Comparing the definitions of $K$-theory and $K$-homology for $C^*$-algebras

In Higson and Roe's Analytic K-homology, for a unital $$C*$$-algebra $$A$$, the definitions of K-theory and K-homology have quite a similar flavor.

Roughly, the group $$K_0(A)$$ is given by the Grothendieck group of homotopy classes of matrix algebra projections over $$A$$.

On the other hand, $$K^0(A)$$ is the Grothendieck group of homotopy classes of even Fredholm modules over $$A$$ (or more correctly unitary classes of Fredholm modules).

Now in contrast to the $$K_0(A)$$ case, every element of $$K^0(A)$$ contains a representative Fredholm module. (For the inverse of the class of $$(H,\rho,F)$$, just take the class of $$(H^{\text{op}},\rho,-F)$$, where op denotes the opposite grading.

Does this means that taking the Grothendieck is not strictly necessarily to define $$K^0(A)$$? Could one just as well define it as the monoid of homotopy classes of Fredholm modules, and then prove that it was a group?

If this is true, then is there any deeper philosophical reason why $$K_0$$ requires us to introduce inverses, while $$K^0$$ does not?

• One can define K^0 using Fredholm operators with dualizable (co)kernels, this will produce a counterpart to the cited definition of K_0, and the resulting model also has inverses defined in a similar way. The very point of Fredholm stuff is to allow inverses in the model by considering formal differences of kernels and cokernels. Mar 13 '19 at 2:28
• The model of Karoubi of K theory involving gradings (as in the definition of K homology cycles) makes non necessary the introduction of formal inverses. See his introductory book on the subject. Mar 13 '19 at 12:03

KK-theory provides one way to eliminate the asymmetry in the definitions of K-theory and K-homology. A cycle in $$KK(A, B)$$ is a triple $$(H, \rho, F)$$ where $$H$$ is a (adjectives) Hilbert $$B$$-module, $$\rho$$ is a (adjectives) representation of $$A$$ on $$H$$, and $$F$$ is a (adjectives) bounded operator on $$H$$ such that $$[F, \rho(a)]$$, $$(F^2 - 1)\rho(a)$$, and $$(F - F^*)\rho(a)$$ are $$B$$-compact for all $$a \in A$$. The relations are analogous to the relations in K-homology, and in particular KK-cycles form an abelian group without the need to invert anything.
If $$B = \mathbb{C}$$ then $$H$$ is an ordinary Hilbert space and the KK-cycles are exactly the same thing as Fredholm modules over $$A$$. It is a fact that $$KK(\mathbb{C}, B)$$ is isomorphic to the K-theory of $$B$$ for all (adjectives) C*-algebras $$B$$, so this gives a model of K-theory of the sort you're looking for.