� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 37, Num. 1 (2022), 57 – 74 doi:10.17398/2605-5686.37.1.57 Available online March 9, 2022 On formal power series over topological algebras M. Weigt 1, I. Zarakas 2 1 Department of Mathematics, Nelson Mandela University Summerstrand Campus (South), Port Elizabeth, 6031, South Africa 2 Department of Mathematics, Hellenic Military Academy, Athens, 19400, Greece martin.weigt@mandela.ac.za , weigt.martin@gmail.com , gzarak@hotmail.gr Received April 21, 2021 Presented by D. Dikranjan Accepted February 9, 2022 Abstract: We present a general survey on formal power series over topological algebras, along with some perspectives which are not easily found in the literature. Key words: Topological algebra, formal power series, algebra homomorphism. MSC (2020): 46H05, 46H10, 46H30. 1. Introduction In our research course in [17], we came across the following question: Let A[τ] be a GB∗-algebra, δ a closed derivation of A with domain D(δ), x ∈ D(δ) and 0 6= λ ∈ C such that (λ1−x)−1 exists in A. Does it follow that (λ1−x)−1 lies in the domain of the derivation? The latter question is answered in the affirmative in the case of C∗-algebras by Bratelli and Robinson in [7], and more generally for pro-C∗-algebras in [17]. The basic tool used in the proof of Bratelli and Robinson’s result is the Neumann series which is complemented with the use of an analytic continuation argument. Given that a GB∗ algebra is an algebra of possibly unbounded operators, the use of Neumann series in the GB∗-algebra setting fails and consequently the Bratelli-Robinson proof cannot be carried over. We were thus led to ask ourselves whether it might be possible to an- swer our initial question, which for brevity we shall call the domain problem, with the use of formal power series. In particular, the following two questions naturally arise: (I) does there exist a unital injective algebra homomorphism φ̃ : D(δ) → D(δ)[[X]], with φ̃(x) = X? (II) Does there exist a unital homo- morphism ψ : C[[X]] → D(δ) such that ψ(X) = x? An affirmative answer to any of the above questions could have the potential to place (λ1−x)−1 inside the domain of the derivation (see Section 3 for details). ISSN: 0213-8743 (print), 2605-5686 (online) c©The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.37.1.57 mailto:martin.weigt@mandela.ac.za mailto:weigt.martin@gmail.com mailto:gzarak@hotmail.gr https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 58 m. weigt, i. zarakas However, observe that the existence of such a unital homomorphism in Question II would imply that x is an element of D(δ) such that SpD(δ)(x) ⊆ {0} (see Section 3). Although this is a severe restriction, this motivates the following more general questions: (I’) Let A[τ] be a topological algebra and let x ∈ A. Does there exist a unital injective algebra homomorphism φ̃ : A → A[[X]], with φ̃(x) = X? (II’) Let A[τ] be a topological algebra and let x ∈ A. Does there exist a unital homomorphism ψ : C[[X]] → A such that ψ(X) = x? These two more general questions are interesting for their own sake, and are the main focus of this paper. In the current expository paper, we visit some well-known results of rel- evance to our purposes in answering the two more general questions in the previous paragraph, which refer to formal power series on Banach and Fréchet algebras. The literature is rich in papers which deal with formal power se- ries on topological algebras and what’s more with their intriguing interplay with derivations. Indicatively, we refer the reader to [16, 4, 8]. Moreover, we present certain byproducts obtained along the way, for the case of generalized B∗-algebras, which give stimulus for further reserach on the topic, and which appear to be interesting for their own sake. 2. Preliminaries Throughout, all algebras are assumed to be over the complex numbers, i.e., all algebras are assumed to be C-algebras. A topological algebra A[τ] is an algebra which is also a topological vector space and has the property that multiplication is separately continuous. In some topological algebras, such as Fréchet algebras, multiplication is not only separately continuous, but jointly continuous [10]. By a Fréchet algebra, we mean a complete metrizable topological algebra. A topological algebra is said to be a locally convex algebra if it is also a locally convex space. An m-convex algebra is a locally convex algebra A[τ] for which the topology τ on A is defined by a family of seminorms {pγ : pγ ∈ Γ} such that pγ(xy) ≤ pγ(x)pγ(y) for all x,y ∈ A and for all γ ∈ Γ. For every γ ∈ Γ, let Nγ = {x ∈ A : pγ(x) = 0}. Then A/Nγ is a normed algebra with respect to the norm ṗγ(x + Nγ) = pγ(x) for all x ∈ A and A = lim←−γ A/Nγ. The completion Aγ of A/Nγ is therefore a Banach algebra for all γ ∈ Γ. If A is, in addition, complete, then A = lim←−γ Aγ up to a topological and algebraic on formal power series 59 ∗-isomorphism, which is called the Arens-Michael decomposition of the com- plete m-convex algebra A. For details, we refer to [10]. A C∗-algebra is a Banach ∗-algebra A such that ‖x∗x‖ = ‖x‖2 for all x ∈ A, where ‖ ·‖ denotes the submultiplicative norm on A defining the topology on A. Definition 2.1. ([2]) Let A[τ] be a unital topological ∗-algebra and let B∗A (denoted by B ∗ if no confusion arises as to which algebra is being consid- ered) be a collection of subsets B of A satisfying the following: (i) B is absolutely convex, closed and bounded; (ii) 1 ∈ B, B2 ⊆ B and B∗ = B. For every B ∈ B∗, denote by A[B] the linear span of B, which is a normed algebra under the gauge function ‖ · ‖B of B. If A[B] is complete for every B ∈B∗, then A[τ] is called pseudo-complete. An element x ∈ A is called bounded, if there exists 0 6= λ ∈ C such that the set {(λx)n : n = 1, 2, 3, . . .} is a bounded subset of A. We denote by A0 the set of all bounded elements in A. A unital topological ∗-algebra A[τ] is called symmetric if, for every x ∈ A, the element (1 + x∗x)−1 exists and belongs to A0. Definition 2.2. ([2]) A symmetric pseudo-complete locally convex ∗-algebra A[τ], such that the collection B∗ has a greatest member, denoted by B0, is called a GB ∗-algebra over B0. Every C∗-algebra is a GB∗-algebra. An example of a GB∗-algebra, which generally need not be a C∗-algebra, is a pro-C∗-algebra. By a pro-C∗-algebra, we mean a complete topological ∗-algebra A[τ] for which the topology τ is defined by a directed family of C∗-seminorms. Hence every pro-C∗-algebra is m-convex, and is therefore an inverse limit of C∗-algebras. An example of a GB∗-algebra which is not a pro-C∗-algebra is the locally convex ∗-algebra Lω([0, 1]) = ∩p≥1Lp([0, 1]) defined by the family of semi- norms { ‖ ·‖p : p ≥ 1 } , where ‖ ·‖p is the Lp-norm on Lp([0, 1]) for all p ≥ 1. A formal power series in the indeterminate X over a unital algebra A is an expression of the form ∑∞ n=0 anX n, where an ∈ A for all n ∈ N. The set A[[X]] of all formal power series over A is an algebra with respect to addition and scalar multiplication defined in the obvious way, and multiplication defined as the usual Cauchy product as with complex power series. This is an immediate consequence of [12, Proposition 5.8]. 60 m. weigt, i. zarakas We assume throughout this paper that all our algebras are unital, i.e., have an identity element 1, unless stated otherwise. If A denotes an algebra, and x ∈ A, then the symbol SpA(x) denotes the set { λ ∈ C : λ1 −x is not invertible in A } throughout. 3. Main results A derivation of an algebra A is a linear mapping δ : D(δ) → A such that δ(xy) = xδ(y) + δ(x)y for all x,y ∈ A, where D(δ) denotes the domain of δ and is a subalgebra of A. If, in addition, A is equipped with an involution ∗, then we say that δ is a ∗-derivation if D(δ) is a ∗-subalgebra of A (i.e., D(δ) is closed under the involution ∗), and δ(x∗) = δ(x)∗ for all x ∈ A. For a topological algebra A[τ], a ∗-derivation δ : D(δ) → A is said to be a closed ∗-derivation if D(δ) is dense in A, and if (xα) is a net in A with xα → x ∈ A and δ(xα) → y ∈ A, then x ∈ D(δ) and y = δ(x). Let A[τ] be a GB∗-algebra, δ a closed ∗-derivation of A with 1 ∈ D(δ), x ∈ D(δ), λ ∈ C −{0} such that (λ1 − x)−1 ∈ A. As already outlined in the introduction, in regards to the domain problem for a GB∗-algebra, the following two questions emerge: (I) Does there exist a unital injective algebra homomorphism φ̃ : D(δ) → D(δ)[[X]], with φ̃(x) = X? (II) Does there exist a unital homomorphism ψ : C[[X]] → D(δ) such that ψ(X) = x? Let us first assume that question (I) has an affirmative answer, and assume also that C[[X]] ⊆ φ̃(D(δ)). Let x ∈ D(δ). Then, since 0 ∈ Sp φ̃(D(δ)) (X) ⊆ SpC[[X]](X) = {0} and φ̃ is an injective algebra homomorphism, we get that SpD(δ)(x) = {0}. So, for our initially assumed λ 6= 0 such that (λ1 − x)−1 exists in A, we conclude that (λ1 −x)−1 ∈ D(δ). The condition SpD(δ)(x) = {0} is a severe restriction. Observe that the condition C[[X]] ⊆ φ̃(D(δ)) is equivalent to the condition that φ̃(D(δ))∩C[[X]] = C[[X]], and this is explored in Propositions 3.1 and 3.2 below. on formal power series 61 Now let us assume that question (II) has a positive answer. Then, for the initially assumed non-zero λ ∈ C, we have that 1 λ ∞∑ k=0 ( 1 λ X )k = ( λ ( 1 − 1 λ X ))−1 = (λ1 −X)−1. Therefore, λ1 − X is invertible in C[[X]] for all λ ∈ C − {0}. Given the assumption of the existence of the unital homomorphism ψ described in question (II), we consequently have that (λ1 −x)−1 ∈ D(δ). It is worth noting that once we assume the validity of questions (I) or (II), with the additional assumption that C[[X]] ⊆ φ̃(D(δ)) in Question I, the domain problem can be immediately answered by straightforward algebraic considerations which hold for any algebra. It is therefore expected that the intrinsic nature of the topology of a GB∗-algebra is to play its role in answering questions (I) and (II). In Question II, however, the existence of such a unital homomorphism implies that x is an element of D(δ) having the property that SpD(δ)(x) = SpD(δ)(Ψ(X)) ⊆ SpC[[X]](X) = {0}. Although the restrictions above are severe, Questions I and II motivate the following more general questions, which appear to be interesting for their own sake. (I’) Let A[τ] be a topological algebra and let x ∈ A. Does there exist a unital injective algebra homomorphism φ̃ : A → A[[X]], with φ̃(x) = X? (II’) Let A[τ] be a topological algebra and let x ∈ A. Does there exist a unital homomorphism ψ : C[[X]] → A such that ψ(X) = x? We start with some considerations with respect to question (I). In [8, Theorem 11.2], it is shown that if θ is an algebra homomorphism of a Fréchet algebra into the formal power series algebra C[[X]], then θ is either continuous or a surjection. In this regard, we have the following result. Proposition 3.1. Let A[τ] be a topological algebra with identity element 1 and let x ∈ A such that (1 −x)−1 exists in A. Assume that φ̃ : A → A[[X]] is a unital algebra homomorphism such that φ̃(A)∩C[[X]] is a Fréchet locally convex algebra in some topology τ1. Let ψ be the identity map of C[[X]] restricted to φ̃(A)∩C[[X]]. Let τ ′ denote the topology of convergence on C[[X]] with respect to all coefficients in C[[X]]. Consider the following statements. 62 m. weigt, i. zarakas (i) The map ψ is τ1 − τ ′ discontinuous. (ii) φ̃(A) ∩ C[[X]] is not m-convex with respect to τ1 and is τ ′-closed in C[[X]]. (iii) φ̃(A) ∩ C[[X]] is not a Q-algebra with respect to τ1 and is τ ′-closed in C[[X]]. Then (i) holds if and only if (ii) holds, and (i) implies (iii). Proof. (i) ⇒ (ii) : If ψ is τ1 − τ ′ discontinuous, then ψ is surjective [8], i.e., φ̃(A) ∩ C[[X]] = C[[X]]. Therefore φ̃(A) ∩ C[[X]] is τ ′-closed in C[[X]], and is commutative, m-convex, Fréchet with respect to the topology τ ′, and Noetherian (this is due to C[[X]] being Fréchet, commutative, m-convex and Noetherian with respect to the topology τ ′ [9]). Since ψ is discontinuous, it is now immediate from [9, Theorem 2.5] that φ̃(A) ∩ C[[X]] is not m-convex with respect to τ1. (ii) ⇒ (i) : Assume that (ii) holds, and suppose that ψ is τ1−τ ′ continuous. Then, since ψ is a surjective homomorphism onto its range φ̃(A)∩C[[X]], which is Fréchet with respect to τ ′ and τ1, it follows from the open mapping theorem that ψ is a τ1−τ ′ topological isomorphism. This cannot be since φ̃(A)∩C[[X]] is m-convex with respect to τ ′, and φ̃(A)∩C[[X]] is not m-convex with respect to τ1. Therefore ψ is τ1 − τ ′ discontinuous. (i) ⇒ (iii) : If ψ is τ1−τ ′ discontinuous, then, as in the proof of (i) ⇒ (ii), it follows that φ̃(A) ∩ C[[X]] is not m-convex with respect to τ1. By [18, Theorem 13.17], φ̃(A) ∩C[[X]] is not a Q-algebra with respect to τ1. Consider a map φ̃ as in question (I’). The following proposition depicts a certain positioning of the image of φ̃, and the proof shows that the map ψ in Proposition 3.1 is always τ1 − τ ′ continuous, and therefore ψ is not surjective. This demonstrates that φ̃(A) ∩ C[[X]] does not satisfy condition (ii) of Proposition 3.1. Proposition 3.2. Let A[τ] be a unital topological algebra with identity 1. Let x ∈ A and φ̃ : A → A[[X]] a unital algebra homomorphism with φ̃(x) = X, such that φ̃(A) ∩ C[[X]] is a Fréchet algebra in some topology τ1. Then φ̃(A) ∩C[[X]] is properly contained in C[[X]]. Proof. Let ψ := id|φ̃(A)∩C[[X]] : φ̃(A) ∩C[[X]] → C[[X]]. We have that φ̃(λ01 + λ1x + · · · + λnxn) = λ01 + λ1X + · · · + λnXn. on formal power series 63 Therefore C[X] ⊂ φ̃(A)∩C[[X]]. Hence, (φ̃(A)∩C[[X]],τ1) is a Fréchet algebra which is a subalgebra of C[[X]] containing C[X]. Then, by [8, Theorem 11.2], the map ψ is τ1 −τ ′ continuous, where τ ′ is the topology of convergence with respect to all coefficients in C[[X]]. Hence, by [8, p. 3], we take that ψ is not onto and hence φ̃(A) ∩C[[X]] is properly contained in C[[X]]. Corollary 3.3. Let A[τ] be a unital topological algebra with identity 1 and δ : D(δ) → A a closed derivation of A such that 1 ∈ D(δ). Let x ∈ D(δ) and φ̃ : D(δ) → D(δ)[[X]] a unital algebra homomorphism with φ̃(x) = X, such that φ̃(D(δ)) ∩ C[[X]] is a Fréchet algebra in some topology τ1. Then φ̃(D(δ)) ∩C[[X]] is properly contained in C[[X]]. The following result is described in [16, first paragraph of p. 2145] and we give its proof for sake of completeness. Proposition 3.4. Let A be a unital algebra and x ∈ A. If θ : A → A[[X]] is a unital injective algebra homomorphism of A into the algebra A[[X]] of all formal power series over A with indeterminate X such that θ(x) = X, then x is not invertible in A and ∩∞n=1x nA = {0}. Proof. We first show that X is not invertible in A[[X]]. Observe that X = ∑∞ k=0 akX k, where ak = 0 for all k 6= 1 and a1 = 1. If we suppose that X is invertible in A[[X]], then there is an element ∑∞ k=0 bkX k in A[[X]] such that X ( ∞∑ k=0 bkX k ) = ( ∞∑ k=0 akX k )( ∞∑ k=0 bkX k ) = ∞∑ k=0 ( k∑ r=0 arbk−r ) Xk = 1 . Hence, ∑k r=0 arbk−r = 0 for all k ≥ 1 and a0b0 = 1. This last equation is in contradiction with the fact that a0 = 0, thus X is not invertible in A[[X]]. So, since θ(x) = X and θ is unital, we conclude that x is not invertible in A. Let now a ∈ ⋂∞ n=1 x nA. Then, for all n ∈ N, there exists bn ∈ A such that a = xnbn. Hence, θ(a) = θ(x nbn) = θ(x) nθ(bn) = X nθ(bn), for all n ∈ N. Then, Xθ(b1) = X2θ(b2), implying that θ(b1) = θ(b2) = 0. Thus, θ(a) = Xθ(b1) = 0. So, ⋂∞ n=1 x nA ⊂ ker θ = {0}. Therefore a = 0, implying that ⋂∞ n=1 x nA = {0}. The following corollary is a direct consequence of Proposition 3.4. It gives us necessary conditions under which question (I) in the beginning of this section has a negative answer. 64 m. weigt, i. zarakas Corollary 3.5. Let δ : D(δ) → A be a closed ∗-derivation of a GB∗- algebra A such that 1 ∈ D(δ). Let x ∈ D(δ) such that (1 − x)−1 exists in A. If x is invertible in A, or ∩∞n=1x nA 6= {0}, then there is no injective unital algebra homomorphism φ̃ : D(δ) → D(δ)[[X]] with φ̃(x) = X. The proof of the following result is similar to that of [16, Theorem 3.10], and we only give a sketch of the proof. Proposition 3.6. Let A[τ] be a unital commutative topological algebra, and let x ∈ A have the following properties: (i) there exists a derivation D : A → A such that D(x) = 1 and ∞∑ k=0 (−1)kDk(a)xk k! τ-converges for all a ∈ A, (ii) ⋂∞ n=0 x nA = {0}. Then there exists a unital injective algebra homomorphism φ̃ : A → A[[X]] such that φ̃(x) = X. Proof. By hypothesis we can form the map θ : A → A by θ(a) = ∞∑ k=0 (−1)kDk(a)xk k! for all a ∈ A. It is easily seen that θ is a unital algebra homomorphism. Also, the kernel of θ is xA. Define the map φ̃ : A → A[[X]] by φ̃(a) = ∞∑ n=0 θ(Dn(a)) n! Xn for all a ∈ A. Since the kernel of θ is xA and D(xn) = nxn−1, for all n ∈ N, we get that φ̃ is injective. Lastly, it is easily seen that φ̃(x) = X. Proposition 3.6 is an optimal result as far as sufficient conditions are con- cerned: Condition (i) of Proposition 3.6 cannot be dropped. The reason is that if condition (ii) is sufficient on its own, then it would imply that our map exists if x is nilpotent (for x nilpotent gives us condition (ii)). This cannot be, since φ̃(x) = X and X is not nilpotent in A[[X]]. So condition (i) is required on formal power series 65 in addition to condition (ii). Also, the condition D(x) = 1 in (i) implies that x is not nilpotent: For if x is nilpotent and x 6= 0 (which we assume, since the case x = 0 is trivial), there is a smallest natural number m > 1 such that xm = 0. Hence 0 = D(xm) = mxm−1 6= 0, a contradiction. Furthermore, x not nilpotent and condition (ii) of Proposition 3.6 imply that x cannot have the property that 0 6= Axm ⊂ Axm+1 (see [4, Corollary 2]). This is not in contradiction to [16, Proposition 3.10]. Also, recall from Proposition 3.4 that condition (ii) is a necessary condition for the existence of our map. Lastly, condition (i) is not redundant: No commutative C∗-algebra has a derivation D with D(x) = 1. Corollary 3.7. Let δ : D(δ) → A be a closed unbounded ∗-derivation with 1 ∈ D(δ), where A[τ] is a commutative GB∗-algebra. Let x ∈ D(δ) such that (1 −x)−1 ∈ A. Suppose that there exists a derivation D : D(δ) → D(δ) such that D(x) = 1 and ∞∑ k=0 (−1)kDk(a)xk k! τ-converges to an element in D(δ) for all a ∈ D(δ). If also ⋂∞ n=0 x nA = {0}, then there exists a map φ̃ : D(δ) → D(δ)[[X]] which is a unital injective algebra homomorphism such that φ̃(x) = X. Remarks 3.8. (i) [16, Theorem 3.10] is stated for a commutative radi- cal Fréchet m-convex algebra. It is interesting to note that the conditions of Proposition 3.6 above ensure that one does not require A[τ] to be a commu- tative radical Fréchet m-convex algebra. In the proof of [16, Theorem 3.10], one requires A[τ] to be a commutative radical Fréchet m-convex algebra in order to apply [16, Proposition 3.7 and Lemma 3.9], which is to ensure that condition (i) of Proposition 3.6 holds in some form. (ii) The results obtained in [16], and therefore Propositions 3.4, 3.6 (above) and 3.9 (below), stem from the paper [15], in which M.P. Thomas proved that the image of a derivation of a commutative Banach algebra is contained in the Jacobson radical of the Banach algebra (commutative Singer-Wermer conjecture). Based on Proposition 3.6 and [16, Lemma 3.9], the following result can be recorded. Proposition 3.9. Let A[τ] be a Fréchet m-convex algebra and {‖.‖i} a sequence of seminorms defining the topology τ. Let x ∈ A and D : A → A 66 m. weigt, i. zarakas be a derivation such that D(x) = 1. Suppose that for every i ∈ N there is mi ∈ N such that xmi ∈ ⋂∞ n=0 x nA ‖.‖i , where ⋂∞ n=0 x nA ‖.‖i denotes the closure of ⋂∞ n=0 x nA with respect to the seminorm ‖.‖i. Then there exists an algebra homomorphism φ̃ : A → A0[[X]], where A0 = A/ ⋂+∞ n=0 x nA, such that φ̃(x) = X. Proof. Based on the proof of [16, Theorem 3.10] there exists an injective algebra homomorphism φ : A/ ⋂∞ k=0 x kA → A0[[X]] such that φ ( a + ∞⋂ k=0 xkA ) = ∞∑ n=0 θ(Dn(a) + ⋂∞ k=0 x kA) n! Xn, where θ : A/ ⋂∞ k=0 x kA → A/ ⋂∞ k=0 x kA is the following algebra homomor- phism: θ ( a + ∞⋂ k=0 xkA ) = ∞∑ n=0 (−1)nDn(a)xn n! + ∞⋂ k=0 xkA. Then, by composing φ with the natural quotient map π :A→A/ ⋂∞ k=0 x kA, we get the desired algebra homomorphism φ̃. The fact that φ̃(x) = X can be easily seen through the following consid- erations: φ̃(x) = (φ◦π)(x) = φ ( x + +∞⋂ k=0 xkA ) = +∞∑ n=0 θ(Dn(x) + ⋂+∞ k=0 x kA) n! Xn. Since D(x) = 1, we get that Dn(x) = 0 for every n ≥ 2. Therefore, (φ◦π)(x) = θ ( x + +∞⋂ k=0 xkA ) + θ ( 1 + +∞⋂ k=0 xkA ) ·X . Moreover, θ ( x + +∞⋂ k=0 xkA ) = +∞∑ n=0 (−1)nDn(x)xn n! + +∞⋂ k=0 xkA = (x−x) + +∞⋂ k=0 xkA = +∞⋂ k=0 xkA, on formal power series 67 and θ ( 1 + +∞⋂ k=0 xkA ) = +∞∑ n=0 (1)nDn(1)xn n! + +∞⋂ k=0 xkA = 1 + +∞⋂ k=0 xkA. So, φ̃(x) = (φ◦π)(x) = +∞⋂ k=0 xkA + ( 1 + +∞⋂ k=0 xkA ) ·X = X , given that on A0(= A/ ⋂+∞ k=0 x kA), the elements ⋂+∞ k=0 x kA and 1 + ⋂+∞ k=0 x kA are the zero and the unit element respectively. With respect to a possible strategy for answering question (II) in the be- ginning of this section, we note the following two routes, denoted by (1) and (2) in what follows. (1) In [3, Theorem 2], G.R. Allan proved the following result for a com- mutative Banach algebra. Theorem 3.10. Let A be a commutative Banach algebra with identity and let x be in the Jacobson radical of A such that 0 6= Axm ⊂ Axm+1 for some m ≥ 1. Then there is a unital injective algebra homomorphism ψ : C[[X]] → A such that ψ(X) = x. The proof of Theorem 3.10 in [3] is entirely algebraic, except in the ap- plication of the Arens-Calderon theorem, which relies on A being a Banach algebra. The Arens-Calderon theorem is given by the following theorem. Theorem 3.11. ([11, Lemma 3.2.8]) Let A be a commutative Banach al- gebra with identity and Jacobson radical R. If n ∈ N and a0,a1, . . . ,an ∈ A are such that a0 ∈ R and a1 is invertible in A, then there exists y ∈ R such that ∑n k=0 aky k = 0. The proof of Theorem 3.11, as given in [11], relies strongly on analytic functions of several complex variables and the implicit function theorem. If we could extend the Arens-Calderon theorem (Theorem 3.11) to commutative Fréchet algebras, then we will have extended Theorem 3.10 to the case where A is a Fréchet algebra, with exactly the same proof as that of [3, Theorem 68 m. weigt, i. zarakas 2]. Therefore Question (II’) would attain a positive answer in the setting of commutative Fréchet algebras. In Theorem 3.10, the condition that x is in the radical of A is strong if A is semi-simple, especially if x 6= 0. Recall that a sequentially complete locally convex algebra is pseudo-complete, and hence has an abundance of Banach subalgebras (see [1, Proposition 4.2]). This motivates the following observation. Corollary 3.12. Let A[τ] be a commutative unital Fréchet locally con- vex algebra and x ∈ A. Assume that there is a unital Banach subalgebra B of A (in some norm) such that x ∈ B and 0 6= Bxm ⊂ Bxm+1 for some m ≥ 1, and such that x is in the radical of B. Then there is an embedding ψ : C[[X]] → B ⊆ A such that ψ(X) = x. If, in the above corollary, B is a commutative unital radical Banach sub- algebra, then the condition x in the radical of B becomes more realistic. For example, a famous theorem of B.E. Johnson [13, Theorem 4] asserts that there are at least some commutative Banach algebras having at least one closed (hence Banach) radical subalgebra. Incidentally, this theorem of Johnson was used by M.P. Thomas in his proof of the commutative Singer-Wermer con- jecture: He only had to settle the conjecture for commutative radical Banach algebras. One word of caution: Not all Fréchet Jacobson semi-simple algebras have closed (hence Fréchet) radical subalgebras. For instance, a C∗-algebra is semi- simple, and every norm closed ∗-subalgebra is again a C∗-algebra, and hence semi-simple. An extension of Theorem 3.10 to Fréchet m-convex algebras is given in [4, Theorem 7]. (2) The following theorem is a special case of [9, Lemma 2], and is the holomorphic functional calculus for complete barrelled m-convex algebras. Theorem 3.13. Let A[τ] be a commutative complete barelled m-convex topological algebra with weakly compact character space. Let a ∈ A. Then there exists a continuous homomorphism g : O(spA(a)) → A, such that g(f) = a, where O(spA(a)) denotes the algebra of all analytic functions on a neighbourhood of spA(a) and f(z) = z for all z ∈ C. The proof of Theorem 3.13 relies on [5, Theorem 5.1], which is the Silov- Arens-Calderon theorem for Banach algebras. In what follows, we give a proof on formal power series 69 of Theorem 3.13 for the case where all Banach algebras in the Arens-Michael decomposition of A are semi-simple. Proof. By the Arens-Michael decomposition of A we have that A[τ] is topo- logically isomorphic to the inverse limit lim←−γ Aγ, where all Aγ are semisimple by assumption. The space O(spA(a)) can be represented as the following inductive limit O(spA(a)) = lim→ H(Ω) where H(Ω) stands for the holomorphic functions on the open subset Ω of C and Ω runs over all open subsets of C which contain the weakly compact subset spA(a). The topology considered on each H(Ω) is that of the uniform convergence on compact subsets of Ω. The topology on O(spA(a)) is the inductive limit topology on O(spA(a) induced by the H(Ω)’s. Let us fix an index γ ∈ Γ. We consider an open subset Ω of C which contains spA(a). Since spA(a) = ⋃ γ∈ΓspAγ (aγ) [10], we have that Ω contains spAγ (aγ) for all γ ∈ Γ. By the Arens-Calderón theorem [6, Corollary II.20.6], there exists a homomorhism g γ Ω : H(Ω) −→ Aγ which is continuous with respect to the topology of uniform convergence on compact subsets of Ω, it extends the natural homomorhism of the complex polynomials P(C) into Aγ and g γ Ω(f) = aγ, for f(z) = z, z ∈ C. By the proof of [6, Corollary II.20.6], we recall that if f ∈ H(Ω), and, say, g γ Ω(f) = lγ ∈ Aγ, then φγ(lγ) = f(φγ(aγ)), for all φγ ∈M(Aγ), where M(Aγ) is the character space of Aγ. We then define the map gγ : O(spA(a)) −→ Aγ as follows: let f be a function in O(spA(a)). Hence there is an open neigh- borhood Ω of spA(a) such that f ∈ H(Ω). We then define gγ(f) := g γ Ω(f). We show that the map gγ is well-defined: for this, consider an open subset Ω′ containing spA(a) such that f ∈ H(Ω′). Then for every φγ ∈ M(Aγ), we have that φγ ( g γ Ω(f) ) = f ( φγ(aγ) ) = φγ ( g γ Ω′(f) ) . 70 m. weigt, i. zarakas Hence, since Aγ is semisimple, we get that g γ Ω(f) = g γ Ω′(f). Hence, the map gγ is well-defined. Moreover, if iΩ is the canonical embed- ding of H(Ω) into O(spA(a)), we clearly have that gγ ◦ iΩ = g γ Ω . Therefore, for every open subset Ω containing spA(a) we have that gγ ◦ iΩ is continuous. So, by [14, (6.1), page 54] the map gγ : O(spA(a)) → Aγ is continuous for every γ ∈ Γ. Next, we consider the map g : O(spA(a)) −→ A, g(f) := ( gγ(f) ) γ∈Γ . In order to show that the map g is well-defined, we need to show that πγδ(gδ(f)) = gγ(f), γ ≤ δ, f ∈ O(spA(a)), where πγ δ : Aδ → Aγ are the ‖.‖-continuous connecting maps of the inverse system. We fix one open neighborhood, say Ω, of the set spA(a). By the preceding paragraphs, we have that gγ(f) = g γ Ω(f) = lγ, gδ(f) = g δ Ω(f) = lδ. So, we shall show that πγ δ(lδ) = lγ. We recall that a = (aλ)λ∈Γ. Since Ω is an open neighborhood of spAδ (aδ), by the proof of [6, Theorem II.20.5], we have that there are aδ2,aδ3, . . . ,aδN ∈ Aδ and Fδ a holomorphic function on an open neighborhood of the polydisc Γδ = { z ∈ CN : |zk| ≤ 1 + pδ(aδk), k = 1, . . .N } (in the above writing of Γδ, aδ1 is identified with aδ), such that f(φδ(aδ)) = Fδ ( φδ(aδ),φδ(aδ2), . . . ,φδ(aδN ) ) , φδ ∈M(Aδ) . (3.1) Moreover, by the proof of [6, Theorem II.20.5], we have that the series∑ k αk a k1 δ a k2 δ2 · · ·a kN δN is ‖.‖δ-convergent to the element lδ and for every φδ ∈M(Aδ), φδ(lδ) = Fδ ( φδ(aδ),φδ(aδ2), . . . ,φδ(aδN ) ) . (3.2) Now, given that πγδ is a ‖.‖-continuous linear map, we have that πγ δ (∑ k αk a k1 δ a k2 δ2 · · ·akNδN ) = ∑ k αk a k1 γ a k2 γ2 · · ·akNγN on formal power series 71 ‖.‖γ-converges to an element in Aγ, say ωγ. For any φγ ∈ M(Aγ) we have that φγ(ωγ) = φγ ( πγ δ (∑ k αka k1 δ a k2 δ2 · · ·akNδN )) = φγ ( πγδ(lδ) ) = Fδ ( (φγ ◦πγ δ)(aδ), . . . , (φγ ◦πγ δ)(aδN ) ) = f ( (φγ ◦πγ δ)(aδ) ) = f ( φγ(aγ) ) , where the second equality in the above string of relations derives from rela- tion (3.2) and from the fact that φγ ◦ πγ δ ∈ M(Aδ), and the third equality results from relation (3.1) above. By the proof of [6, Corollary II.20.6], we have that for every φγ ∈M(Aγ): φγ(lγ) = f ( φγ(aγ) ) . Therefore, we have that φγ(ωγ) = φγ(lγ) for every φγ ∈M(Aγ). Since Aγ is semisimple, we get that ωγ = lγ, i.e., πγ δ(lδ) = lγ. The map g is clearly a homomorphism, such that g(f) = ( gγ(f) ) γ = (aγ)γ = a, for f(z) = z, z ∈ C . Moreover, if by πγ we denote the canonical projections πγ : A → Aγ, then we have that πγ ◦ g = gγ and we have already seen that gγ is continuous. Hence, by [14, (5.2), p. 51], we get that g is continuous. Let now A be a commutative complete barelled locally m-convex algebra which is also a Q-algebra. Then, by [9, Lemma 0.1], A[τ] has weakly compact character space. Let x ∈ A, and f(z) = z for all z ∈ C. Then f is an entire function and thus analytic on spA(x), that is f ∈ O(spA(x)). Define a map φ1 : C[X] →O(spA(x)) by φ1 ( n∑ i=1 λiX i ) = g , where g(z) = n∑ i=1 λiz i. Then φ1 is a homomorphism. By Theorem 3.13 there is a continuous ho- momorphism φ2 : O(spA(x)) → A such that φ2(f) = x. Let ψ0 = φ2 ◦ φ1. 72 m. weigt, i. zarakas Then, ψ0(X) = x. If φ1 is continuous when C[X] is equipped with the co- ordinatewise topology coming from C[[X]], then ψ0 is continuous and hence extends via continuity to an algebra homomorphism ψ : C[[X]] → A such that ψ(X) = x. For the case of a general commutative algebra A, the I-adic topology on A with respect to an ideal I of A plays an important role in embedding C[[X]] into A as the following result shows. Recall that a subset U of A is open with respect to the I-adic topology if and only if for all x ∈ U, there exists n ∈ N such that x + In ⊆ U. Theorem 3.14. Let A be a unital commutative algebra, I an ideal in A and x ∈ I. If A is I-adic complete and Hausdorff with respect to the I-adic topology, then there is a unital homomorphism ψ : C[[X]] → A such that ψ(X) = x. Proof. Let An = A/I n, for all n ∈ N. By [12, Theorem 5.5], there is a unital homomorphism ψ0,m : C[X] → Am such that ψ0,m(X) = x + Im. Then ψ0,m ( n∑ i=0 λiX i ) = n∑ i=0 λi(x + I m)i for all m ∈ N. This induces the map ψ̃0,m : C[X]/〈X〉−→ Am for all m ∈ N, where 〈X〉 is the two-sided ideal of C[X] generated by X. Since A is, by hypothesis, I-adic complete and Hausdorff, by taking the inverse limit of the afforementioned maps ψ̃0,m, we get the desired mapping ψ (see [12, Theorem 5.5] for details). Remark 3.15. Note that the ideal I in Theorem 3.14 cannot be the whole algebra A. For then, there would exist a unital algebra homomorphism ψ : C[[X]] → A such that ψ(X) = 1, i.e., ψ(1 − X) = 0. The latter equality contradicts the fact that 1 −X is invertible in C[[X]]. Lemma 3.16. Let A be a unital commutative algebra and I an ideal of A such that A is Hausdorff, has jointly continuous multiplication and is advert- ibly complete with respect to the I-adic topology. If x ∈ A, then (1 − x)−1 exists and is in A. on formal power series 73 Proof. Let à be the completion of A with respect to the I-adic topology. 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