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FROM FINITELY GENERATED PROJECTIVE MODULES TO A GENERALIZATION OF SERRE-SWAN-MALLIOS THEOREM Mart Abel University of Tartu

• J. P. Serre, Modules projectifs et espaces fibr´

es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23, pp. 531–543: .

• J. P. Serre, Modules projectifs et espaces fibr´

es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23, pp. 531–543: There is an one-to-one correspondence between the finite dimensional fibers of a vector bundle over a connected affine algebraic manifold V and (some special) projective A-modules, where A is a coordinate ring

• f V over an algebraically closed field.

.

• J. P. Serre, Modules projectifs et espaces fibr´

es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23, pp. 531–543: There is an one-to-one correspondence between the finite dimensional fibers of a vector bundle over a connected affine algebraic manifold V and (some special) projective A-modules, where A is a coordinate ring

• f V over an algebraically closed field.
• R. G. Swan, Vector Bundles and Projective Modules, Trans. Amer.
• Math. Soc. 105 (1962), pp. 264–277:

.

• J. P. Serre, Modules projectifs et espaces fibr´

es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23, pp. 531–543: There is an one-to-one correspondence between the finite dimensional fibers of a vector bundle over a connected affine algebraic manifold V and (some special) projective A-modules, where A is a coordinate ring

• f V over an algebraically closed field.
• R. G. Swan, Vector Bundles and Projective Modules, Trans. Amer.
• Math. Soc. 105 (1962), pp. 264–277:

There is an equivalence between the category of K-valued vector bundles over compact Hausdorff space and the category of finitely generated projective C(X, K)-modules (where K is either R, C or H). .

• J. P. Serre, Modules projectifs et espaces fibr´

es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23, pp. 531–543: There is an one-to-one correspondence between the finite dimensional fibers of a vector bundle over a connected affine algebraic manifold V and (some special) projective A-modules, where A is a coordinate ring

• f V over an algebraically closed field.
• R. G. Swan, Vector Bundles and Projective Modules, Trans. Amer.
• Math. Soc. 105 (1962), pp. 264–277:

There is an equivalence between the category of K-valued vector bundles over compact Hausdorff space and the category of finitely generated projective C(X, K)-modules (where K is either R, C or H). Generally speaking, Serre-Swan (-Type) Theorem gives us an equivalence between the category of A-vector bundles over X and the category of finitely generated projective C(X, A)-modules with different conditions on A and X.

Different versions of Serre-Swan Theorem:

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C.

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C. Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C.

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C. Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C. Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C.

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C. Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C. Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C. Karoubi (1978): X is a compact topological space and A is R or C.

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C. Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C. Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C. Karoubi (1978): X is a compact topological space and A is R or C. Goodearl (1984): X is a paracompact topological space and A is R, C

• r H.

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C. Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C. Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C. Karoubi (1978): X is a compact topological space and A is R or C. Goodearl (1984): X is a paracompact topological space and A is R, C

• r H.

Mallios (1983): X is a compact Hausdorff space and A is a unital locally m-convex Q-algebra.

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C. Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C. Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C. Karoubi (1978): X is a compact topological space and A is R or C. Goodearl (1984): X is a paracompact topological space and A is R, C

• r H.

Mallios (1983): X is a compact Hausdorff space and A is a unital locally m-convex Q-algebra. Vaserstein (1986): X is an arbitrary topological space and A is R, C or H.

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C. Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C. Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C. Karoubi (1978): X is a compact topological space and A is R or C. Goodearl (1984): X is a paracompact topological space and A is R, C

• r H.

Mallios (1983): X is a compact Hausdorff space and A is a unital locally m-convex Q-algebra. Vaserstein (1986): X is an arbitrary topological space and A is R, C or H. Kawamura (2003): considers Hilbert C ∗-modules over unital non-commutative C ∗-algebras and considers module bundles instead of vector bundles.

My goal: to get as general result as possible for topological algebras A.

My goal: to get as general result as possible for topological algebras A. Starting point: Serre-Swan-Mallios Theorem in paper

• A. Mallios, Vector Bundles and K-Theory over Topological Algebras, J.
• Math. Anal. Appl. 92, No. 2, 1983, pp. 452–506.

My goal: to get as general result as possible for topological algebras A. Starting point: Serre-Swan-Mallios Theorem in paper

• A. Mallios, Vector Bundles and K-Theory over Topological Algebras, J.
• Math. Anal. Appl. 92, No. 2, 1983, pp. 452–506.

This talk is based on 3 papers:

My goal: to get as general result as possible for topological algebras A. Starting point: Serre-Swan-Mallios Theorem in paper

• A. Mallios, Vector Bundles and K-Theory over Topological Algebras, J.
• Math. Anal. Appl. 92, No. 2, 1983, pp. 452–506.

This talk is based on 3 papers:

• M. Abel, A. Mallios, On finitely generated projective modules, Rend.
• Circ. Mat. Palermo, 54 (2005), serie II, pp. 145–166.

My goal: to get as general result as possible for topological algebras A. Starting point: Serre-Swan-Mallios Theorem in paper

• A. Mallios, Vector Bundles and K-Theory over Topological Algebras, J.
• Math. Anal. Appl. 92, No. 2, 1983, pp. 452–506.

This talk is based on 3 papers:

• M. Abel, A. Mallios, On finitely generated projective modules, Rend.
• Circ. Mat. Palermo, 54 (2005), serie II, pp. 145–166.
• M. Abel, On a category of module bundles, submitted to Rend. Circ.
• Mat. Palermo.

My goal: to get as general result as possible for topological algebras A. Starting point: Serre-Swan-Mallios Theorem in paper

• A. Mallios, Vector Bundles and K-Theory over Topological Algebras, J.
• Math. Anal. Appl. 92, No. 2, 1983, pp. 452–506.

This talk is based on 3 papers:

• M. Abel, A. Mallios, On finitely generated projective modules, Rend.
• Circ. Mat. Palermo, 54 (2005), serie II, pp. 145–166.
• M. Abel, On a category of module bundles, submitted to Rend. Circ.
• Mat. Palermo.
• M. Abel, A generalization of the Serre-Swan-Mallios Theorem,

submitted to Rend. Circ. Mat. Palermo.

A is an algebra with zero element θA.

A is an algebra with zero element θA. K is one of the fields R and C.

A is an algebra with zero element θA. K is one of the fields R and C. A left A-module M is a linear space over K with bilinear map (a, m) → am from A × M into M, satisfying (ab)m = a(bm) for all a, b ∈ A and all m ∈ M.

A is an algebra with zero element θA. K is one of the fields R and C. A left A-module M is a linear space over K with bilinear map (a, m) → am from A × M into M, satisfying (ab)m = a(bm) for all a, b ∈ A and all m ∈ M. An A-module M is finitely generated, if there exist k ∈ N and elements e1, . . . , ek ∈ M such that for every m ∈ M there are elements a1, . . . , ak ∈ A and numbers λ1, . . . , λk ∈ K, for which m =

k

• i=1

aiei +

k

• i=1

λiei.

A is an algebra with zero element θA. K is one of the fields R and C. A left A-module M is a linear space over K with bilinear map (a, m) → am from A × M into M, satisfying (ab)m = a(bm) for all a, b ∈ A and all m ∈ M. An A-module M is finitely generated, if there exist k ∈ N and elements e1, . . . , ek ∈ M such that for every m ∈ M there are elements a1, . . . , ak ∈ A and numbers λ1, . . . , λk ∈ K, for which m =

k

• i=1

aiei +

k

• i=1

λiei. An A-module P is projective, if for every A-linear map f : P → N and for every surjective A-linear map g : M → N, there exists an A-linear map h : P → M such that f = g ◦ h.

An element a ∈ A is quasi-invertible in A, if there exists an element b ∈ A such that a ◦ b = b ◦ a = θA, where a ◦ b = a + b − ab for each a, b ∈ A and θA is the zero element of A.

An element a ∈ A is quasi-invertible in A, if there exists an element b ∈ A such that a ◦ b = b ◦ a = θA, where a ◦ b = a + b − ab for each a, b ∈ A and θA is the zero element of A. Topological algebra is a topological vector space in which the multiplication is associative and jointly continuous.

An element a ∈ A is quasi-invertible in A, if there exists an element b ∈ A such that a ◦ b = b ◦ a = θA, where a ◦ b = a + b − ab for each a, b ∈ A and θA is the zero element of A. Topological algebra is a topological vector space in which the multiplication is associative and jointly continuous. A topological algebra A is:

An element a ∈ A is quasi-invertible in A, if there exists an element b ∈ A such that a ◦ b = b ◦ a = θA, where a ◦ b = a + b − ab for each a, b ∈ A and θA is the zero element of A. Topological algebra is a topological vector space in which the multiplication is associative and jointly continuous. A topological algebra A is: 1) a Q-algebra, if the set q-invA of quasi-invertible elements of A is

• pen;

An element a ∈ A is quasi-invertible in A, if there exists an element b ∈ A such that a ◦ b = b ◦ a = θA, where a ◦ b = a + b − ab for each a, b ∈ A and θA is the zero element of A. Topological algebra is a topological vector space in which the multiplication is associative and jointly continuous. A topological algebra A is: 1) a Q-algebra, if the set q-invA of quasi-invertible elements of A is

• pen;

2) a Waelbroeck algebra, if A is a Q-algebra, in which the quasi-inversion (that is, the map a → a−1

q ) is continuous;

An element a ∈ A is quasi-invertible in A, if there exists an element b ∈ A such that a ◦ b = b ◦ a = θA, where a ◦ b = a + b − ab for each a, b ∈ A and θA is the zero element of A. Topological algebra is a topological vector space in which the multiplication is associative and jointly continuous. A topological algebra A is: 1) a Q-algebra, if the set q-invA of quasi-invertible elements of A is

• pen;

2) a Waelbroeck algebra, if A is a Q-algebra, in which the quasi-inversion (that is, the map a → a−1

q ) is continuous;

3) a locally m-convex algebra, if A has a base of neighbourhoods of zero, consisting of convex (i.e. O + O ⊂ 2O) and idempotent (i.e. OO ⊂ O) neighbourhoods.

An element a ∈ A is quasi-invertible in A, if there exists an element b ∈ A such that a ◦ b = b ◦ a = θA, where a ◦ b = a + b − ab for each a, b ∈ A and θA is the zero element of A. Topological algebra is a topological vector space in which the multiplication is associative and jointly continuous. A topological algebra A is: 1) a Q-algebra, if the set q-invA of quasi-invertible elements of A is

• pen;

2) a Waelbroeck algebra, if A is a Q-algebra, in which the quasi-inversion (that is, the map a → a−1

q ) is continuous;

3) a locally m-convex algebra, if A has a base of neighbourhoods of zero, consisting of convex (i.e. O + O ⊂ 2O) and idempotent (i.e. OO ⊂ O) neighbourhoods. It is known result that the quasi-inversion is continuous in every locally m-convex algebra, thus every locally m-convex Q-algebra is also a Waelbroeck algebra.

An A-module M is a topological A-module, if there has been given a topology on M such that M is a topological vector space over K, in which the multiplication over A (as a bilinear map A × M → M), is separately continuous.

An A-module M is a topological A-module, if there has been given a topology on M such that M is a topological vector space over K, in which the multiplication over A (as a bilinear map A × M → M), is separately continuous. The set of all finitely generated projective (topological) A-modules is denoted by P(A).

An A-module M is a topological A-module, if there has been given a topology on M such that M is a topological vector space over K, in which the multiplication over A (as a bilinear map A × M → M), is separately continuous. The set of all finitely generated projective (topological) A-modules is denoted by P(A). Mm,n(A) is the set of all A-valued m × n-matrices, m, n ∈ N.

An A-module M is a topological A-module, if there has been given a topology on M such that M is a topological vector space over K, in which the multiplication over A (as a bilinear map A × M → M), is separately continuous. The set of all finitely generated projective (topological) A-modules is denoted by P(A). Mm,n(A) is the set of all A-valued m × n-matrices, m, n ∈ N. Mn(A) := Mn,n(A).

An A-module M is a topological A-module, if there has been given a topology on M such that M is a topological vector space over K, in which the multiplication over A (as a bilinear map A × M → M), is separately continuous. The set of all finitely generated projective (topological) A-modules is denoted by P(A). Mm,n(A) is the set of all A-valued m × n-matrices, m, n ∈ N. Mn(A) := Mn,n(A). A× K is the unitization of A.

Theorem 1. A topological algebra A is a Waelbroeck algebra if and only if Mn(A) is a Waelbroeck algebra for every n ∈ N.

Theorem 1. A topological algebra A is a Waelbroeck algebra if and only if Mn(A) is a Waelbroeck algebra for every n ∈ N. Proposition 1. A topological algebra A is locally m-convex if and only if Mn(A) is locally m-convex for every n ∈ N.

Theorem 1. A topological algebra A is a Waelbroeck algebra if and only if Mn(A) is a Waelbroeck algebra for every n ∈ N. Proposition 1. A topological algebra A is locally m-convex if and only if Mn(A) is locally m-convex for every n ∈ N. An algebra A belongs to the class W (i.e., is finitely generated), if there are k ∈ N and e1, . . . , ek ∈ A such that: for every a ∈ A there exists a set of elements a1, . . . , ak ∈ A and a set

• f numbers λ1, . . . , λk ∈ K, such that

a =

k

• i=1

aiei +

k

• i=1

λiei.

Theorem 1. A topological algebra A is a Waelbroeck algebra if and only if Mn(A) is a Waelbroeck algebra for every n ∈ N. Proposition 1. A topological algebra A is locally m-convex if and only if Mn(A) is locally m-convex for every n ∈ N. An algebra A belongs to the class W (i.e., is finitely generated), if there are k ∈ N and e1, . . . , ek ∈ A such that: for every a ∈ A there exists a set of elements a1, . . . , ak ∈ A and a set

• f numbers λ1, . . . , λk ∈ K, such that

a =

k

• i=1

aiei +

k

• i=1

λiei. Proposition 2. Let A be an algebra from W and n, m ∈ N. Then there exists a bijection b : HomA(An, Am) → Mkn,m(A).

A topological algebra A belongs to the class S, if there are k ∈ N and e1, . . . , ek ∈ A such that: 1) A belongs to W; 2) for every neighbourhood U of zero in A there exists a neighbourhood V of zero in A such that V ⊂ k

• i=1

biei : bi ∈ U, for every i ∈ Nk

• .

A topological algebra A belongs to the class S, if there are k ∈ N and e1, . . . , ek ∈ A such that: 1) A belongs to W; 2) for every neighbourhood U of zero in A there exists a neighbourhood V of zero in A such that V ⊂ k

• i=1

biei : bi ∈ U, for every i ∈ Nk

• .

Theorem 2. Let A be a topological algebra from the class S and let n, m ∈ N. Then every A-linear map u : An → Am is continuous. In particular, every A-module direct summand of An is a closed subspace of An in the product topology.

Proposition 3. Let E be a finitely generated A-module, where A is a given topological algebra. Then E can be made into a topological A-module in a unique way, independently of how we consider it as finitely generated. In particular, in case E is also a projective (A × K)-module this is the relative topology on it, when we consider E as a direct summand of some (A × K)l.

Proposition 3. Let E be a finitely generated A-module, where A is a given topological algebra. Then E can be made into a topological A-module in a unique way, independently of how we consider it as finitely generated. In particular, in case E is also a projective (A × K)-module this is the relative topology on it, when we consider E as a direct summand of some (A × K)l. Corollary 1. Let A be a topological algebra and E, F finitely generated projective A-modules. Then every A-linear map u : E → F is continuous.

Proposition 3. Let E be a finitely generated A-module, where A is a given topological algebra. Then E can be made into a topological A-module in a unique way, independently of how we consider it as finitely generated. In particular, in case E is also a projective (A × K)-module this is the relative topology on it, when we consider E as a direct summand of some (A × K)l. Corollary 1. Let A be a topological algebra and E, F finitely generated projective A-modules. Then every A-linear map u : E → F is continuous. Theorem 3. Let A be a Waelbroeck algebra and M a finitely generated projective A-module endowed with the respective relative topology. Then the algebra HomA(M, M) ≡ EndA(M) of all continuous A-linear endomorphisms of M is a Waelbroeck algebra.

Let A be a topological algebra, E a set, X a topological space and π : E → X an onto map.

Let A be a topological algebra, E a set, X a topological space and π : E → X an onto map. For every A-module M, every x ∈ X and every bijection b : π−1(x) → {x} × M which sends e ∈ π−1(x) to (x, me), we define the operations as usual:

Let A be a topological algebra, E a set, X a topological space and π : E → X an onto map. For every A-module M, every x ∈ X and every bijection b : π−1(x) → {x} × M which sends e ∈ π−1(x) to (x, me), we define the operations as usual: e + f := b−1(x, me + mf ) = b−1[(x, me) + (x, mf )],

Let A be a topological algebra, E a set, X a topological space and π : E → X an onto map. For every A-module M, every x ∈ X and every bijection b : π−1(x) → {x} × M which sends e ∈ π−1(x) to (x, me), we define the operations as usual: e + f := b−1(x, me + mf ) = b−1[(x, me) + (x, mf )], λe := b−1(x, λme) = b−1[λ(x, me)],

Let A be a topological algebra, E a set, X a topological space and π : E → X an onto map. For every A-module M, every x ∈ X and every bijection b : π−1(x) → {x} × M which sends e ∈ π−1(x) to (x, me), we define the operations as usual: e + f := b−1(x, me + mf ) = b−1[(x, me) + (x, mf )], λe := b−1(x, λme) = b−1[λ(x, me)], ae := b−1(x, ame) = b−1[a(x, me)] for every e, f ∈ π−1(x), λ ∈ K and a ∈ A.

Let A be a topological algebra, E a set, X a topological space and π : E → X an onto map. For every A-module M, every x ∈ X and every bijection b : π−1(x) → {x} × M which sends e ∈ π−1(x) to (x, me), we define the operations as usual: e + f := b−1(x, me + mf ) = b−1[(x, me) + (x, mf )], λe := b−1(x, λme) = b−1[λ(x, me)], ae := b−1(x, ame) = b−1[a(x, me)] for every e, f ∈ π−1(x), λ ∈ K and a ∈ A. The triplet t = (U, φ, M) is a chart of the triplet (E, π, X), if U is an

• pen subset of X, M is a finitely generated projective A-module and

φ : π−1(U) → U × M is a bijection which satisfies the conditions:

Let A be a topological algebra, E a set, X a topological space and π : E → X an onto map. For every A-module M, every x ∈ X and every bijection b : π−1(x) → {x} × M which sends e ∈ π−1(x) to (x, me), we define the operations as usual: e + f := b−1(x, me + mf ) = b−1[(x, me) + (x, mf )], λe := b−1(x, λme) = b−1[λ(x, me)], ae := b−1(x, ame) = b−1[a(x, me)] for every e, f ∈ π−1(x), λ ∈ K and a ∈ A. The triplet t = (U, φ, M) is a chart of the triplet (E, π, X), if U is an

• pen subset of X, M is a finitely generated projective A-module and

φ : π−1(U) → U × M is a bijection which satisfies the conditions: 1) π(φ−1(x, h)) = x for every x ∈ U and every h ∈ M;

Let A be a topological algebra, E a set, X a topological space and π : E → X an onto map. For every A-module M, every x ∈ X and every bijection b : π−1(x) → {x} × M which sends e ∈ π−1(x) to (x, me), we define the operations as usual: e + f := b−1(x, me + mf ) = b−1[(x, me) + (x, mf )], λe := b−1(x, λme) = b−1[λ(x, me)], ae := b−1(x, ame) = b−1[a(x, me)] for every e, f ∈ π−1(x), λ ∈ K and a ∈ A. The triplet t = (U, φ, M) is a chart of the triplet (E, π, X), if U is an

• pen subset of X, M is a finitely generated projective A-module and

φ : π−1(U) → U × M is a bijection which satisfies the conditions: 1) π(φ−1(x, h)) = x for every x ∈ U and every h ∈ M; 2) φ|π−1(x) : π−1(x) → U × M is A-linear for every x ∈ U.

Two charts t = (U, φ, M) and s = (V , ψ, N) of (E, π, X) are compatible, if for every x ∈ U ∩ V the respective maps φx := prM ◦ φ|π−1(x), ψx := prN ◦ ψ|π−1(x) (here prM and prN are the usual projections to M and N, respectively) satisfy the following conditions:

Two charts t = (U, φ, M) and s = (V , ψ, N) of (E, π, X) are compatible, if for every x ∈ U ∩ V the respective maps φx := prM ◦ φ|π−1(x), ψx := prN ◦ ψ|π−1(x) (here prM and prN are the usual projections to M and N, respectively) satisfy the following conditions: 1) ψx ◦ φ−1

x

∈ HomA(M, N);

Two charts t = (U, φ, M) and s = (V , ψ, N) of (E, π, X) are compatible, if for every x ∈ U ∩ V the respective maps φx := prM ◦ φ|π−1(x), ψx := prN ◦ ψ|π−1(x) (here prM and prN are the usual projections to M and N, respectively) satisfy the following conditions: 1) ψx ◦ φ−1

x

∈ HomA(M, N); 2) the map f : U ∩ V → HomA(M, N), defined by f (x) = ψx ◦ φ−1

x , is

continuous.

Two charts t = (U, φ, M) and s = (V , ψ, N) of (E, π, X) are compatible, if for every x ∈ U ∩ V the respective maps φx := prM ◦ φ|π−1(x), ψx := prN ◦ ψ|π−1(x) (here prM and prN are the usual projections to M and N, respectively) satisfy the following conditions: 1) ψx ◦ φ−1

x

∈ HomA(M, N); 2) the map f : U ∩ V → HomA(M, N), defined by f (x) = ψx ◦ φ−1

x , is

continuous. A family of charts A = {(Uα, φα, Mα)}α∈I

• f (E, π, X) is an atlas of (E, π, X), if all the charts of A are mutually

compatible so that X =

• α ∈ I

Uα and Mα ∩ Mβ = ∅, if α = β.

Two atlases A and B of (E, π, X) are equivalent, if A ∪ B = {(U, φ, M) : (U, φ, M) ∈ A or (U, φ, M) ∈ B} is still an atlas of (E, π, X). This notion gives us an equivalence relation.

Two atlases A and B of (E, π, X) are equivalent, if A ∪ B = {(U, φ, M) : (U, φ, M) ∈ A or (U, φ, M) ∈ B} is still an atlas of (E, π, X). This notion gives us an equivalence relation. Given a topological algebra A and a map π : E → X as above, we say that the triple (E, π, X) is an A-bundle (or an A-module bundle) over X, in case we are given an equivalence class of atlases of (E, π, X).

Proposition 4. Let A be a topological algebra, (E, π, X) an A-bundle and A = {Uα, φα, Mα}α∈ I an atlas of (E, π, X). Then, there exists a unique topology on E making π continuous and the φα’s, α ∈ I, homeomorphisms, in such a way that each ”fiber” of E, i.e. the set Ex := π−1(x), x ∈ X, is uniquely endowed with the structure of a (finitely generated projective) topological A-module, the topology of which coincides with the subset topology of E.

Proposition 4. Let A be a topological algebra, (E, π, X) an A-bundle and A = {Uα, φα, Mα}α∈ I an atlas of (E, π, X). Then, there exists a unique topology on E making π continuous and the φα’s, α ∈ I, homeomorphisms, in such a way that each ”fiber” of E, i.e. the set Ex := π−1(x), x ∈ X, is uniquely endowed with the structure of a (finitely generated projective) topological A-module, the topology of which coincides with the subset topology of E. EA(X) will be the category of A-module bundles:

Proposition 4. Let A be a topological algebra, (E, π, X) an A-bundle and A = {Uα, φα, Mα}α∈ I an atlas of (E, π, X). Then, there exists a unique topology on E making π continuous and the φα’s, α ∈ I, homeomorphisms, in such a way that each ”fiber” of E, i.e. the set Ex := π−1(x), x ∈ X, is uniquely endowed with the structure of a (finitely generated projective) topological A-module, the topology of which coincides with the subset topology of E. EA(X) will be the category of A-module bundles:

• bjects of EA(X) are all A-bundles (E, πE, X) over (fixed topological

space) X, their base sets E being topologized by the Proposition 4;

Proposition 4. Let A be a topological algebra, (E, π, X) an A-bundle and A = {Uα, φα, Mα}α∈ I an atlas of (E, π, X). Then, there exists a unique topology on E making π continuous and the φα’s, α ∈ I, homeomorphisms, in such a way that each ”fiber” of E, i.e. the set Ex := π−1(x), x ∈ X, is uniquely endowed with the structure of a (finitely generated projective) topological A-module, the topology of which coincides with the subset topology of E. EA(X) will be the category of A-module bundles:

• bjects of EA(X) are all A-bundles (E, πE, X) over (fixed topological

space) X, their base sets E being topologized by the Proposition 4; morphisms of EA(X) are determined by continuous maps α : E → F with πF ◦ α = πE for any pair of A-bundles (E, πE, X) and (F, πF, X).

Proposition 4. Let A be a topological algebra, (E, π, X) an A-bundle and A = {Uα, φα, Mα}α∈ I an atlas of (E, π, X). Then, there exists a unique topology on E making π continuous and the φα’s, α ∈ I, homeomorphisms, in such a way that each ”fiber” of E, i.e. the set Ex := π−1(x), x ∈ X, is uniquely endowed with the structure of a (finitely generated projective) topological A-module, the topology of which coincides with the subset topology of E. EA(X) will be the category of A-module bundles:

• bjects of EA(X) are all A-bundles (E, πE, X) over (fixed topological

space) X, their base sets E being topologized by the Proposition 4; morphisms of EA(X) are determined by continuous maps α : E → F with πF ◦ α = πE for any pair of A-bundles (E, πE, X) and (F, πF, X). We assume, that αx := α |π−1

E (x): π−1

E (x) = Ex → Fx = π−1 F (x)

is an A-linear map for every x ∈ X.

A category A is additive, if A has a zero object, any two objects of A have a product in A, and all the morphism sets A(A, B) := {α : A → B : α is a morphism in A} are Abelian groups such that the composition A(A, B) × A(B, C) → A(A, C)

• f morphisms is bilinear.

A category A is additive, if A has a zero object, any two objects of A have a product in A, and all the morphism sets A(A, B) := {α : A → B : α is a morphism in A} are Abelian groups such that the composition A(A, B) × A(B, C) → A(A, C)

• f morphisms is bilinear.

A kernel of a morphism α ∈ A(A, B) is a pair (ν, K), where ν ∈ A(K, A) is a morphism such that

A category A is additive, if A has a zero object, any two objects of A have a product in A, and all the morphism sets A(A, B) := {α : A → B : α is a morphism in A} are Abelian groups such that the composition A(A, B) × A(B, C) → A(A, C)

• f morphisms is bilinear.

A kernel of a morphism α ∈ A(A, B) is a pair (ν, K), where ν ∈ A(K, A) is a morphism such that 1) α ◦ ν = θKB, where ν : K → A is a morphism and θKB is a zero element of A(K, B);

A category A is additive, if A has a zero object, any two objects of A have a product in A, and all the morphism sets A(A, B) := {α : A → B : α is a morphism in A} are Abelian groups such that the composition A(A, B) × A(B, C) → A(A, C)

• f morphisms is bilinear.

A kernel of a morphism α ∈ A(A, B) is a pair (ν, K), where ν ∈ A(K, A) is a morphism such that 1) α ◦ ν = θKB, where ν : K → A is a morphism and θKB is a zero element of A(K, B); 2) every morphism m ∈ A(K ′, A), for which α ◦ m = θK ′B, is uniquely represented in a form m = ν ◦ n, where n ∈ A(K ′, K).

A category A is pseudo-Abelian, if for every object O of A and for any idempotent endomorphism α : O → O (i.e. such α that α2 = α), the kernel of α exists (and is an object of A).

A category A is pseudo-Abelian, if for every object O of A and for any idempotent endomorphism α : O → O (i.e. such α that α2 = α), the kernel of α exists (and is an object of A). Theorem 4. Let A be a Waelbroeck algebra. Then the set EA(X) is an additive pseudo-Abelian category.

Lemma 1. Let A be a topological algebra, X a topological space, (E, πE, X), (F, πF, X) any two objects of the category EA(X) with {(Ui, φi, Mi)}i∈I an atlas of (E, πE, X) and α : E → F a morphism such that the partial A-linear maps αx : Ex → Fx are bijective for every x ∈ X. If the inversion is continuous on EndA(Mi) for every i ∈ I, then α is an isomorphism in the category EA(X).

Lemma 1. Let A be a topological algebra, X a topological space, (E, πE, X), (F, πF, X) any two objects of the category EA(X) with {(Ui, φi, Mi)}i∈I an atlas of (E, πE, X) and α : E → F a morphism such that the partial A-linear maps αx : Ex → Fx are bijective for every x ∈ X. If the inversion is continuous on EndA(Mi) for every i ∈ I, then α is an isomorphism in the category EA(X). Corollary 2. Let A be a locally m-convex or a Waelbroeck algebra, X a topological space, (E, πE, X), (F, πF, X) any two objects of the category EA(X) and α : E → F a morphism. If the partial A-linear maps αx : Ex → Fx are bijective for every x ∈ X, then α is an isomorphism in the category EA(X).

Lemma 2. Let A be a topological algebra, X a topological space, (E, πE, X), (F, πF, X) any two objects of the category EA(X) with {(Vj, ψj, Nj)}j∈ J an atlas of (F, πF, X) and α : E → F a morphism such that the partial A-linear maps αx : Ex → Fx are surjective for every x ∈ X. If the inversion is continuous on EndA(Nj) for every j ∈ J, then there exists a morphism β : F → E such that α ◦ β = idF.

Lemma 2. Let A be a topological algebra, X a topological space, (E, πE, X), (F, πF, X) any two objects of the category EA(X) with {(Vj, ψj, Nj)}j∈ J an atlas of (F, πF, X) and α : E → F a morphism such that the partial A-linear maps αx : Ex → Fx are surjective for every x ∈ X. If the inversion is continuous on EndA(Nj) for every j ∈ J, then there exists a morphism β : F → E such that α ◦ β = idF. Corollary 3. Let A be a locally m-convex or a Waelbroeck algebra, X a topological space, (E, πE, X), (F, πF, X) any two objects of the category EA(X) and α : E → F a morphism. If the partial A-linear maps αx : Ex → Fx are surjective for every x ∈ X, then there exists a morphism β : F → E such that α ◦ β = idF.

An A-bundle (E, π, X) is suitable, if there exists an atlas {(Uα, φα, Mα)}α∈I of (E, π, X) such that sup

α∈I

nα < ∞, where nα denotes the number of generators of Mα for every α ∈ I.

An A-bundle (E, π, X) is suitable, if there exists an atlas {(Uα, φα, Mα)}α∈I of (E, π, X) such that sup

α∈I

nα < ∞, where nα denotes the number of generators of Mα for every α ∈ I. Lemma 3. Let X be a topological space, A a Waelbroeck algebra and (E, πE, X) a suitable A-bundle. Then there exists a suitable A-bundle (F, πF, X) and a number n ∈ N such that E ⊕ F = X × (A × K)n, within an isomorphism in EA(X).

Let ES

A(X) be a subcategory of EA(X), the objects of which are all

suitable A-bundles and the morphisms are the respective morphisms between suitable A-bundles from the set of morphisms in EA(X).

Let ES

A(X) be a subcategory of EA(X), the objects of which are all

suitable A-bundles and the morphisms are the respective morphisms between suitable A-bundles from the set of morphisms in EA(X). Theorem 5 (The generalization of the Serre-Swan-Mallios Theorem). Let X be a topological space and A a Waelbroeck algebra. Then we have ES

A(X) = P(C(X, A × K))

within an equivalence of categories.