This book examines some recent developments in the theory of $C^*$-algebras, which are algebras of operators on Hilbert spaces. An elementary introduction to the technical part of the theory is given via a basic homotopy lemma concerning a pair of almost commuting unitaries. The book presents an outline of the background as well as some recent results of the classification of simple amenable $C^*$-algebra, otherwise known as the Elliott program. This includes some stable uniqueness theorems and a revisiting of Bott maps via stable homotopy. Furthermore, $KK$-theory related rotation maps are...

A unital separable $C^ast$-algebra, $A$ is said to be locally AH with no dimension growth if there is an integer $d>0$ satisfying the following: for any $epsilon >0$ and any compact subset ${mathcal F}subset A,$ there is a unital $C^ast$-subalgebra, $B$ of $A$ with the form $PC(X, M_n)P$, where $X$ is a compact metric space with covering dimension no more than $d$ and $Pin C(X, M_n)$ is a projection, such that $mathrm(a, B)

A unital separable $C^ast$-algebra, $A$ is said to be locally AH with no dimension growth if there is an integer $d>0$ satisfying the following: for a...