Separability of the C*-algebra in the definition of K-homology

Question

There are (at least) two approaches to K-homoology: one is via the so called dual algebra which is due to Paschke. The second is via the Fredholm modules and is due to Kasparov. In Nigel Higson's book there is the note that the second approach is more flexible. I wonder what exactly does it mean: to be more precise, in the first treatment there is a assumption that the underlying $C^*$-algebra is separable and as far as I understood some properties and proofs, this assumption is heavily used. Namely, one needs the so called ample representation in order to prove that the dual algebra (essentially) doesn't depend on the representation.
When it comes to discussing Fredholm modules, author assumes also that the underlying $C^*$-algebra is separable however he points out that the definition of Fredholm modules can be stated for general $C^*$-algebras. So my question is the following:
Question: is it true that one can define $K$-homology groups for all $C^*$-algebras but only via Fredholm modules. So in other words, the dual algebra aproach is not proper to define $K$-homology groups for general $C^*$-algebras?

Answer 1

The main problem with dual algebras for non-separable C*-algebras is that they need not be functorial. Given representations $\rho_A \colon A \to \mathcal{B}(H_A)$ and $\rho_B \colon B \to \mathcal{B}(H_B)$ and a $*$-homomorphism $\phi \colon A \to B$, one wants to define the induced map $\phi_* \colon \mathcal{D}(B,\rho_B) \to \mathcal{D}(A, \rho_A)$ by $\phi_*(T) = VTV^*$ where $V \colon H_B \to H_A$ is an isometry such that $V^* \rho_A(a) V$ is equal to $\rho_B(\phi(a))$ modulo compact operators. Such isometries are constructed using Voiculescu's theorem, which requires that $A$ is separable. Without functoriality the whole theory pretty much collapses; for instance, one proves that K-homology is independent of the ample representation used to define the dual algebra by arguing that the K-theory map induced by $\phi_* \colon \mathcal{D}(B, \rho_B) \to \mathcal{D}(A, \rho_A)$ is independent of the isometry $V$.

For Kasparov's model of K-homology, you don't run into this problem: if $(\rho, H, F)$ is a Fredholm module over $B$ then $(\rho \circ \phi, H, F)$ is a Fredholm module over $A$. Still, there are problems: without Voiculescu's theorem you don't have the excision theorem in either model (at least the standard proof doesn't work) and without excision you don't have long exact sequences.

None of this should be too much of a concern, however - most C*-algebras for which K-theory is useful are separable (e.g. $C(X)$ for $X$ compact, crossed products of $C(X)$ by a locally compact group action). The main source of non-separable C*-algebras is the theory of von Neumann algebras, and K-theory/homology doesn't tell you anything about von Neumann algebras anyway.