## The tower of path algebras associated to a tower of finite dimensional $C^*$-algebras is isomorphic to the original tower

Let $$A_0\subseteq A_1\subseteq…$$ be an infinite tower of unital inclusions of finite dimensional $$C^*$$-algebras and $$B_0\subseteq B_1\subseteq …$$ be its associated infinite tower of path algebras. Then how to show that these two towers are tower isomorphic ?

## 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?