Change of base for operator space modules

We prove a change of base theorem for operator space modules over C*-algebras, analogous to the change of rings for algebraic modules. We demonstrate how this can be used to show that the category of (right) matrix normed modules and completely bounded module maps has enough injectives. Mathematics Subject Classification. 46H25, 46L05, 46L07, 46M10, 46M18.

Hahn-Banach theorem states that B(H), the space of bounded operators on a Hilbert space H, is injective even in Op 1 (which has the same objects as Op ∞ but the morphisms are the complete contractions), and injectivity in an additive category of h-modules has been investigated by Frank and Paulsen [9]. Injective matrix normed modules (see Definition 3.1 below) seem to be less well understood; we will discuss this in Sections 3 and 4 and prove that every injective operator space provides us with an injective matrix normed module in a canonical way (Corollary 4.11).
The main topic of this paper is a change of base procedure for matrix normed modules over unital C*-algebras. Change of base, also called 'change of rings' in algebraic module theory, is an adjunction of module categories and as such more general than equivalence. Hence, it extends Morita equivalence of rings. The 'extension of scalars' (Definition 4.5) makes use of the module operator space projective tensor product, which we therefore recall in Section 3. The paper's main theorem is Theorem 4.9; as a consequence of it, we find that every matrix normed module can be completely isometrically embedded into an injective one (Proposition 4.13). Change of base is a fundamental tool in algebraic geometry for sheaves of modules over ringed spaces and, in a similar vein, for operator module sheaves over C*-ringed spaces in [1].

Terminology, notations, and conventions.
Our standard references are [2] and [8]; in particular, we will always assume that operator spaces are complete. The category Op ∞ consists of objects: the operator spaces, and morphisms: completely bounded linear maps. The subcategory Op 1 consist of the same objects, but the morphisms are the complete contractions. When m, n ∈ N and E is a vector space, if we want to emphasise indexing, we will sometimes identify elements in the vector space M nm (E) of (nm × nm)-matrices as [[x kl ij ] (i,j) ] (k,l) ∈ M m (M n (E)) where i, j ∈ {1, . . . , n} and k, l ∈ {1, . . . , m}. If E, F are operator spaces, we will denote the operator space consisting of all completely bounded linear maps from E to F by CB(E, F ). We write the completely bounded norm of an element φ ∈ CB(E, F ) as φ cb . If moreover, E and F are right Banach A-modules for a C*-algebra A, then the space where each vertical arrow is a bijection. If the morphism sets for both A and B are objects in a third category C , the adjoint pairs considered are those such that the associated commutative diagram is of morphisms in C . For more information on adjoint pairs, see, e.g., [12, Chapter IV].

Injective operator space modules.
In this section, we fix a unital C*algebra A, and all modules will henceforth be assumed to be unital. We will discuss two types of modules over A, the h-modules and the matrix normed modules. Both are related to different notions of multilinear mappings on operator spaces and appeared in independent papers by Blecher-Paulsen [3] and Effros-Ruan [7].
We say φ is jointly completely bounded if there exists K > 0 such that, for each n, m ∈ N, and [ We denote the infimum of all K in equation (3.1) (respectively, in equation (3.2)) by φ mcb (resp., φ jcb ). If φ mcb ≤ 1 (resp., φ jcb ≤ 1), then φ is multiplicatively (resp., jointly) completely contractive. Suppose E is a right A-module that is also an operator space. If the module action E × A → E, (x, a) → x · a, is multiplicatively (resp., jointly) completely contractive, then E is a right h-module over A (resp., a right matrix normed A-module). There are similar definitions for left modules and bimodules.
For our modules, we have used the naming conventions of [2]; we will however avoid the term "operator module" as it has been used for matrix normed modules elsewhere in the past. We use "operator space module" as the overarching terminology. It is not difficult to show that any multiplicatively completely contractive map is also jointly completely contractive so any right hmodule over A is also a right matrix normed A-module. We also note that the operator spaces are precisely the right h-modules over C which agree with the right matrix normed C-modules. It has become customary to drop the adverb 'multiplicatively' whereas 'jointly completely bounded' was called 'matricially bounded' at a time [7].
For operator spaces E, F, G, the space MCB(E × F , G) of all multiplicatively completely bounded bilinear maps (resp., JCB(E × F , G) of all jointly completely bounded bilinear maps) becomes a Banach space with norm · mcb (resp., · jcb ). Definition 3.1 can be reformulated in terms of the following operator space tensor products.

Definition 3.2.
Let E and F be operator spaces. The operator space projective tensor product (resp., Haagerup tensor product), denoted E ⊗ F (resp., E ⊗ h F ), is the unique (up to completely isometric isomorphism) operator space such that there is a jointly (resp., multiplicatively) completely contractive map θ : For the existence of each of these tensor products, see [8,Chapters 7 and 9]. They involve the completion of the algebraic tensor product E ⊗ F with respect to particular matrix norms. We denote an elementary tensor in E ⊗ F by x ⊗ y, where x ∈ E, y ∈ F. A right module E that is also an operator space is a right h-module (resp., matrix normed module) over A precisely when the module action induces a complete contraction E ⊗ h A − → E (resp., E ⊗ A − → E). A benefit of working with the operator space projective tensor product is its relationship to spaces of completely bounded linear mappings.
Using the language of Section 2, this states that the tensor product functor −⊗ F and the hom-functor CB(F, −) form an adjoint pair on the category Op ∞ as Mor Op ∞ (E, F ) = CB(E, F ). There is no such relation for the Haagerup tensor product; on the other hand, h-modules are characterised by the CES-theorem ([2, Theorem 3.3.1]) which allows for a common completely isometric representation of the C*-algebra and the module on one Hilbert space.
Turning towards injectivity, we note that the usual reference to all monomorphisms as the 'embeddings of a subobject' often is not the right choice and it is therefore desirable to specify a particular class of morphisms as in the following definition.

Definition 3.4.
Let M be a class of monomorphisms in a category A (which is typically assumed to be closed under composition and contains all isomorphisms). An object I ∈ A is M -injective if, for all E, F ∈ A and morphisms μ ∈ M (E, F ) and any morphism f ∈ Mor(E, I), there exists a morphism f ∈ Mor(F, I) such that f = fμ. That is, there exists a filler arrow making the following diagram commutative.
A is the category whose objects are the right h-modules over A, whose morphisms are the completely contractive A-module mappings, and M 1 A is the class of completely isometric A-module maps. This relies on Wittstock's result [16,Theorem 4 In the additive category hMod ∞ A , of h-modules over A with completely bounded A-module maps, the class M ∞ A of embeddings to consider are the completely bounded injective module maps with closed range and completely bounded inverse from the range. The corresponding M ∞ A -injectives are less studied; however, there are some nice results of Frank and Paulsen on when A is M ∞ A -injective as an h-module over itself [9]. It is clear that every Thus the afore-mentioned CES-theorem yields the following.
Matrix normed modules are not necessarily representable; therefore we need a different device to ensure that there are enough injectives among them. This will be done in the next section.

Change of base.
Restriction and extension of scalars for matrix normed modules turn out to form a pair of adjoint functors in the sense of diagram (2.1). We will use this to establish the existence of enough injectives in the appropriate module category.
In this section, we fix a unital *-homomorphism π : A → B between unital C*-algebras A and B. The category whose objects are the right matrix normed A-modules and whose morphisms are the completely bounded A-module maps will be denoted mnMod ∞ A . There is also the similarly defined category of left matrix normed A-modules A mnMod ∞ . If E, F ∈ mnMod ∞ A , we denote the set of morphisms from E to F by CB A (E, F ), it is clear that this forms a closed subspace of CB(E, F ). For left matrix normed A-modules, we use the notation A CB(E, F ). A matrix normed A-B-bimodule is an operator space that is both an object in A mnMod ∞ and mnMod ∞ B with the associativity condition (a · x) · b = a · (x · b). The corresponding category is denoted by A mnMod ∞ B and here the morphism sets are denoted by A CB B (E, F ). Thus B ∈ A mnMod ∞ B via the left action of A given by a · b = π(a)b, a ∈ A, b ∈ B, and the canonical right action of B on itself.
Like in the theory of modules over rings, we make use of module tensor products. Our source for operator space module tensor products is [2, Section 3.4].

Definition 4.1. Let E and F be right and left A-modules, respectively. Suppose
The module operator space projective tensor product is the quotient of E⊗ F by the closure of the subspace spanned by {(x · a) ⊗ y − x ⊗ (a · y) | x ∈ E, y ∈ F, a ∈ A}. This is denoted by E ⊗ A F . We still use the notation x ⊗ y, now for the coset in E⊗ A F of the elementary tensor x ⊗ y.
It is not difficult to show, via the universal property of the operator space projective tensor product described in Definition 3.2, that the above tensor product 'linearises jointly completely bounded balanced bilinear maps'. That is, we have the following: (y · b), and the closed subspace spanned by

the canonical jointly completely contractive bilinear mapping. Given an operator space G and a jointly completely contractive A-balanced bilinear map φ : E × F → G, there exists a unique completely contractive linear map
is a submodule. Thus the quotient module E ⊗ A F is a right matrix normed B-module (for details, see [2, paragraph 3.4.9]). In order to define the extension of scalars, we need the following property for the module operator space projective tensor product.

Lemma 4.3 ([2, Lemma 3.4.5]). Let
We shall now introduce the main concepts of this paper.
Definition 4.5. The extension of scalars from A to B is the covariant functor Let E ∈ mnMod ∞ B ; then we can consider E as a right matrix normed Amodule when we define the module action x · a = x · π(a). We will write E with this module action as E π . Note then that the module action E × B → E is a jointly completely contractive A-balanced map and so by Proposition 4.2, we have a canonical completely contractive map E ⊗ A B → E.
is called the restriction of scalars from B to A.
That the extension and restriction of scalars are, in fact, functors is routinely checked. We aim to make use of a matrix normed module version of Proposition 3.3 involving the module operator space projective tensor product and use that for the change of base.

Proposition 4.7. Let A and B be unital C*-algebras. Suppose that E ∈
Before we arrive at our main result, we record the following.
The following theorem, the first part of which appears in [2, Proposition 3.5.9], is the main result of this paper. We will use the notation introduced in Section 2.

(ii) Moreover, if E ∈ mnMod ∞
A and G ∈ mnMod ∞ B with φ ∈ CB A (E, E ) and ψ ∈ CB B (G, G ), we have the following commutative diagram of operator spaces and completely bounded linear maps. Let Ψ denote ΘΦ. Let Ψ EF G be Ψ restricted to CB B (E ⊗ A F , G).
To prove the first statement of the theorem, all that remains is to show that Ψ EF G is surjective. Suppose κ ∈ CB A (E, CB B (F, G)). Then κ ∈ CB(E, CB(F, G)) and so there exists, by Proposition 3.3, a map f ∈ Therefore, by the universal property of the module operator space projective tensor product, there exists ν ∈ CB(E ⊗ A F , G) such that νq = f . Then Ψ(ν) = κ and it is easy to check that ν is a B-module map as required.
For each x ∈ E, y ∈ F , we have with φ ∈ CB A (E, E ) and ψ ∈ CB B (G, G ), the following commutative diagram of operator spaces and bounded linear maps as required.
Remark 4.10. The only connection between the C*-algebras A and B that we require in parts (i) and (ii) is that there exists some F ∈ A mnMod ∞ B . When we specialise the above to the canonical embedding C → B, where B is a unital C*-algebra, the restriction of scalars becomes the forgetful functor mnMod ∞ B → Op ∞ and the extension of scalars the functor On the other hand, noting that mnMod ∞ C and Op ∞ are isomorphic categories, we can take F = A in Theorem 4.9, part (i), and B = C. Combining this with Propositions 4.4 and 4.7, we obtain, for every E ∈ mnMod ∞ A and every operator space G, the following chain of isometric isomorphisms CB(A, G)). (4.2) We apply this to prove the existence of enough injectives in the category of matrix normed modules. Recall, from Section 3, that M ∞ denotes the class of all (not necessarily surjective) completely bounded isomorphisms and M ∞ A the subclass of those which are A-module maps.  CB(A, I)). By (4.2) above, f corresponds to a unique g ∈ CB(E, I) with f cb = g cb such that f (x)(a) = g(x · a) for x ∈ E, a ∈ A. As I is injective and μ ∈ M ∞ (E, F ), there existsg ∈ CB(F, I) such that g =g μ. Settingf (y)(a) =g(y·a), y ∈ F , a ∈ A, we get the extension as an A-module map from F into CB(A, I) of f as desired.

Remark 4.12.
Suppose I is an M 1 -injective object in Op ∞ (what is usually called an 'injective operator space'). Then an extension of g ∈ CB(E, I) can be found such that the cb-norm of g is preserved, provided μ is completely isometric. In this case, f cb = g cb = g cb = f cb so that CB(A, I) is M 1 -injective in mnMod 1 A . Proposition 4.13. Let A be a unital C*-algebra. Then every matrix normed right A-module can be completely isometrically embedded into an injective one. In particular, mnMod ∞ A has enough injectives. Proof. Let E ∈ mnMod ∞ A . There exists a Hilbert space H such that E inherits its operator space structure as a closed subspace of B(H). As B(H) is injective in Op ∞ , CB(A, B(H)) is injective in mnMod ∞ A by Corollary 4.11. By Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/ licenses/by/4.0/.
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