The Algebra of Affiliated Operators

Abstract

In this chapter we introduce and study the algebra U(G) of operators affiliated to N(G) for a group G. A G-operator f: dom(f) ⊂ V → W of Hilbert N(G)-modules is an operator whose domain dom(f) is a linear G-invariant subspace and which satisfies f(gx) = gf(x) for all x ∈ dom(f) and g ∈ G. The algebra U(G) consists of densely defined closed G-operators a: dom(a) ⊂ l ²(G) → l ²(G) and contains N(G) as a subalgebra. It is constructed in such a way that an element f : l ²(G) → l ²(G) in N(G) is a weak isomorphism if and only it is invertible in U(G). This is reflected algebraically by the fact that U(G) is the Ore localization of N(G) with respect to the set of non-zero divisors (see Theorem 8.22 (1)). It does not come with a natural topology anymore but has nice ring theoretic properties. Namely, U(G) is von Neumann regular, i.e. for any r ∈ U(G) there is s ∈ U(G) with rsr = r, or, equivalently, any finitely generated submodule of a projective U(G)-module is a direct summand (see Theorem 8.22 (3)). We have already mentioned in Example 6.12 that N(G) behaves in several ways like a principal ideal domain except that N(G) is not Noetherian and has zero-divisors if G is infinite. Any principal ideal domain R has a quotient field F, and in this analogy U (G) should be thought of as F. Recall that the (extended) dimension of an arbitrary R-module M is the same as the F-dimension of the F-vector space F ⊗ _R M (see Example 6.12).