(1) Ando proved the following result:

Let A, B be self adjoint n by n matrices.

Then A is majorized by B if and only if A is an element of the convex hull

of the unitary orbit of B.

In this talk, we discuss infinite generalizations of the above result.

(2) Theorem: If B is a nonunital separable simple C*-algebra with strict comparison, almost divisibility,

stable rank one and quasicontinuous scale, then M(B)/B has real rank zero.

The above generalizes various results in the literature, with connections

to classical generalizations of the Weyl--von Neumann theorem as well as

attempts to generalize BDF Theory. The above

utilizes the following (earlier) result:

Theorem: Let B be a nonunital simple separable C*-algebra with strict

comparison, almost divisibility and stable rank one.

Then the following are equivalent:

(i.) B has quasicontinuous scale.

(ii.) M(B) has strict comparison.

(iii.) M(B)/B is purely infinite.

(iv.) M(B)/I_{min} is purely infinite.

(v.) M(B) has finitely many ideals.

(vi.) I_{min} = I_{fin}.