Two extensions of a
C∗-algebra A by a C∗-algebra B can be added, by the method of Brown, Douglas,
and Fillmore, whenever the quotient multiplier algebra M(A)∕A contains two
isometries with orthogonal ranges. If A is stable (i.e., if A≅A ⊗𝒦) then such
isometries can be found already in M(A), but if A has a tracial state then this is not
possible. (Hence in the case that A is a separable AF algebra, this is possible only if
A is stable.) Here it is shown that, in the case that A is a separable AF algebra
(assumed to have no nonzero unital quotient), there exist two isometries in M(A)∕A
with orthogonal ranges if, and only if, the space T(A) of tracial states of A is
compact.
