Fréchet algebras of finite type

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1 Fréchet algebras of finite type MK Kopp Abstract The main objects of study in this paper are Fréchet algebras having an Arens Michael representation in which every Banach algebra is finite dimensional. We shall classify these algebras using a theorem which enssures that the image of any continuous linear map of a Fréchet space of finite type (i.e. for which the defining seminorms have a finite dimensional cokernel) into any Fréchet space is in fact closed. 1 Preliminaries Throughout this paper, a Fréchet algebra shall be a locally multiplicatively convex, complete, topological algebra whose topology arises from an increasing countable collection {p j j N} of submultiplicative seminorms satisfying (the Hausdorff condition) ker(p j ) = {0}. j N Any Fréchet algebra can be realized in the following way (see [3] and [9]). Let f 1 f 2 χ : B 1 B2... be an inverse (projective) limit sequence of Banach algebras such that the f j s are continuous algebra homomorphisms with dense range. If limχ is equipped with the weakest topology making all the universal maps π j : limχ B j continuous, then it is in fact a Fréchet algebra. Conversely, given any Fréchet algebra F, if the topology of F is defined by the increasing sequence (p j ) j N of submultiplicative seminorms, then define B j to be the completion of F/ ker(p j ) equipped with the norm p j. Then not only is F = lim χ in the category of C-algebras alone, but if limχ is equipped with the Fréchet algebra topology outlined above, then it is a Fréchet algebra which is Fréchet algebra isomorphic to F. We shall say that an inverse limit sequence χ is an Arens Michael representation for F if F and limχ are isomorphic as Fréchet algebras. We shall be interested in the following special type of Fréchet algebras Mathematical Subject Classification 46H40, 46J05, 46M40. This work is part of the research project of the European Research Training Network Analysis and Operators, contract HPRN-CT , funded by the European Commission. 1

2 Definition 1 A Fréchet algebra is said to be of finite type if its topology can be defined by an increasing collection of submultiplicative seminorms all of which have a finite dimensional cokernel. Alternatively, we can say that a Fréchet algebra F is of finite type if each continuous seminorm has a finite dimensional cokernel. It is easy to see that any closed subalgebra of a Fréchet algebra of finite type is of finite type as well. Examples. 1. Consider the ring C[[X]] of formal power series equipped with its (unique) Fréchet algebra topology given by the submultiplciative seminorms p k : C[[X]] R +, λ j X j j=0 k λ j. Each C[[X]]/ ker(p k ) is finite dimensional and so the ring of formal power series is a Fréchet algebra of finite type. 2. Similarly, it is easy to show that power series rings in several variables are Fréchet algebras of finite type, as is the ring C N [[X]] (see [7]) of formal power series in countably many variables. We shall show that these examples are the basic building blocks to describe all commutative local Fréchet algebras of finite type. An important tool in showing this is the theorem in the next section which shows that any any continuous linear map from a Fréchet space of finite type into any Fréchet space has closed range. j=0 2 A Theorem in the spirit of G.R. Allan The following is a linear analogue to Fréchet algebras of finite type. Definition 2 A Fréchet space is said to be of finite type if its topology is given by an increasing sequence of seminorms all of whose cokernels are finite dimensional. In Corollary 6 of [2], G.R. Allan proved that if C[[X]] is a dense subalgebra in some commutative Fréchet algebra F, then F is in fact equal to C[[X]]. The main tool used in the proof of this result (see Proposition 4 of [2]) is the fact that C[[X]] is a principal ideal domain and so certain inclusions about kernels of seminorms follow. The following theorem is in some sense an extension of Allan s result. Theorem 1 Let F and G be real or complex Fréchet spaces and let F be of finite type. Suppose further that f : F G is a continuous linear map. Then Im(f) is closed. Proof. We note that by quotienting out the (closed) kernel of f we may assume that f be injective. Suppose that (p n ) n N is an increasing sequence of seminorms defining the complete locally convex Hausdorff vector space topology on F such that F / ker(p j ) 2

3 is a finite dimensional vector space for each j N. Let (q n ) n N be an increasing sequence of seminorms defining the Fréchet space topology on G. Since f is continuous we may assume without loss of generality (thinning out (p j ) and rescaling the p j s if necessary) that holds for all j N. q j f p j ( ) If we denote the quotient map F F / ker(p j ) by π j, then, noting that we assumed F/ ker(p j ) finite dimensional, (π j (ker(q m f))) m N = (ker(q m f) + ker(p j )) m N defines a decreasing sequence of subspaces of F/ ker(p j ) which must stabilize. Thus for each j N there exists some integer N j > 0 such that π j (ker(q Nj f)) = π j (ker(q Nj +1 f)) =.... Without loss of generality we may choose the N j s in such a way that (N j ) j N is strictly increasing. Claim. For each j N ker(q Nj f) ker(p j ) holds. Proof of claim. The following argument is due to Jean Esterle and represents a simplification of our original proof. In particular it avoids rather complicated global diagram arguments and a use of the Arens Michael representation theorem. By construction of the N j s, ker (q Nk f) ker ( q Nk+1 f ) + ker(p k ) ( ) holds for all all k N. Given any x ker(q Nj f), ( ) allows us to write x = y 1 + z 1 with y 1 ker(p j ) and z 1 ker(q Nj+1 f). Applying ( ) again on z 1, z 1 = y 2 + z 2 with y 2 ker(p j+1 ) and z 2 ker ( q Nj+2 f ). Continuing by applying ( ) to z 2, then z 3 etc., we obtain two sequences (y n ) n N and (z n ) n N such that n N x = y 1 + y y n + z n n N y n ker(p j+n 1 ) ker(p j n N z n ker ( q Nj+n f ) We note that for each m, n, M N with n m > j + M ( n ) p j+m y t = 0 t=m 3

4 and so the series n=1 y n converges in F and n=1 y n ker(p j ). Hence the sequence (z n ) n N converges in F and we shall denote its limit by z. Since each ker(q n f) is closed (as f is continous) and since we assumed f injective, z n N ker ( q Nj+n f ) = {0} which shows that ( ) ( ) x = y n + z = y n ker(p j ). This establishes the claim. n=1 n=1 Fix some j N. Then we may pick b 1,..., b m in F such that b 1 + ker(p j ),..., b m + ker(p j ) form a basis for F/ ker(p j ). If V denotes the linear span of b 1,..., b m in F, then F = V ker(p j ) is a linear decomposition of F. By the claim above, ker(q Nj f) is a linear subspace of ker(p j ) which by ( ) must contain ker(p Nj ). As the latter subspace has finite codimension in F, so does ker(q Nj f) and thus there exists a finite dimensional subspace W of ker(p j ) such that is a linear decomposition of F. By construction defines a norm on V W. Moreover defines a norm on V. As F = V W ker(q Nj f) }{{} =ker(p j ). : V W R +, x q Nj (f(x)). : V R +, v p j (v) θ : (V W,. ) (V,. ), v + w v is a linear map between finite dimensional vector spaces, it must be continuous and hence there exists a constant c j > 0 such that for any v V and w W c j p j (v) ( q Nj f ) (v + w). ( ) But for any x F = V W ker(q Nj f), if x = v + w + k with v V, w W and k ker(q Nj f) then c j p j (v + w + k) = c j p j (v) q Nj f(v + w + k) = q Nj f(v + w) 4

5 and hence c j p j q Nj f holds on F by ( ). But this implies that the sequence of increasing seminorms ( qnj f ) j N defines the same Fréchet space topology on F as the p j s do. The result thus follows. Corollary 1 Let F be a Fréchet algebra of finite type and let a 1,..., a n and b 1,..., b n a collection of elements in F. We define F to be F if F is unital or to be the usual unitization otherwise. Then both sets n a j F b j and are closed in F. n a j F b j Proof. Considering the continuous linear maps φ 1 : F n F n, (f 1,..., f n ) a j f j b j and φ 2 : F n F, (f 1,..., f n ) n a j f j b j. As the domain of both maps are Fréchet algebras of finite type and as Im(φ 1 ) = n a j F b j and Im(φ 2 ) = the result follows from Theorem 1. n a j F b j Remark. In particular any finitely generated (left or right) ideal in a Fréchet algebra of finite type is closed. 3 A classification theorem The aim of this section is to give a simple structure theorem for all Fréchet algebras of finite type. We shall first of all consider two examples which will turn out to be important. 5

6 1. Consider the C-algebra M nj (C) j N (where (n j ) j N is a given sequence of positive integers) equipped with the product topology. A defining collection of submultiplicative seminorms (p n ) for this topology is given by p m : m M nj (C) R +, (A 1,...) A j Mnj (C) m N j N (. Mnj (C) denotes a given (submultiplicative) norm on M nj (C)) and so the algebra above is a Fréchet algebra of finite type. 2. Let S ℵ0 denote the free semigroup generated by a countable collection of symbols, i.e. S ℵ0 = n N S n where S n denotes the free semigroup on n symbols and where S n S n+1 is obtained by identifying the n generating symbols of the former, in order, with the first n generating symbols of the latter. Denote by the set of formal C-linear sums C 0 [[S ℵ0 ]] g S ℵ0 λ g g which can be given a C-algebra structure via the product λ f f µ f f = f S ℵ0 f S ℵ0 f S ℵ0 g,h S ℵ0 ;gh=f λ g µ h f (the sum in brackets on the right hand side is defined as each element in S ℵ0 a finite product of elementary symbols). For each n N, define p n : C 0 [[S ℵ0 ]] R +, λ f f λ f f S ℵ0 f C n consists of where C n denotes the collection of all elements in S n which can be written as a product of no more than n (not necessarily distinct) basic symbols. It is easy to see that the p n s define submultiplicative seminorms and that the locally multiplicatively convex algebra obtained by equipping C 0 [[S ℵ0 ]] with these seminorms is a Fréchet algebra. The algebra C 0 [[S ℵ0 ]] is in fact a radical Fréchet algebra. This follows from the fact that the canonical image of any element f C 0 [[S ℵ0 ]] is quasi-regular and hence, by Theorem 5.2 part a) in [9], f must be quasi-regular. Theorem 2 Let F be a Fréchet algebra of finite type. Then F is radical if and only if F is Fréchet algebra isomorphic to a quotient of C 0 [[S ℵ0 ]] by a closed ideal. 6

7 Proof. We note that the if part is immediate from Theorem 5.2 part a) in [9] and from the fact that the Arens Michael representation for C 0 [[S ℵ0 ]] consists of nilpotent finite dimensional Banach algebras. For the only if part, suppose that f 1 f 2 B 1 B2... ( ) is an Arens Michael representation of F such that each B j is a finite dimensional C-vector space. Suppose further that (q n ) n N is an increasing sequence of submultiplicative seminorms associated with ( ), i.e. B j = F/ ker(q j ). We pick a sequence (a n ) n N in F in the following way. We let a 1,..., a N1 be such that {π 1 (a j ) j {1,..., N 1 }} is a basis for the vector space B 1. Then we pick a N1 +1,..., a N2 in F such that {π 2 (a j ) j {1,..., N 2 }} is a linear basis for B 2. This automatically implies that {π 2 (a j )) j {N 1 + 1,..., N 2 }} spans ker(f 1 ) and so π 1 (a j ) = 0 for each j {N 1 + 1,..., N 2 }. If we continue doing this we obtain a sequence (a n ) n N and an increasing sequence (N j ) j N of non-negative integers such that π j (a k ) = 0 for all k > N j and such that the closed linear span of {a n n N} equals F. Suppose now that S = j N N j and that for any sequence (b n ) n N in F and any (i 1,..., i k ) S we define b (i 1,...,i k ) = b i1 b i2... b ik. As each B j is a finite dimensional and radical algebra it is in fact nilpotent. Thus, taking into account the defining property π j (a k ) = 0 for all k > N j, we obtain that λ a S converges in F for any (λ ) S C S. Furthermore, if we denote the countable non-commuting basic symbols generating the free semigroup S ℵ0 by {X j j N} then, upon writing for any = (i 1,..., i k ) S X = X i1 X i2... X ik, we define a map Γ : C 0 [[S ℵ0 ]] F, S λ X S λ a 7

8 where the λ s are arbitrary scalars indexed by S. This is certainly an algebra homomorphism and it is also continuous. For given any j N, let N N be larger than dim C B j and also larger than the degree of nilpotency of B j. If S N denotes the union N {1,..., N}j then ( ) ( N q j λ a max ( 1, q j (a k ) N)) λ S k=1 S N ( N ) λ X k=1 max ( 1, q j (a k ) N)) p N ( S where the p j s are the defining submultiplicative seminorms for the Fréchet algebra topology on C 0 [[S ℵ0 ]] given in example 2. at the beginning of this section. It follows from Theorem 1 that Γ is in fact surjective. The proof of Theorem 4 we shall present here is a shorter version of our original proof. It relies on the following Lemma which was suggested to us by Jean Esterle and which will spare us from a rather elaborate matrix unit lifting procedure. Lemma 3 Let A 1 and A 2 be finite dimensional C-algebras and let π : A 2 A 1 be a surjective algebra homomorphism. Let R 1 and R 2 denote the Jacobson radicals of A 1 and A 2 respectively and assume that B 1 is a semisimple sub-algebra of A 1 such that A 1 = B 1 R 1 (direct sum of vector spaces). Then π(r 2 ) = R 1 and there exists a semisimple subalgebra B 2 of A 2 such that A 2 = B 2 R 2 (direct sum of vector spaces) and such that π(b 2 ) = B 1. Proof. By Wedderburn s principle theorem (see [8],p.127) there exists a semisimple subalgebra U of A 2 such that A 2 = U R 2 as a direct sum of vector spaces. Since π(u) is isomorphic to a quotient of a finite dimensional semisimple algebra, Wedderburn s structure theorem for finite dimensional semisimple algebras implies that π(u) is semisimple. As π(u) R 1 is a two-sided ideal in π(u) which is contained in rad(π(u)) (the Jacobson radical of π(u)) it follows that π(u) R 1 = {0} (#) Since π is surjective π(r 2 ) is a two-sided ideal of B 1. Since each element in π(r 2 ) is also nilpotent we deduce that π(r 2 ) R 1 Let x R 1. Since π is surjective there exist u U and r R 2 such that x = π(u + r) = π(u) + π(r). 8

9 But then π(u) π(u) R 1 and so x = π(r) by (#). Since x R 1 was arbitrary this shows π(r 2 ) = R 1. We note that π 1 (B 1 ) is a subalgebra of A 2 and, by the above Wedderburn s principle theorem implies that A 2 = π 1 (B 1 ) + R 2. π 1 (B 1 ) = B 2 H where B 2 is a semisimple subalgebra of π 1 (B 1 ) and H is its Jacobson radical. We note that H + R 2 is in fact a two-sided ideal in A 2 as for any h H, s, r R 2 and any b π 1 (B 1 ) (b + s)(h + r) = }{{} bh + br } + sh {{ + sr } H R 2 and (h + r)(b + s) = }{{} hb H + rb } + hs {{ + rs }. R 2 Since H is the Jacobson radical of π 1 (B 1 ) all its elements must be nilpotent. Thus given h H and r R 2, there exists some k N such that h k = 0. But then (h + r) k R 2 and so (h + r) is nilpotent. Thus H + R 2 is a nilpotent two-sided ideal and must thus be contained in R 2. But this means that H + R 2 = R 2 and so A 2 = B 2 R 2. and π(b 2 ) = B 1 since π(h) B 1 R 1 = {0}. Theorem 4 Suppose that F is a Fréchet algebra of finite type and let R denote the radical of F. Then there exists a closed subalgebra S of F such that F = R S, i.e. F decomposes as a vector space into a direct sum of a closed semisimple subalgebra S and the closed ideal R. Moreover, S is Fréchet algebra isomorphic to M nj (C) j N for some sequence (n j ) j N of non-negative integers equipped with the product topology. The ideal R is isomorphic to a quotient of C 0 [[S ℵ0 ]] by a closed ideal. Proof. Suppose that f 1 f 2 f 3 B 1 B2 B3... is an Arens Michael representation for F. Then by Wedderburn s principal theorem and a repeated application of Lemma 3 above there exists a decomposition for each B j B j = R j S j 9

10 such that R j is the Jacobson radical of B j, S j is a semisimple subalgebra and such that both f j (S j+1 ) = S j and f j (R j+1 ) = R j. We now define ( R = lim R 1 ( S = lim S 1 f 1 R2 R2 S 1 S2 S2 ) f 2 R3... ) f 2 S3... and equip these algebras with the weakest topology which makes all the universal maps R R j and S S j continuous. These topologies are both Fréchet algebra topologies (by [9]) and so S and R are Fréchet algebras of finite type. As subalgebras of F both R and S inherit subspace topologies. By definition, the identity map defined from R equipped with the Fréchet algebra topology defined above into R with its subspace topology from F is trivially continuous. Thus Theorem 1 implies that these two topologies coincide. Similarly the subspace topology on S coincides with the Fréchet algebra topology defined above. Moreover, Theorem 5.2 part a) in [9] implies that R = rad(f ). Concerning the statement about S, we note that each S j is isomorphic to M (j) n (C)... M (j) 1 n (C) k j for some k j Z + and some n (j) 1,..., n(j) k j Z +. Since f j (S j+1 ) = S j and since each matrix algebra M n (C) is semisimple we may relabel the n (j+1) 1,..., n (j+1), if necessary, such that n (j) k for all k {1,..., k j }. The result thus follows. = n (j+1) k In [7] the algebra C N [[X]] consisting of formal power series in countably many variables is defined. It is a local commutative algebra and its maximal ideal will be denoted by C 0,N [[X]]. It is not hard to see that this ideal is algebra isomorphic to the quotient of C 0 [[S ℵ0 ]] by the closure of the ideal generated by all commutation relations X i X j X j X i where i j; i, j N and where the X j s denote the basic non-commuting symbols generating S ℵ0. Corollary 1 Let F be a commutative Fréchet algebra of finite type. Then there exists a closed ideal I in C 0,N [[X]] and some ordinal α ℵ 0 {ℵ 0 } such that F is algebra isomorphic to the vector space direct sum C α C 0,N [[X]]/I where C 0,N [[X]] is subalgebra of C N [[X]] consisting of all formal power series in countably many variables whose constant term is zero. Remark. We note that our proofs yield analogous classification results for real Fréchet algebras of finite type. References [1] G. R. Allan:Embedding the algebra of formal power series in a Banach algebra, Proc. London Math. Soc. (3) 25 (1972), k j+1 10

11 [2] G. R. Allan:Fréchet algebras and formal power series, Studia Math. 119 (3), (1996), [3] R. F. Arens:Dense Inverse Limit Rings,Michigan Math. J. 5 (1958), [4] M.F. Atiyah, I.G. MacDonald: Introduction to Commutative Algebra, Addison-Wesley, 1969 [5] H. G. Dales: Banach algebras and automatic continuity, London Math. Soc. Monographs. New Series, 24. Oxford Science Publications 2000 [6] J. Esterle: Elements of a Classification of a Commutative Radical Banach Algebra,in Radical Banach Algebras and Automatic Continuity, Proceedings of the Long Beach Conference 1981, Springer Notes in Mathematics 975, pp.4-66 [7] J. Esterle:Picard s theorem, Mittag-Leffler methods, and continuity of characters on Fréchet algebras, Ann. Sci. Ecole Norm. Sup. (4) 29 (1996), no.5, [8] I. Kaplansky: Fields and Rings, Chicago Lectures in Mathematics, Second Edition (1972) [9] E. A. Michael:Locally multiplicatively convex topological algebras, Mem. Amer. Math. Soc. 11 (1953; third printing 1971) Laboratoire d Analyse et Géométrie UMR 5467 Université Bordeaux Cours de la libération Talence, Cedex France kopp@math.u-bordeaux.fr 11