EFFECTIVE DESCENT MORPHISMS FOR BANACH MODULES 7 BACHUKIMESABLISHVILI 1 0 Abstract. It isproved that anorm-decreasinghomomorphismof commuta- 2 tive Banach algebras is an effective descent morphism for Banach modules if n andonlyifitisaweakretract. a J 9 1 1. INTRODUCTION ] The present note is a continuation of the previous works on the problem of de- A scribing effective descent morphisms in various monoidal categories [10], [11], [12], F [13], [14] and aims to study the descent problem for the symmetric monoidal cate- . h gory of Banach spaces (with linear contractions as morphisms, and the projective t a tensor product). Recall that Grothendieck’s descent theory for modules in a sym- m metricmonoidalcategoryV =(V,⊗,I)isthestudyofwhichmorphismsι:A→B [ ofV-monoidsareeffectivedescentmorphismsformodulesinthesensethatthecor- responding extension-of-scalars functor B ⊗A − : AV → BV from the category of 1 (left) A-modules to the category of (left) B-modules is comonadic. In this note v we prove that effective descent morphisms for Banach modules are precisely those 0 3 norm-decreasing homomorphisms of commutative Banach algebra which are weak 3 retracts. 5 As background to the subject, we refer to S. MacLane [9] for generalities on 0 category theory, to [3] and [15] for terminology and general results on Banach . 1 spaces and to G. Janelidze and W. Tholen [5], [6] and [7] for descent theory. 0 7 2. PRELIMINARIES 1 : SupposethatV isafixedsymmetricmonoidalclosedcategorywithtensorprod- v uct ⊗, unit object I, and internal-hom [−,−]. Recall ([9]) that a monoid A in i X V (or V-monoid) consists of an object A of V endowed with a multiplication r m : A⊗A → A and unit morphism e : I → A such that the usual identity a A A and associative conditions are satisfied. A monoid is called commutative if the multiplication map is unchanged when composed with the symmetry. Recall further that, for any V-monoid A = (A,eA,mA), a left A-module is a pair (V,ρ ), where V is an object of V and ρ :A⊗V →V is a morphism in V, V V called the action (or the A-action) on V, such that ρV(mA ⊗V) = ρV(A⊗ρV) andρV(eA⊗V)=1. For a givenV-monoidA, the left A-modules are the objects of a category AV. A morphism f : (V,ρV) → (W,ρW) is a morphism f : V → W in V such that ρW(A⊗f)= fρV. Analogously, one has the category VA of right A-modules. 2010 Mathematics Subject Classification. 46H25,46M15,18D10. Key words and phrases. Banachmodules,weakretracts,effective descentmorphisms. TheworkwaspartiallysupportedbytheShotaRustaveliNationalScienceFoundationGrants DI/18/5-113/13 andFR/189/5-113/14. 1 2 BACHUKIMESABLISHVILI If V admits coequalizers, then each morphism ι : A → B of V-monoids gives rise to two functors: • the restriction–of–scalars functor ι! : BV → AV, where for any (left) B- module (V,̺V), ι!(V,̺V) is a (left) A-module via the action A⊗V −ι−⊗b−V→ B⊗V −̺−→V V; b • the extension–of–scalars functor B⊗A−: AV →BV, where for any (left) A-module (W,ρW), B⊗AW is a (left) B-module via the action B⊗B⊗AW −m−B−−⊗−A−W→B⊗AW. Itiswell-knownthattherestriction–of–scalarsfunctorisrightadjointtotheextension– of–scalarsfunctor. ι:A→ Biscalledaneffectivedescentmorphism (for modules) if the extension–of–scalarsfunctor B⊗A−:AV →BV is comonadic. We henceforth suppose that V is a symmetric monoidal closed category with equalizers and coequalizers. An inspection of the proof of [13, Theorem 3.7] shows that the theorem holds true also for morphisms ι : A → B of V-monoids which are central in the sense that the diagram A⊗B τA,B // B⊗A B⊗ι //B⊗B ι⊗B mB (cid:15)(cid:15) (cid:15)(cid:15) B⊗B // B, mB whereτ is the symmetryofthe monoidalcategory,commutes(see, [12]). Hence we can improve [13, Theorem 3.7] slightly as follows. (Recall that a regular injective object in a category is an object which has the extension property with respect to regular monomorphisms.) 2.1. THEOREM. Let V have a regular injective object Q such that the functor [−,Q]: V →Vop is comonadic, and let ι: A → B be a central morphism of monoids in V. The following are equivalent: (i) ι: A→B is an effective descent morphism; (ii) ι: A → B is a pure morphism in AV; that is, for any A-module V, the morphism ι⊗AV : V =A⊗AV →B⊗AV is a regular monomorphism; (iii) the morphism [ι,Q]:[B,Q]→[A,Q] is a split epimorphism in AV; Note that, if ι satisfies any (and hence all) of the above equivalent conditions, then it is a monomorphismandthe centrality then implies that A is commutative. 3. THE MAIN RESULT Let K denote either the field of real numbers R or the field of complex numbers C. Write Ban1 for the categorywhose objects areBanachspaces overK and whose morphisms are linear contractions. It is well-known (e.g, see [4], [16]) that Ban1 is a symmetric monoidal category with tensor product of two Banach spaces being theirprojective tensor product ⊗(see[3])andtheunitforthistensorproductbeing b EFFECTIVE DESCENT MORPHISMS FOR BANACH MODULES 3 K. Moreover, there is a bifunctor [−,−] : (Ban1)op ×Ban1 → Ban1 (the internal Hom) making the category Ban1 into a symmetric closed monoidal category. For two Banach spaces V and W, [V,W] is the Banach space whose elements are the bounded linear transformations V →W quipped with the operator norm. In Ban1 all small limits and all small colimits exist (e.g. [1]). Recall (for example, from [4], [16]) that (commutative) unital Banach algebras are exactly (commutative) monoids in the symmetric monoidal category Ban1, and that, for any unital Banach algebra A, an object of ABan1 is a (left) Ban1-module over A, that is, a Banach space V together with Ban1-morphism A⊗V →V, a⊗v →av b b such that a(bv) = (ab)v and eAv = v (a,b ∈ A, v ∈ V). Since the action is a morphism in Ban1, the map (a,v) → av is bilinear and satisfies the condition kavkV ≤ kakA ·kvkV. The morphisms in ABan1 are morphisms in Ban1 which are A-linear. If A is a unital Banachalgebraand V is a A-module, then the dual space V∗ = [V,K] of V has the structure of a BanachA-module, where the action A⊗[V,K]→ b [V,K] is given by a⊗f 7−→(v →f(av)). b Moreover, for any morphism f : V → W in ABan1, the map f∗ : W∗ → V∗ is again a morphism in ABan1. And one says that f is a weak retract if f∗ is a split epimorphism in ABan1. The main result of this note is the following theorem. 3.1.THEOREM. Let A be a commutativeunital Banach algebra, and ι: A→B a norm-decreasing central homomorphism of unital Banach algebras. Then the following conditions are equivalent: (i) ιisaneffectivedescentmorphismforBanachmodules;thatis,theextension– of–scalars functor B⊗A−:ABan1 →BBan1 is comonadic; b (ii) ι is a ⊗–pure morphism in ABan1; that is, for any Banach A-module V, b the morphism ι⊗AV : V =A⊗AV →B⊗AV is an isometric inclusion; b b b (iii) ι is a weak retract in ABan1. Proof. Since the regular monomorphisms in Ban1 are precisely the isometric inclusions (e.g. [1, 4.3.10.e]), K is regular injective in Ban1 by the Hahn-Banach Theorem. Moreover,the functor [−,K]: (Ban1)op →Ban1 is monadic by [8]. Hence [−,K],seenasafunctor[−,K]: Ban1 →(Ban1)op,iscomonadic. Onenowconcludes the proof by applying Theorem 2.1. ⊔⊓ Since any morphism of commutative unital Ban1-monoids is easily seen to be central, a corollary follows immediately: 3.2.COROLLARY. Given a norm-decreasing homomorphism ι: A→B of com- mutative unital Banach algebras, the following conditions are equivalent: (i) ι is an effective descent morphism; (ii) ι is a ⊗–pure morphism in ABan1; b (iii) ι is a weak retract in ABan1. 3.3. EXAMPLE. Let c0 be the Banach space of all sequences a = (an)n∈N of scalars converging to zero with the supremum norm kak∞ = supn∈N{|an|}, ℓ1 the 4 BACHUKIMESABLISHVILI space of all sequences for which the norm kbk1 = P∞n=1|bn| is finite, and ℓ∞ the space of all bounded sequences of scalars with the some supremum norm as c0. Then(c0)∗ isisometricallyisomorphictoℓ1 andℓ∞ to(ℓ1)∗ (e.g.,[15]). Withthese isometrical isomorphisms, the canonical isometric inclusion of c0 into its double dual can be identified with the usual inclusion c0 → ℓ∞ of spaces of sequences. Since both of c0 and ℓ∞ with element-wise algebra operations are commutative unital Banach algebras, it follows from Theorem 3.1 that the canonical inclusion c0 →ℓ∞ of unital Banach algebras is an effective descent morphism. We conclude the note by giving a result which shows how to construct an effec- tive descent morphism for Banach modules from any commutative unital Banach algebra. Let A be an arbitrary unital Banach algebra. 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