Continuous homomorphisms of Arens-Michael algebras

It is shown that every continuous homomorphism of Arens-Michael algebras can be obtained as the limit of a morphism of certain projective systems consisting of Fr\'{e}chet algebras. Based on this we prove that a complemented subalgebra of an uncountable product of Fr\'{e}chet algebras is topologically isomorphic to the product of Fr\'{e}chet algebras. These results are used to characterize injective objects of the category of locally convex topological vector spaces. Dually, it is shown that a complemented subspace of an uncountable direct sum of Banach spaces is topologically isomorphic to the direct sum of ({\bf LB})-spaces. This result is used to characterize projective objects of the above category.


Introduction
Arens-Michael algebras are limits of projective systems of Banach algebras (or, alternatively, closed subalgebras of (uncountable) products of Banach algebras). Quite often, when dealing with a particular Arens-Michael algebra, at least one projective system arises naturally (for instance, as a result of certain construction) and, in most cases, it does contain the needed information about its limit. The situation is somewhat different if an Arens-Michael algebra is given arbitrarily and there is no particular projective system associated with it in a canonical way.
Below (Definition 3.1) we introduce the concept of a projective Fréchet system and show (Theorem 3.3) that every continuous homomorphism of Arens-Michael agebras can be obtained as the limit homomorphism of certain morphism of cofinal subsystems of the corresponding Fréchet systems. This result applied to the identity homomorphism obviously implies that any Arens-Michael algebra has essentially unique Fréchet system associated with it. Consequently, any information about an Arens-Michael algebra is contained in the associated Fréchet system. The remaining problem of restoring this information is, of course, still non-trivial, but sometimes can be successfully handled by using a simple but effective method. This method is based on Proposition 2.5 1 .
Applying such an approach we obtain the following, in a sense dual, statements (precise statements are recorded in Theorems 4.2 and 6.2 respectively).
Retracts of uncountable products of Banach spaces. A retract of an uncountable product of Banach spaces is the product of retracts of countable subproducts.
Retracts of uncountable coproducts of Banach spaces. A retract of an uncountable coproduct of Banach spaces is the coproduct of retracts of countable subcoproducts.
Based on these results we present a complete description of injective and projective objects of the category LCS of locally convex topological vector spaces (over the field of complex numbers). Obviously products (coproducts) of arbitrary collections of injective (projective) objects of any category are again injective (projective). Also the class of injectives (projectives) is stable under retractions. Sometimes in a relatively nice category there naturally exists a class of "simple" injectives (projectives) so that any other injective (projective) can be obtained from simple ones by applying the above mentioned operations, i.e. by forming products (coproducts) and by passing to retracts. Consider some examples: 1. Injectives in the category COMP of compact Hausdorff spaces are precisely retracts of products of copies of the closed unit segment [2], whereas projectives are retracts of coproducts of copies of the singleton [4, Problem 6.3.19(a)]. The cone of the uncountable power of the segment is an example of an injective which is not the product of simpler injectives. 2. Injectives in the category COMP ABGR of compact Hausdorff abelian topological groups are precisely the tori, i.e. products of copies of the circle group T (see [1] for a topological characterization of arbitrary tori). 3. Projectives in the category R-MOD of left R-modules (where R is an associative ring with unit) are coproducts of countably generated projectives [6]. In these examples injectives are retracts of products and projectives are retracts of coproducts of simpler objects. On the other hand, as shows the first example, injectives need not be products.
The following two results (Theorems 5.2 and ?? respectively) provide the full solution of the corresponding problems in the category LCS.
Characterization of injectives. The following conditions are equivalent for a locally convex topological vector space X: (1) X is an injective object of the category LCS.
(2) X is isomorphic to the product {F t : t ∈ T }, where each F t , t ∈ T , is a complemented subspace of the product {ℓ ∞ (J tn ) : n ∈ ω}.

Preliminaries
2.1. Projective systems and their morphisms. Below we consider projective systems S X = {X α , p β α , A} consisting of topological algebras X α , α ∈ A, and continuous homomorphisms p β α : X β → X α , α ≤ β, α, β ∈ A (A is the directed indexing set of S X ). The limit lim ← − S X of this system is defined as the closed subalgebra of the Cartesian product {X α : α ∈ A} (with coordinatewise defined operations) consisting of all threads of S X , i.e.
We refer the reader to [8], [9] for general properties of projective systems.
Suppose we are given two projective systems S X = {X α , p β α , A} and S Y = {Y γ , q δ γ , B} consisting of topological algebras X α , α ∈ A, and Y γ , γ ∈ B. A morphism of the system S X into the system S Y is a family {ϕ, {f γ : γ ∈ B}}, consisting of a nondecreasing function ϕ : B → A such that the set ϕ(B) is cofinal in A, and of continuous homomorphisms f γ : ϕ(γ) , whenever γ, δ ∈ B and γ ≤ δ. In other words, we require (in the above situation) the commutativity of the following diagram

Any morphism
{ϕ, {f γ : γ ∈ B}} : S X → S Y induces a continuous homomorphism, called the limit homomorphism of the morphism To see this, assign to each thread x = {x α : α ∈ A} of the system S X the point y = {y γ : γ ∈ B} of the product {Y γ : γ ∈ B} by letting It is easily seen that the point y = {y γ : γ ∈ B} is in fact a thread of the system S Y . Therefore, assigning to Straightforward verification shows that this map is a continuous homomorphism.
Morphisms of projective systems which arise most frequently in practice are those defined over the same indexing set. In this case, the map ϕ : A → A of the definition of morphism is taken to be the identity. Below we shall mostly deal with such situations and use the following notation: A} be a projective system and X be a topological algebra. Suppose that for each α ∈ A a continuous homomorphism f α : X → Y α is given in such a way that f α = q β α f β whenever α, β ∈ A and α ≤ β. Then there exists a natural continuous homomorphism f : X → lim ← − S Y (the diagonal product △{f α : α ∈ A}) satisfying, for each α ∈ A, the condition f α = q α f .
Proof. Indeed, we only have to note that X, together with its identity map id X , forms the projective system S. So the collection {f α : α ∈ A} is in fact a morphism S → S Y . The rest follows from the definitions given above.

2.2.
Arens-Michael algebras. We recall some definitions [5]. A polynormed space X is a topological linear space X furnished with a collection {|| · || ν , Λ} of seminorms generating the topology of X. This simply means that the collection forms a subbase of the topology of X. A polynormed algebra is a polynormed space X wich admits a separately continuous multiplication bioperator m : X × X → X. A multinormed algebra is a polynormed algebra such that ||xy|| ν ≤ ||x|| ν · ||y|| ν for each ν ∈ Λ and any (x, y) ∈ X × X. Finally, an Arens-Michael algebra is a complete (and Hausdorff) multinormed algebra.
Theorem 2.2. The following conditions are equivalent for a multinormed algebra X: (a) X is an Arens-Michael algebra. (b) X is limit of a certain projective system of Banach algebras. (c) X is topologically isomorphic to a closed subalgebra of the Cartesian product of a certain family of Banach algebras.

Set-theoretical facts.
For the reader's convenience we begin by presenting necessary set-theoretic facts. Their complete proofs can be found in [2]. Let A be a partially ordered directed set (i.e. for every two elements α, β ∈ A there exists an element γ ∈ A such that γ ≥ α and γ ≥ β). We say that a subset A 1 ⊆ A of A majorates another subset A 2 ⊆ A of A if for each element α 2 ∈ A 2 there exists an element α 1 ∈ A 1 such that α 1 ≥ α 2 . A subset which majorates A is called cofinal in A. A subset of A is said to be a chain if every two elements of it are comparable. The symbol sup B , where B ⊆ A, denotes the lower upper bound of B (if such an element exists in A). Let now τ be an infinite cardinal number. A subset B of A is said to be τ -closed in A if for each chain C ⊆ B, with |C| ≤ τ , we have sup C ∈ B, whenever the element sup C exists in A. Finally, a directed set A is said to be τ -complete if for each chain C of elements of A with |C| ≤ τ , there exists an element sup C in A.
The standard example of a τ -complete set can be obtained as follows. For an arbitrary set A let exp A denote, as usual, the collection of all subsets of A. There is a natural partial order on exp A: A 1 ≥ A 2 if and only if A 1 ⊇ A 2 . With this partial order exp A becomes a directed set. If we consider only those subsets of the set A which have cardinality ≤ τ , then the corresponding subcollection of exp A, denoted by exp τ A, serves as a basic example of a τ -complete set. Proposition 2.3. Let {A t : t ∈ T } be a collection of τ -closed and cofinal subsets of a τ -complete set A. If | T |≤ τ , then the intersection ∩{A t : t ∈ T } is also cofinal (in particular, non-empty) and τ -closed in A .
Then the set of all L-reflexive elements of A (an element α ∈ A is L-reflexive if (α, α) ∈ L) is cofinal and τ -closed in A.

Continuous homomorphisms of Arens-Michael algebras
The following statement is needed in the proof of Theorem 3.3. In the case when all X α 's are Banach algebras its proof can be extracted from [5, Proposition 0.1.9] (see also [5, Proof of Proposition V.1.8]).
A} be a projective system consisting of Arens-Micheael algebras X α , α ∈ A, Y be a Banach algebra and f : lim ← − S → Y be a continuous homomorphism. Then there exist an index α ∈ A and a continuous homomorphism Proof. The continuity of f and the definition of the topology on lim ← − S guarantee that there exists an index α ∈ A and an open subset V α ⊆ X α such that where | · | denotes the norm of the Banach space Y .
Since X α is an Arens-Michael algebra, X α can be identified with a closed subalgebra of the product {B t : t ∈ T } → {B t : t ∈ S} denote the natural projection onto the corresponding subproduct. If S ⊆ T is a finite subset of T , then ||{x t : t ∈ S}|| S = max{||x t || t : t ∈ S} for each {x t : t ∈ S} ∈ {B t : t ∈ S}. Since V α is open in X α , the definition of the product topology guarantees the existence of a finite subset S ⊆ T and of a number ǫ > 0 such that It then follows that if x ∈ X and ||π S (p α (x))|| S ≤ 1, then ||π S (p α (ǫx))|| S = ǫ||π S (p α (x))|| S ≤ ǫ and consequently Let us now show that the map Assuming the contrary, suppose that for some z ∈ π S (p α (X)) there exist two points This contradiction shows that the map h : which implies that the map h is linear. Next consider points z ∈ π S (p α (X)) and x ∈ X such that π S (p α (x)) = z. By (3.5), This shows that h is bounded and, consequently, continuous. Next let us show that h : π S (p α (X)) → Y is multiplicative. Let (x ′ , y ′ ) ∈ π S (p α (X)) × π S (p α (X)) and consider a point (x, y) ∈ X×X such that π S (p α (x)) = x ′ and π S (p α (y)) = y ′ . Then, by (3.6), Since (Y, | · |) is complete, h admits the linear continuous extension Since the multiplication on {B t : t ∈ T f } is jointly continuous, we conclude that g is also multplicative. Finally, define the map f α as the composition Obviously f α is a continuous homomorphism satisfying the required equality f α • p α = f .
Next we introduce the concept of projective Fréchet system. Definition 3.1. Let τ ≥ ω be a cardinal number. A projective system S X = {X α , p β α , A} consisting of topological algebras X α and continuous homomorphisms p β α : X β → X α , α ≤ β, α, β ∈ A, is a τ -system if: 1. X α is a closed subalgebra of the product of at most τ Banach algebras, α ∈ A.

The indexing set
Fréchet systems are defined as projective ω-systems.

Proposition 3.2. Every Arens-Micheal algebra X can be represented as the limit of a projective Fréchet system
Conversely, the limit of a projective Fréchet system is an Arens-Michael algebra.
Proof. By Theorem 2.2, X can be identified with a closed subalgebra of the product {X t : t ∈ T } of some collection of Banach algebras. If |T | ≤ ω, then X itself is a Fréchet algebra and therefore our statement is trivially true. If |T | > ω, then consider the set exp ω T of all countable subsets of T . Clearly, denotes the natural projection onto the corresponding subproduct. Also let A} be a projective Fréchet system. Clearly, lim ← − S X can be identified with a closed subalgebra of the product {X α : α ∈ A} (see Subsection 2.1). Each X α , α ∈ A, can obviously be identified with a closed subalgebra of the product {B α n : n ∈ T α } of a countable collection of Banach algebras B α n . Then lim ← − S X , as a closed subalgebra of the product {B α n : n ∈ T α } : a ∈ A is, according to Theorem 2.2, an Arens-Michael algebra.
The following statement is one of our main results.
Proof. We perform the spectral search by means of the following relation Let us verify the conditions of Proposition 2.5. Existence Condition. By assumption, Y α is a Fréchet algebra. Therefore Y α can be identified with a closed subspace of a countable product {B n : n ∈ ω} of Banach algebras. Let π n : {B n : n ∈ ω} → B n denote the n-th natural projection. For each n ∈ ω, by Lemma 3.1, there exist an index β n ∈ A and a continuous homomorphism f βn : X βn → B n such that π n q α f = f βn p βn . By Corollary 2.4, there exists an index β ∈ A such that β ≥ β n for each n. Without loss of generality we may assume that β ≥ α. Let f n = f βn p β βn , n ∈ ω. Next consider the diagonal product The verification of this condition is trivial. Indeed, it suffiecs to consider the composition f γ α = f β α p γ β . ω-closeness Condition. Suppose that for some countable chain C = {α n : n ∈ ω} in A with α = sup C and for some β ∈ A with β ≥ α, the maps f β αn : X β → Y αn have already been defined is such a way that f β αn p β = q αn f for each n ∈ ω (in other words, (α n , β) ∈ L for each n ∈ ω). Next consider the composition ). This shows that (α, β) ∈ L and finishes the verification of the ω-closeness condition.
Now denote by B f the set of all L-reflexive elements in A. By Proposition 2.5, B f is a cofinal and ω-closed subset of A. One can easily see that the Lreflexivity of an element α ∈ A is equivalent to the existence of a continuous is a morphism of the cofinal and ω-closed subspectrum S X |B f of the spectrum S X into the cofinal and ω-closed subspectrum S Y |B f of the spectrum S Y . It only remains to remark that the original map f is induced by the constructed morphism. This finishes the proof of the first part of our Theorem.
The second part of this theorem can be obtained from the first as follows. Let f : lim ← − S X → lim ← − S Y be a topological isomorphism. Denote by f −1 : lim ← − S Y → lim ← − S X the inverse of f . By the first part proved above, there exist a cofinal and ω-closed subset B f of A and a morphism Similarly, there exist a cofinal and ω-closed subset B f −1 of A and a morphism Therefore, in order to complete the proof, it suffices to show that for each α ∈ B the map f α : X α → Y α is a topological isomorphism. Indeed, take a point x α ∈ p α lim ← − S X ⊆ X α . Also choose a point x ∈ lim ← − S X such that x α = p α (x). Then This proves that g α f α |p α lim ← − S X = id pα(lim ← − S X ) . Similar considerations show is a topological isomorphism (whose inverse is g α ).
Remark 3.1. A similar statement remains true (with the identical proof) for projective τ -systems for any cardinal number τ > ω. : C 2n × C 2 → C 2n and π 2(n+1)+1 2n+1 denote the natural projections. Clearly the limits lim ← − S even and lim ← − S odd of these projective systems are topologically isomorphic (both are topologically isomorphic to the countable infinite power C ω of C), but S even and S odd do not contain isomorphic cofinal subsystems.
Corollary 3.4. Let S X = {X α , p β α , A} be a projective Fréchet system. If lim ← − S is a Fréchet algebra, then there exists an index α ∈ A such that the β-th limit projection p β : lim ← − S → X β is a topological isomorphism for each β ≥ α. Proof. Consider a trivial projective Fréchet system S ′ = {X α , q β α , A}, where X α = lim ← − S and q β α = id lim ← − S for each α, β ∈ A. By Theorem 3.3 (applied to the identity homomorphism f = id lim ← − S ), there exist an index α ∈ A and a continuous homomorphism g α : X α → lim ← − S such that id lim ← − S = g α • p α . Clearly, in this situation, p α | lim ← − S : lim ← − S → X α is an embedding with a closed image. But this image p α (lim ← − S) is dense in X α (condition (4) of Definition 3.1). Therefore p α (and, consequently p β for any β ≥ α) is a topological isomorphism.

Corollary 3.5. Suppose that X is a Fréchet subalgebra of an uncountable product
{B t : t ∈ T } of Fréchet (Banach) algebras, then there exists a countable subset T X of the indexing set T such that the restriction π T X |X : X → π T X (X) of the natural projection π T X : 3.1. Arens-Michael * -algebras. The concept of projective Fréchet system can naturally be adjusted to handle variety of situations. Below it will always be completely clear in what content this concept is being used. Let us consider Arens-Michael * -algebras, i.e. Arens-Michael algebras with a continuous involution. It is known that every such an algebra can be identified with a closed * -subalgebra of the product of Banach * -algebras (see, for instance, [5,Proposition V.3.41]). Therefore one can obtain an alternative description of such algebras as limits of projective systems consisting of Banach * -algebras and continuous * -homomorphisms (compare with Theorem 2.2). This, as in Proposition 3.2, leads us to the conclusion recorded in the following statement.
The analog of Theorem 3.3 is also true.
Proof. By Theorem 3.3, there exists a cofinal and ω-closed subset B f of A and a morphism Let us show that f α , α ∈ B f is actually a * -homomorphism. Indeed, let x α ∈ p α (X) and x ∈ X such that p α (x) = x α . Then

Complemented subalgebras of uncountable products of Fréchet algebras
In this section we show (Theorem 4.2) that complemented subalgebras of (uncountable) products of Fréchet algebras are products of Fréchet algebras. We begin with the following lemma.
Lemma 4.1. Let p : X → Y be a surjective continuous homomorphism of topological algebras and suppose that X is a closed subalgebra of the product Y × B, where B is a a topological algebra. Assume also that there exists a continuous homomorphism r : Y × B → X satisfying the following conditions: Then there exists a topological isomorphism h : X → Y × ker p such that π Y h = p.
Proof. If x ∈ X, then p (x − r(p(x), 0)) = p(x) − p(r(p(x), 0)) This shows that the formula h(x) = (p(x), x − r(p(x), 0)) , x ∈ X, defines a continuous linear map h : X → Y × ker p. Moreover, h is a topological isomorphism between X and Y × ker p considered as topological vector spaces (to see this observe that the continuous and linear map g : Y × ker p → X defined by letting g(y, x) = r(y, 0) + x for each (y, x) ∈ Y × ker p, has the following properties: g • h = id X and h • g = id Y ×ker p ). We now show that h is an isomorophism of the category of topological algebras as well.
Let x 1 , x 2 ∈ X. We need to show that h( (a 1 , b 1 ) − r(a 1 , 0) This shows that h is a homomorphism and, consequently, a topological isomorphism as required.

Theorem 4.2. A complemented subalgebra of the product of uncountable family of Fréchet algebras is topologically isomorphic to the product of Fréchet algebras. More formally, if X is a complemented subalgebra of the product
Proof. Let X be a complemented subalgebra of the uncountable product B = {B t : t ∈ T } of Fréchet algebras B t , t ∈ T , where T is an indexing set with |T | = τ > ω. There exists a continuous homomorphism r : B → X such that r(x) = x for each x ∈ X. A subset S ⊆ T is called r-admissible if π S (r(z)) = π S (z) for each point z ∈ π −1 S (π S (X)). Claim 1. The union of an arbitrary family of r-admissible sets is r-admissible. Let {S j : j ∈ J} be a collection of r-admissible sets and S = {S j : j ∈ J}. Let z ∈ π −1 S (π S (X)). Clearly z ∈ π −1 S j π S j (X) for each j ∈ J and consequently π S j (r(z)) = π S j (z) for each j ∈ J. Obviously, π S (z) ∈ π S (r(z)) and it therefore suffices to show that the set π S (r(z)) contains only one point. Assuming that there is a point y ∈ π S (r(z)) such that y = π S (z) we conclude (remembering that S = {S j : j ∈ J}) that there must exist an index j ∈ J such that π S S j (y) = π S S j (π S (z)). But this is impossible π S S j (y) ∈ π S S j (π S (r(z))) = π S j (r(z)) = π S j (z) = π S S j (π S (z)) . Claim 2. If S ⊆ T is r-admissible, then π S (X) is a closed subalgebra of B S = {B t : t ∈ S}. Indeed, let i S : B S → B be the canonical section of π S (this means that i S = id B S △0 : B S → B S × B T −S = B). Consider a continuous linear map r S = π S • r • i S : B S → π S (X). Obviously, i S (y) ∈ π −1 S (π S (X)) for any point y ∈ π S (X). Since S is r-admissible the latter implies that y = π S (i S (y)) = π S (r (i S (y))) = r S (y).
This shows that π S (X) is closed in B S . Claim 3. Let S and R be r-admissible subsets of T and S ⊆ R ⊆ T . Then the map π R S : π R (X) → π S (X) is topologically isomorphic to the natural projection π : π S (X) × ker π R S |π R (X) → π S (X).
Application of Lemma 4.1 (with X = π R (X), Y = π S (X), B = B R−S , p = π R S |π R (X) and r = r R ) finishes the proof of Claim 3. Claim 4. Every countable subset of T is contained in a countable r-admissible subset of T .
Let A be a countable subset of T . Our goal is to find a countable r-admissible subset C such that A ⊆ C. By Theorem 3.3, there exist a countable subset C of T and a continuous homomorphism r C : B C → cl B C (π C (X)) such that A ⊆ C and π C • r = r C • π C . Consider a point y ∈ π C (X). Also pick a point x ∈ X such that π C (x) = y. Then y = π C (x) = π C (r(x)) = r C (π C (x)) = r C (y).
This shows that r C |π C (X) = id π C (X) . It also follows that π C (X) is closed in In order to show that C is r-admissible let us consider a point z ∈ π −1 C (π C (X)). By the observation made above, r C (π C (z)) = π C (z). Finally π C (z) = r C (π C (z)) = π C (r(x)) which implies that C is r-admissible.
Since |T | = τ , we can write T = {t α : α < τ }. Since the collection of countable r-admissible subsets of T is cofinal in exp ω T (see Claim 4), each element t α ∈ T is contained in a countable r-admissible subset A α ⊆ T . According to Claim 1, the set T α = {A β : β ≤ α} is r-admissible for each α < τ . Consider the projective system Tα |π T α+1 (X) for each α < τ. Since T = {T α : α < τ }, it follows that X = proj lim S. Obvious transfinite induction based on Claim 3 shows that Since, by the construction, A α is a countable r-admissible subset of T , it follows from Claim 2 that X 0 and ker (p α+1 α ), α < τ , are Fréchet algebras. This finishes the proof of Theorem 4.2.

Injective objects of the category LCS
In this section we investigate injective objects of the category LCS of locally convex Hausdorff topological vector spaces and their continuous linear maps.
Recall that an object X of the category LSC is injective if any continuous linear map f : A → X, defined on a linear subspace of a space B, admits a continuous linear extension g : B → X (i.e. g|A = f ). Here is the corresponding diagram We start with the metrizable case. The following statement is probably known (consult with [7,Theorem C.3.4], [3, pp.6-9]).
Proposition 5.1. The following conditions are equivalent for any Banach space X: (1) X is an injective object of the category LCS.
(2) X is an injective object of the category BAN.
(3) X is isomorphic to a complemented subspace of ℓ ∞ (J) for some set J.
(2) =⇒ (3). The Banach space X can be identified with a closed linear subspace of the space ℓ ∞ (J) for some set J. By condition (2), there exists a linear continuous map r : ℓ ∞ (J) → X such that r(x) = x for each x ∈ X. This obviously implies that X is a complemented subspace of ℓ ∞ (J).
(3) =⇒ (1). First let us show that the Banach space ℓ ∞ (J) (for any set J) is an injective object of the category BAN . Let where C j , j ∈ J, stands for a copy of C, be a continuous linear map defined on a closed linear subspace X of a Banach space Y . Since f is bounded, for each j ∈ J we have This shows that ||π j • f || ≤ ||f ||, where π j : ℓ ∞ (J) → C j denotes the canonical projection onto the j-th coordinate. By the Hahn-Banach Theorem, the linear map π j • f : X → C j , j ∈ J, admits a continuous linear extension g j : Y → C j such that ||g j || = ||f • π j || ≤ ||f ||. Consequently, This shows that the map g : Y → ℓ ∞ (J), given by letting is well defined. Obviously g is continuous, linear and extends the map f . Therefore ℓ ∞ (J) is indeed an injective object of the category BAN . Next consider a complete locally convex topological vector space Y , its closed linear subspace Z and a continuous linear map f : Z → X, where X is a complemented subspace of the Banach space ℓ ∞ (J) for some set J. We need to show that f admits a continuous linear extension g : Y → X. Identify Y with a closed linear subspace of the product Clearly Z is a closed linear subspace of {B t : t ∈ T }. By Lemma 3.1, there exist a finite subset T f and a continuous linear map f ′ : . The first part of the proof of this implication, coupled with condition (3), implies that X is an injective object of the category BAN. Therefore the map f ′ admits a continuous linear extension g ′ : cl B T f π T f (Y ) → X. It only remains to note that the map g = g ′ • π T f |Y is a continuous linear extension of f . (1) X is an injective object of the category LCS.
(2) X is isomorphic to the product Proof.
(2) =⇒ (1). By Proposition 5.1, ℓ ∞ (J) is an injective object of the category LCS for any set J. Obviously the product of an arbitrary collection of injective objects of the category LCS is also an injective object of this category. Consequently, the Fréchet space F t , t ∈ T , as a complemented subspace of {ℓ ∞ (J tn ) : n ∈ ω} is injective. Finally, the space X, as a product of injectives, is an injective object of the category LCS.
(1) =⇒ (2). The space X can be identified with a closed linear subspace of the product {B t : t ∈ T } of Banach spaces B t , T ∈ T . Each of the spaces B t can in turn be identified with a closed linear subspace of the space ℓ ∞ (J t ) for some set J t , t ∈ T . Condition (1) implies in this situation that X is a complemented subspace of the product {ℓ ∞ (J t ) : t ∈ T }. The required conclusion now follows from Theorem 4.2.
6. Projective objects of the category LCS 6.1. Complemented subspaces of uncountable sums of Banach spaces. In this section we consider coproducts in the category of complete locally convex topological vector spaces. These are called locally convex direct sums ([9, p. 55] or simply direct sums [8, p. 89]. We are mainly interested in direct sums {B t : t ∈ T } of (uncountable) collections of Banach spaces B t , t ∈ T . Let us recall corresponding definitions. Let {B t : t ∈ T } be an arbitrary collection of (Banach) spaces. For each t ∈ T the space B t is identified (via an obvious isomorphism) with the following subspace of the product {B t : t ∈ T } of (Banach) spaces B t , t ∈ T . As a set the direct sum {B t : t ∈ T } coincides with the subset of the product {B t : t ∈ T } consisting of points with finitely many non-zero coordinates (in other words, The topology on {B t : t ∈ T } is the finest locally convex topology for which each of the natural embeddings B t ֒→ {B t : t ∈ T } is continuous. Categorical description of this construction is also useful. Let B be a locally convex topological vector space containing each of the space B t , t ∈ T , as a subspace. Suppose that for any choice of continuous linear maps f t : B t → C, where C is a locally convex topological vector space, there exists an unique continuous linear map f : B → C such that f |B t = f t for each t ∈ T . Then B is canonically isomorphic to the direct sum {B t : t ∈ T }. A straightforward verification shows that if S ⊆ T , then {B t : t ∈ S} can be canonically identified with the subspace of {B t : t ∈ T } consisting of points defined by letting is continuous and linear.
Proof. Consider the following relation L S on the set (exp ω (T − S)) 2 : Next we perform the spectral search with respect to L S . Existence Condition. We have to show that for each A ∈ exp ω (T − S) there exists C ∈ exp ω (T − S) such that (A, C) ∈ L S .
We begin with the following observation.
The unit ball X = {x ∈ B j : ||x|| j ≤ 1} (here || · || j denotes a norm of the Banach space B j ) being bounded in B j is, by [9, Theorem 6.3], bounded in {B t : t ∈ T }. Continuity of f guarantees that f (X) is also bounded in {B t : t ∈ T }. Applying [9, Theorem 6.3] once again, we conclude that there exists a finite subset C j ⊆ T such that f (X) ⊆ {B t : t ∈ C j }. Finally the Let now A ∈ exp ω (T − S). For each j ∈ A, according to Claim, there exists a finite subset The linearity of f guarantees in this situation that which means that (A, D) ∈ L S . ω-closeness Condition. Let (A i , C) ∈ L S , i ∈ ω, where {A i : i ∈ ω} is a countable chain in exp ω (T − S). We need to show that (A, C) ∈ L S , where A = Consequently, by the linearity of f , By Proposition 2.5, the set of L S -reflexive elements of exp ω (T − S) is cofinal and ω-closed in exp ω (T − S). It only remains to note that an element A ∈ exp ω (T − S) is L S -reflexive if and only if A ∈ K S . Remark 6.1. Proposition 6.1 is valid only for uncountable sums (compare with Remark 3.2). In order to see this consider an uncountable sum {B t : t ∈ T } of Banach spaces and suppose that f : is a topological isomorphism. Proposition 6.1, applied to the map f , guarantees the existence of a cofinal and ω-closed subset K ∅ (f ) of exp ω T satisfying the following property: f {B t : t ∈ A} ⊆ {B t : t ∈ A} for each A ∈ K ∅ (f ). The same Proposition 6.1, applied to the map f −1 (recall that f is a topological isomorphism), guarantees the existence of a cofinal and ω-closed subset K ∅ (f −1 ) of exp ω T satisfying the following property: is still cofinal and ω-closed in exp ω T . Obviously, for each A ∈ K the restriction is a topological isomorphism. Again, for countable sums such a phenomenom is impossible. In order to see this consider the inductive sequences denote the natural embeddings. Clearly the limit spaces lim − → S even and lim − → S odd of these inductive systems are topologically isomorphic, but S even and S odd do not contain isomorphic cofinal subsystems. Proof. Let X be a complemented subspace of an uncountable sum {B t : t ∈ T } of Banach spaces B t , t ∈ T . Clearly there exists a continuous linear map r : {B t : t ∈ T } → X such that r(x) = x for each x ∈ X. A subset S ⊆ T is called r-admissible if r {B t : t ∈ S} ⊆ {B t : t ∈ S}.
For a subset S ⊆ T , let X S = r {B t : t ∈ S} . We state properties of r-admissible subsets needed below. Claim 1. If S ⊆ T is an r-admissible, then X S = X {B t : t ∈ S} .
Indeed, if y ∈ X S , then there exists a point x ∈ {B t : t ∈ S} such that r(x) = y. Since S is r-admissible, it follows that Clearly, y ∈ X. This shows that X S ⊆ X {B t : t ∈ S} .
Conversely. If y ∈ X {B t : t ∈ S} , then y ∈ X and hence, by the property of r, y = r(y). Since y ∈ {B t : t ∈ S}, it follows that y = r(y) ∈ r {B t : t ∈ S} = X S . Claim 2. The union of an arbitrary collection of r-admissible subsets of T is r-admissible.
This is matter of a straightforward verification of the definition of the r-admissibility.
Claim 3. Every countable subset of T is contained in a countable r-admissible subset of T .
This follows from Proposition 6.1 applied to the map r. Claim 4. If S ⊆ T is an r-admissible subset of T , then r S (x) = x for each point x ∈ X S , where r S = r {B t : t ∈ S} : {B t : t ∈ S} → X S . This follows from the corresponding property of the map r. Claim 5. If S and R are r-admissible subset of T and S ⊆ R, then X R is isomorphic to the sum X S ker f R S , where the map f R S : X R → X S is defined by letting 3 f R S (x) = r S π R S (x) for each x ∈ X R . Observe that f R S (x) = x for each x ∈ X S . Claim 6. If S and R are r-admissible subsets of T and S ⊆ R, then there exists a continuous linear map i R S : ker f R S → ker π R S such that r R • i R S = id ker(f R S ) . Let x ∈ ker f R S . Then r S π R S (x) = 0 and consequently, π R S x − π R S (x) = π R S (x) − π R S π R S (x) = π R S (x) − π R S (x) = 0. This shows that by letting i R S (x) = x − π R S (x) for each x ∈ ker f R S , we indeed define a map i R S : ker f R S → ker π R S . Finally observe that In other words, r R S • i R S = id ker(f R S ) as required.
Since, by construction, the set T 0 is countable, Claim 4 guarantees that X 0 is an (LB)-space. Similarly, since the set T α+1 − T α = A α+1 is countable, Claim 6 guarantees that the space ker f T α+1 Tα is an (LB)-space for each α < τ . This completes the proof of Theorem 6.2.