Remarks on essential codimension (Q2173280)

In this paper, the authors seek generalizations of the following theorem [\textit{L. G. Brown} et al., Lect. Notes Math. 345, 58--128 (1973; Zbl 0277.46053)]. {Theorem.} Given two projections \(P,Q\) in \(\mathbb{B}(\ell^2)\) with compact difference (i.e., \(P-Q\in\mathbb{K}(\ell^2)\)), there exists a unitary \(U\in \mathbb{C}1+\mathbb{K}(\ell^2)\) such that \(UPU^*=Q\) if and only if the \textit{essential codimension} of \(Q\) in \(P\) is zero. Essential codimension can be defined concretely in terms of Fredholm index theory, but it is useful to rephrase it in the modern language of KK-theory, as this points naturally towards possible generalizations. The essential codimension of \(Q\) in \(P\) is the class \([\phi,\psi]\in KK(\mathbb{C},\mathbb{K})\) defined by the \(\ast\)-homomorphisms \(\phi,\psi\) sending \(1\) to \(P,Q\), respectively. It is now natural to replace the pair \((\mathbb{B},\mathbb{K})\) by (\(\mathcal{M}(B),B)\) for some stable separable \(C^*\)-algebra \(B\), and to take general morphisms \(\phi,\psi\colon A\to \mathcal{M}(B)\) where \(A\) is unital and separable. The present paper investigates this more general definition in essentially two cases: (1) \(A\) is nuclear and \(B\) is either \(\mathbb{K}\) or simple purely infinite; (2) \(A\) is simple and nuclear, \(B\) is simple, with a nonzero projection, with strict comparison of positive elements, \(T(B)\) has only finitely many extreme points, and finally there is an embedding \(A\hookrightarrow B\). Previous work along these lines includes [\textit{L. G. Brown} and \textit{H. H. Lee}, Can. J. Math. 64, No. 4, 755--780 (2012; Zbl 1256.46033); \textit{H. H. Lee}, J. Funct. Anal. 260, No. 1, 135--145 (2011; Zbl 1210.46052); J. Funct. Anal. 265, No. 6, 926--940 (2013; Zbl 1288.46041)]. As a preliminary result, the authors study the question of \(K_1\)-injectivity (i.e., whether or not the map \(\mathcal{U}(A)/\mathcal{U}_0(A)\to K_1(A)\) is \(1\)-\(1\)) for the \textit{Paschke dual algebra} of \(A\) relative to \(B\). Given \(\phi\colon A\to \mathcal{M}(B)\), this is defined as the commutant algebra of the image of \(A\) inside the corona algebra \(\mathcal{M}(B)/B\). Notice that we may view \(\phi\) as an extension of \(C^*\)-algebras, and in this sense the definition of the Paschke algebra assumes the extension to be unital, absorbing, and trivial. The word ``dual'' is used in light of the well-known isomorphism \(K_i(A^d_B)\cong KK_{i+1}(A,B)\). However, generalizing previous work by \textit{A. Valette} [Pac. J. Math. 109, 247--255 (1983; Zbl 0488.46061)], the authors here show that the Paschke algebra is ``dual'' in yet another sense, namely the following identities hold: \[ A^\prime= A^d_B\qquad (A^d_B)^\prime =A \] (here we assume that both algebras sit inside the corona algebra of \(B\)). The main results read as follows. {Theorem.} Given \(C^*\)-algebras \(A\) and \(B\), suppose we are either in case 1 or 2 (described above). Suppose \(\phi,\psi\colon A \to \mathcal{M}(B)\) are unital, trivial, (essential) extensions whose difference belongs to \(B\). Then the induced class \([\phi,\psi]\in KK(A,B)\) is zero if and only if \(\phi\) and \(\psi\) are asymptotically unitarily equivalent (via a path \(u_t\) which belongs to \(\mathbb{C}1+B\) for all \(t\)'s).