Change of base for operator space modules
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Archiv der Mathematik
Change of base for operator space modules Michael Rosbotham
Abstract. 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. Keywords. Operator space modules, C*-algebras, Restriction of scalars, Extension of scalars, Change of base, Enough injectives.
1. Introduction. Let A be a Banach algebra. A right A-module E is called a right Banach A-module if it is a Banach space and, for each x ∈ E, a ∈ A, we have x · a ≤ xa. The book of Helemskii [10], e.g., deals with homological algebra in this setting. For C*-algebras, there are good reasons to employ modules which also carry an operator space structure; the most common ones are the h-modules and the matrix normed modules. In [14], Paulsen introduced the notions of relative injectivity and relative projectivity for h-modules, and various pieces of homological algebra, including results on homological dimension, have been obtained in this setting; see, e.g., [11, Section 7] and [17]. The categories that appear in functional analysis, such as Ban∞ and Op∞ whose objects are the (complex) Banach spaces and the operator spaces with bounded and with completely bounded linear mappings, respectively, as morphisms, are not abelian. Thus the standard homological algebra does not apply. The more general setting of exact categories in the sense of Quillen, see [5], has been successfully applied to Ban∞ in [6], and exploited in [15] to define a cohomological dimension for C*-algebras, and in [1] to develop a sheaf cohomology theory; cf. also [13]. In particular in the latter setting, there are too few projective objects; this is why we focus on injectives. The Arveson–Wittstock
M. Rosbotham
Arch. Math.
Hahn–Banach theorem states that B(H), the space of bounded operators on a Hilbert space H, is injective even in Op1 (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
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