## Classification of Splitting Interval Algebras and the C* Exponential Length

##### Abstract

Let A = lim n→∞ (An,ɸn,m) be a C∗ algebra where An =⨁kn i=1 Ain , Ain are splitting interval algebras. Suppose that A has the ideal property: each closed two-sided ideal is generated by the projections inside the ideal, as a closed two-sided ideal. In Chapter 1, we show that the scaled ordered K0 group and the ordered vector spaces AﬀT(eAe) with maps between AﬀT(e'Ae') and AﬀT(eAe) are the complete invariant for the classiﬁcation of this class of C∗-algebras, where eAe := {eae| a ∈ A}, and e, e' are certain projections in A with e' <e. We call this invariant Stevens’ Invariant.
In Chapter 2, we enlarge the Elliott’s invariant by considering the inﬁnite traces. And we show that if A and B have isomorphic Stevens’ Invariant, then they have isomorphic Elliott Invariant and vise versa, where A and B are two C ∗ -algebras with the ideal property. Moreover, for Z-absorbing C∗ -algebra, we give a characterization of Cuntz comparability by lower semi-continuous dimension functions.
In Chapter 3, we talk about the C ∗ exponential length. Let X be a compact Hausdorﬀ space. We give an example to show that there is u ∈ C(X) ⊗ Mn with det(u(x)) = 1 for all x ∈ X and u ∼h 1 such that the C∗ exponential length of u (denoted by cel(u)) can not be controlled by π. Moreover, for any ε> 0, we can ﬁnd a simple inductive limit C ∗ -algebra (simple AH algebra), say A, and a unitary u ∈ CU(A) with u ∼h 1 and cel(u) ≥ 2π − ε.