebra. Recall that the total variation µ of µ is defined by µ = sup µ(e k ), k= the supremum being taken over all partitions of H [n], that is over all countable collections (E k ) k N of Borel subsets of H [n] such that E k Em = whenever m k and k N E k = H [n]. The identity with respect to convolution in M(H [n] ) is the Dirac measure δ n in Rn supported at, given by δ n (E) = { if E, if E, where E is any Borel subset of H [n]. Definition.2. For µ M(H [n] ), define the measures µ [k] M(H [k] ), k = n,n,...,2,, inductively as follows. Set µ [n] = µ. Suppose µ [k] M(H [k] ) has been defined. Then µ [k ] M(H [k ] ) is defined by µ [k ] (E) = µ({} E), where E is any Borel subset of H [k ]. Given θ [,) and µ M(H [k] ), the measure µ θ M(H [k] ) is defined by () µ θ (E) = ( θ) t dµ(t), where E is any Borel subset of H [k]. If θ =, and k >, then while if k =, then set µ = µ({})δ. E µ := δ µ[k ], Notation.3. If R is a complex commutative unital Banach algebra, then X(R) denotes the maximal ideal space of R. Thus X(R) is the set of all nonzero complex homomorphisms from R to C. X(R) is endowed with the Gelfand topology, that is, the weak- topology induced from the dual space L(R; C) of the Banach space R. If R is any Banach subalgebra of M(H [n] ) which satisfies a mild assumption, namely Property (P) in Theorem.5 below, then we will show that X(R) is contractible. The notion of contractibility of a topological space is recalled below. Definition.4. A topological space X is said to be contractible if there exists a continuous map H : X [,] X and an x X such that for all x X, H(x,) = x and H(x,) = x. Our main result is the following: Theorem.5. Suppose that R is a Banach subalgebra of M(H [n] ) satisfying the property (P) For all µ R and all θ [,], µ θ, δ µ[n ] θ,..., δ n µ [] θ R. Then the maximal ideal space X(R) equipped with the Gelfand topology is contractible. In particular, by a result proved in [3], the above implies that R is a projective free ring. The definition of a projective free ring is given below.

3 CONTRACTIBILITY OF THE MAXIMAL IDEAL SPACE 3 Definition.6. A commutative ring R with identity is said to be projective free if every finitely generated projective R-module is free. Recall that if M is an R-module, then () M is free if M = R d for some integer d ; (2) M is projective if there is an R-module N and an integer d such that M N = R d. In terms of matrices (with entries from R), the ring R is projective free iff for every square matrix P satisfying P 2 = P, there exists an invertible matrix G such that see [4, Proposition 2.6]. GPG = [ Ik For example, it can be seen from the matricial definition that any field F is projective free, since matrices P satisfying P 2 = P are diagonalizable over F. Quillen and Suslin independently proved, that the polynomial ring over a projective free ring is again projective free (see [5]), and so in particular, the polynomial ring F[x,...,x n ] is projective free, settling Serre s conjecture from 955. In the context of Banach algebras, the following result was shown recently [3, Corollary.4.()]: Proposition.7. Let R be a semisimple complex commutative unital Banach algebra. If the maximal ideal space X(R) (equipped with the Gelfand topology) of the Banach algebra R is contractible, then R is a projective free ring. Recall that a commutative unital Banach algebra is said to be semisimple if its radical (that is, the intersection of all maximal ideals) is. Proposition.8. Every Banach subalgebra R of M(H [n] ) is semisimple. This will be proved at the end of Section 2. In light of Proposition.7, the main result given in Theorem.5 then implies the following. Corollary.9. Let R be a Banach subalgebra of M(H [n] ) satisfying the property (P) from Theorem.5. Then R is projective free. At the end of this article, we give examples of subalgebras R of M(H [n] ) which satisfy the property (P), which include several well-known classical convolution algebras of measures. Thus we have (with the notation explained in Section 4): Corollary.. Let R be one of the Banach algebras L (H [n] ) + Cδ n, A(H[n] ), APWΣ n or APΣ n. Then the maximal ideal space X(R) is contractible. In particular, R is projective free. The motivation for investigating whether or not convolution algebras of measures are projective free rings also arises from control theory, in the problem of stabilization of linear systems, since if R is a projective free ring, then every stabilizable plant with a transfer function over the field of fractions of R has a doubly coprime factorization. The reader is referred to [7], [3] for details. The proof of Theorem.5 is given in Section 3, while examples are given in Section 4. But first, a few technical results used in the sequel are proved in Section Preliminaries In this section, a few auxiliary facts needed to prove the main result are shown. Lemma 2.. Let µ M(H [k] ). Then ] ;

4 4 AMOL SASANE () µ θ M(H [k] ). (2) µ θ µ. (3) (δ k) θ = δ k for all θ [,] and all k =,...,n. Proof. () and (3) follow immediately from the definitions. The inequality in (2) is shown below. Note that µ θ = sup µ θ (E i ), the supremum being taken over all partitions (E i ) i N of H [k]. There exists a Borel measurable function w such that d µ (t) = e iw(t) dµ(t). So µ θ (E i ) = ( θ) t dµ(t) = e E i E iw(t) ( θ) t e iw(t) dµ(t) i = e iw(t) ( θ) t d µ (t) d µ (t) = µ (E i ). E i E i Hence µ θ (E i ) µ (E i ) = µ (H [k] ) = µ. Lemma 2.2. If µ,ν M(H [k+] ) and k, then (µ ν) [k] = µ [k] ν [k]. Proof. Let E H [k] be a Borel set. Then (µ ν) [k] (E) = (µ ν)({} E) = = = This completes the proof. {} E E {} E µ({} (E τ))dν [k] (τ) µ(({} E) t)dν(t) µ [k] (E τ)dν [k] (τ) = (µ [k] ν [k] )(E). Lemma 2.3. If µ,ν M(H [k+] ) where k, then (δ µ[k] ) (δ ν[k] ) = δ (µ[k] ν [k] ). Proof. (The notation Fµ is used for the Fourier transform of µ: (Fµ)(w) = e iwt dµ(t), w R). For w R and ω R k, F((δ µ [k] ) (δ ν [k] ))(w,ω) = (F(δ µ [k] ))(w,ω) (F(δ ν [k] ))(w,ω) = (Fδ )(w ) (Fµ [k] )(ω) (Fδ )(w ) (Fν [k] )(ω) = (Fµ [k] )(ω) (Fν [k] )(ω) = (Fµ [k] )(ω) (Fν [k] )(ω) = (F(µ [k] ν [k] ))(ω) = (F(µ [k] ν [k] ))(ω) = (Fδ )(w ) (F(µ [k] ν [k] ))(ω) = (F(δ (µ [k] ν [k] )))(w,ω). Taking the inverse Fourier transform, the claim follows. Proposition 2.4. If µ,ν M(H [k] ), then for all θ [,], (µ ν) θ = µ θ ν θ. Proof. Let us first suppose that θ [,). If E is a Borel subset of H, then (µ ν) θ (E) = ( θ) t d(µ ν)(t) = ( θ) σ+τ dµ(σ)dν(τ). σ+τ E E σ,τ H [k]

5 On the other hand, (µ θ ν θ )(E) = = CONTRACTIBILITY OF THE MAXIMAL IDEAL SPACE 5 µ θ (E τ)dν θ (τ) = τ H [k] τ H [k] ( θ) σ+τ dµ(σ)dν(τ). σ+τ E σ,τ H [k] ( θ) σ dµ(σ) dν σ E τ θ (τ) σ H [k] Now consider the case when θ =. If k =, the claim follows immediately, since (µ ν) = (µ ν)({})δ = µ({}) ν({})δ = (µ({})δ ) (ν({})δ ) = µ ν. If k >, then µ ν = (δ µ [k ] ) (δ ν [k ] ) = δ (µ [k ] ν [k ] ) = δ (µ ν) [k ] = (µ ν). This completes the proof. The following result says that for a fixed µ M(H [k] ), the map θ µ θ : [,] M(H [k] ) is continuous. Proposition 2.5. If µ M(H [k] ) and θ [,], then lim θ θ µ θ = µ θ in M(H [k] ). Proof. Consider first the case when θ [,). Given an ǫ >, first choose an R > large enough so that µ (B) < ǫ, where B = {t R k t 2 R}. Let θ [,). There exists a Borel measurable function w such that d(µ θ µ θ )(t) = e iw(t) d µ θ µ θ (t). Thus µ θ µ θ = µ θ µ θ (H [k] ) = e iw(t) d(µ θ µ θ )(t) H [k] = e iw(t) d(µ θ µ θ )(t) = e H H iw(t) (( θ) t ( θ ) t )dµ(t). [k] [k] Hence µ θ µ θ e iw(t) (( θ) t ( θ ) t )dµ(t) + e iw(t) (( θ) t ( θ ) t )dµ(t) B H [k] H [k] \B ( max ( θ) t ( θ ) t ) µ (B) + 2 µ (H [k] \ B) t B H ( [k] max ( θ) t ( θ ) t ) µ (H [k] ) + 2ǫ. t B H [k] But by the mean value theorem applied to the function θ ( θ) t, ( θ) t ( θ ) t = (θ θ ) t ( c) t = (θ θ ) t ( c)t c, for some c (depending on t = t, θ and θ ) in between θ and θ. Since c lies between θ and θ, and since both θ and θ lie in [,), and t R, it follows that ( c) t and { } c max θ,. θ

6 6 AMOL SASANE Thus using the above, and the fact that t R, max t B H [k] ( θ) t ( θ ) t = Hence (( lim sup θ θ µ θ µ θ lim sup θ θ lim sup θ θ = R max θ θ t ( c) t t B H [k] c { } θ θ R max θ,. θ max ( θ) t ( θ ) t ) ) µ (H [k] ) + 2ǫ t B H ( [k] { } ) θ θ R max θ, µ (H [k] ) + 2ǫ θ θ µ (H [k] ) + 2ǫ = + 2ǫ = 2ǫ. Since ǫ > was arbitrary, it follows that lim sup µ θ µ θ =. Also µ θ µ θ, and so θ θ lim µ θ µ θ =. θ θ 2 Now consider the case when θ =. Assume for the moment that k > and µ [k ] =. We will show that lim µ θ = in M(H [k] ). Given an ǫ >, first choose a r > small enough θ so that µ (B) < ǫ, where B = {t R k t 2 r}. (This is possible, since µ [k ] =.) There exists a Borel measurable function w such that dµ θ (t) = e iw(t) d µ θ (t). Thus µ θ = µ θ (H [k] ) = e iw(t) dµ θ (t) = e iw(t) ( θ) t dµ(t) = e iw(t) ( θ) t dµ(t) H [k] H [k] H [k] e iw(t) ( θ) t dµ(t) + e iw(t) ( θ) t dµ(t) B H [k] H [k] \B µ (B) + ( θ) r µ (H [k] \ B) ǫ + ( θ) r µ (H [k] ). Consequently, lim sup µ θ µ θ ǫ. But ǫ > was arbitrary, and so lim sup µ θ =. Since θ µ θ, it follows that lim µ θ =. θ If µ k, then define ν := µ δ µ[k ] M(H [k] ). It is clear that ν [k ] = and ν θ = µ θ δ µ[k ]. From the above, lim ν θ =, and so lim µ θ = δ µ[k ] = µ in θ θ M(H [k] ). 3 The case when θ = and k = is analogous to 2 above. Finally we prove that every Banach subalgebra R of M(H [n] ) is semisimple. Proof of Proposition.8. If s C, Re(s), and k {,...,n}, then Φ s [k], given by Φ [k] s (µ) = e st k dµ(t) (µ R), {t t=(,τ) R k H [n k] } is an element of X(R), and so the kernel of Φ s [k] is a maximal ideal in R. But if Φ s [k] (µ) = for all s and all k, then µ is zero on H [n]. So the radical of R is. θ

7 CONTRACTIBILITY OF THE MAXIMAL IDEAL SPACE 7 3. Contractibility of X(R) In this section we will prove our main result. Proof of Theorem.5. Define H : X(R) [,] X(R) as follows. If θ [,], Φ X(R) and µ R, then Φ(µ nθ ) θ < n, (H(Φ,θ))(µ) = Φ(δ k µ[n k] nθ k ) k n θ < k+ n, k =,...,n, Φ(µ({})δ n ) = µ({}) θ =. We show that H is well-defined. From the definition, H(Φ,) X(R) for all Φ X(R). If θ [,), then the linearity of H(Φ,θ) : R C is obvious. Continuity of H(Φ,θ) follows from the fact that Φ is continuous and µ θ µ for θ [,]. That H(Φ,θ) is multiplicative is a consequence of Proposition 2.4, and the fact that Φ respects multiplication. Finally (H(Φ,θ))(δ n ) = Φ((δn ) θ) = Φ(δ n ) =. It is obvious that H(,) is the identity map and H(,) is a constant map. Finally, we show below that H is continuous. Since X(M(H [n] )) is equipped with the Gelfand topology, we just have to prove that for every convergent net (Φ i,θ i ) i I with limit (Φ,θ) in X(M(H [n] )) [,], there holds that (H(Φ i,θ i ))(µ) (H(Φ,θ))(µ). We have (H(Φ i,θ i ))(µ) (H(Φ,θ))(µ) (H(Φ i,θ i ))(µ) (H(Φ i,θ))(µ) + (H(Φ i,θ) (H(Φ,θ))(µ), and from the definition of H, it is immediate that (H(Φ i,θ) (H(Φ,θ))(µ). So it remains to show that (H(Φ i,θ i ))(µ) (H(Φ i,θ))(µ). There is no loss of generality in assuming that all the θ i s belong to one of the intervals [, n) [, n, n) 2 [,..., n n,). But then Proposition 2.5 yields the desired result: for example if θ i [ k n, k+ k+ n ) and θ = n, then (H(Φ i,θ i ))(µ) (H(Φ i,θ))(µ) This completes the proof. = Φ i (δ k µ [n k] nθ i k δk (δ µ [n k ] )) Φ i δ k µ nθi k δ µ [n k ] µ nθi k δ µ[n k ]. Our definition of the map H is based on the following consideration, in the case of n =, when H [n] = H [] = [,+ ). The result given below can be thought of as a generalization of the Riemann-Lebesgue Lemma for functions f a L (,+ ) (that the limit as s + of the Laplace transform of f a is ): Proposition 3.. If µ M(H [] ), then lim s + + e st dµ(t) = µ({}). The set X(M(H [] )) contains the half plane C := {s C Re(s) } in the sense that each s C, gives rise to the corresponding complex homomorphism Φ s : M(H [] ) C, given simply by point evaluation of the Laplace transform of µ at s: µ Φ s (µ) = + e st dµ(t), µ M(H [] ).

8 8 AMOL SASANE If we imagine s tending to + along the real axis we see from Proposition 3., that Φ s starts looking more and more like the complex homomorphism Φ + given by µ Φ + (µ) := µ({}), µ M(H [] ). So we may define H(Φ s,θ) = Φ s log( θ), which would suggest that at least the part C of X(M(H [] )) is contractible to Φ +. But we see that we can view the action of H(Φ s,θ) defined above as follows: (H(Φ s,θ))(µ) = Φ s log( θ) (µ) = + e (s log( θ))t dµ(t) = + e st ( θ) t dµ(t) = Φ s (ν), where ν is the measure such that dν(t) = ( θ) t dµ(t). This motivates the definition of µ θ given in (), and the definition of H in the proof of Theorem Examples As specific examples of R in Theorem.5, we have the following: 4.. The algebra L (H [n] )+Cδ n. Consider the Banach subalgebra L (H [n] )+Cδ n of M(H[n] ), consisting of all complex Borel measures of the type µ a + αδ n, where µ a is absolutely continuous (with respect to the Lebesgue measure) and α C. It can be checked that this Banach subalgebra of M(H [n] ) has the property (P) in the statement of Theorem The algebra A(H [n] ). The Banach subalgebra A(H [n] ) of M(H [n] ) consists of all complex Borel measures that do not have a singular non-atomic part. Then it can be verified that A(H [n] ) also possesses the property (P). (So in the case when n =, we recover the main result in [], but this time without recourse to the explicit description of the maximal ideal space.) 4.3. Algebras of almost periodic functions. The algebra AP n of complex valued (uniformly) almost periodic functions is, by definition, the smallest closed subalgebra of L (R n ) (with all operations defined pointwise), that contains all the functions e λ (x) := e i λ,x. Here the variable x = (x,...,x n ), the parameter λ = (λ,...,λ n ) R n, and λ,x := n k= λ kx k. For any f AP n, its Bohr-Fourier series is defined by the formal sum λ f λe i λ,x (x R n ), where f λ := lim N (2N) n e i λ,x a(x)dx, λ R n, [ N,N] n and the sum λ f λe i λ,x is taken over the set σ(f) := {λ R n f λ }, called the Bohr- Fourier spectrum of f. The Bohr-Fourier spectrum of every f AP n is at most a countable set. The almost periodic Wiener algebra APW n is defined as the set of all AP n such that the Bohr-Fourier series λ f λe i λ,x of f converges absolutely. The almost periodic Wiener algebra is a Banach algebra with pointwise operations and the norm f := λ R n f λ. Let be a nonempty subset of R n. Denote AP n = {f AP n σ(f) } APW n = {f APW n σ(f) }. If is an additive subset of R n, then AP n (respectively APWn ) is a Banach subalgebra of AP n (respectively APW n ). Moreover, if, then AP n and APWn are also unital.

9 CONTRACTIBILITY OF THE MAXIMAL IDEAL SPACE 9 Let Σ H [n] be an additive semigroup (if λ,µ Σ, then λ + µ Σ) and suppose Σ. The Banach algebra APWΣ n is isomorphic to the following Banach subalgebra R of M(H[n] ): { R = f λ δ n (λ) } f λ e i λ,x APWΣ n. λ In the above, δ n(λ) M(H[n] ) denotes the Dirac measure supported at λ H [n], that is, { (δ n if λ E, (λ))(e) = if λ E, λ where E is any Borel subset of H [n]. It can be seen that the subalgebra R has the property (P). Thus the maximal ideal space of APWΣ n is contractible. The maximal ideal spaces of APΣ n and APWn Σ are homeomorphic as topological spaces; see for example [, Theorem 3.]. So the maximal ideal space of APΣ n is contractible as well. So we recover the main result from [8]. (In [8], instead of the canonical half space H [n], more general half spaces S were considered. There a subset S of R n was called a half space in R n if it satisfied the properties S ( S) = R n, S ( S) = {}, x + y S for all x,y S, αx S for all x S and α. However, it was shown in [8, Proposition.2] that any such half space S is of the form ZH [n] for an invertible matrix Z R n n.) Summarizing the results of this section, we have shown Corollary. as a particular consequence of our main result Theorem.5. References [] A. Böttcher. On the corona theorem for almost periodic functions. Integral Equations Operator Theory, no. 3, 33: , 999. [2] A. Brudnyi. Contractibility of maximal ideal spaces of certain algebras of almost periodic functions. Integral Equations Operator Theory, no. 4, 52: , 25. [3] A. Brudnyi and A.J. Sasane. Sufficient conditions for the projective freeness of Banach algebras. Submitted, March 29. [4] P.M. Cohn. Free Rings and their Relations. Second edition. London Mathematical Society Monographs 9, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 985. [5] T.Y. Lam. Serre s Conjecture. Lecture Notes in Mathematics 635, Springer-Verlag, Berlin-New York, 978. [6] V. Ya. Lin. Holomorphic fiberings and multivalued functions of elements of a Banach algebra. Functional Analysis and its Applications, no. 2, 7:22-28, 973, English translation. [7] A. Quadrat. The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. II. Internal stabilization. SIAM Journal on Control and Optimization, no., 42:3-32, 23. [8] L. Rodman and I.M. Spitkovsky. Algebras of almost periodic functions with Bohr-Fourier spectrum in a semigroup: Hermite property and its applications. Journal of Functional Analysis, no., 255: , 28. [9] A.J. Sasane. Extension to an invertible matrix in convolution algebras of measures. Proceedings of the International Workshop on Operator Theory and its Applications, Operator Theory: Advances and Applications series, J. Ball, V. Bolotnikov, J. Helton, L. Rodman, I. Spitkovsky (Editors), Birkhäuser Verlag, 29. [] A.J. Sasane. The Hermite property of a causal Wiener algebra used in control theory. To appear in Complex Analysis and Operator Theory. Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom. address: A.J.Sasane@lse.ac.uk