Approximate homotopy of homomorphisms from $C(X)$ into a simple $C^*$-algebra

In this paper we prove Generalized Homotopy Lemmas. These type of results play an important role in the classification theory of $*$-homomorphisms up to asymptotic unitary equivalence.

Let $X$ be a finite CW complex and let $h_1, h_2: C(X)\to A$ be two unital homomorphisms, where $A$ is a unital $C^*$-algebra. We study the problem when $h_1$ and $h_2$ are approximately homotopic. We present a $K$-theoretical necessary and sufficient condition for them to be approximately homotopic under the assumption that $A$ is a unital separable simple $C^*$-algebra, of tracial rank zero, or $A$ is a unital purely infinite simple $C^*$-algebra. When they are approximately homotopic, we also give an upper bound for the length of the homotopy.

Suppose that $h: C(X)\to A$ is a monomorphism and $u\in A$ is a unitary (with $[u]=0$ in $K_1(A)$). We prove that, for any $\epsilon>0,$ and any compact subset ${\mathcal F}\subset C(X),$ there exist $\delta>0$ and a finite subset ${\mathcal G}\subset C(X)$ satisfying the following: if $\Vert[h(f), u]\Vert<\delta$ for all $f\in {\mathcal G}$ and Bott$(h,u)=0,$ then there exists a continuous rectifiable path $\{u_t: t\in [0,1]\}$ in $A$ such that

$u_0=u, u_1=1_A {\text{and}} \Vert[h(g),u_t]\Vert<\epsilon {\rm for all} g\in {\mathcal F} {\text{and}} t\in [0,1].$

(1)

Moreover,

${\rm {Length}}(\{u_t\})\le 2\pi+\epsilon.$

(2)

We show that if dim$X\le 1,$ or $A$ is purely infinite simple, then $\delta$ and ${\mathcal G}$ are universal (independent of $A$ or $h$). In the case that ${\rm dim} X=1,$ this provides an improvement of the so-called Basic Homotopy Lemma of Bratteli, Elliott, Evans and Kishimoto for the case that $A$ is as mentioned above. Moreover, we show that $\delta$ and ${\mathcal G}$ cannot be universal whenever dim$X\ge 2.$ Nevertheless, we also found that $\delta$ can be chosen to be dependent on a measure distribution but independent of $A$ and $h.$ The above version of the so-called Basic Homotopy is also extended to the case that $C(X)$ is replaced by an AH-algebra.

We also present some general versions of the so-called Super Homotopy Lemma.