  We study properties of the central sequence algebra of a C*-algebra, and we
present an alternative approach to a recent result of Matui and Sato. They
prove that every unital separable simple nuclear C*-algebra, whose trace
simplex is finite dimensional, tensorially absorbs the Jiang-Su algebra if and
only if it has the strict comparison property. We extend their result to the
case where the extreme boundary of the trace simplex is closed and of finite
topological dimension. We are also able to relax the assumption on the
C*-algebra of having the strict comparison property to a weaker property, that
we call local weak comparison. Namely, we prove that a unital separable simple
nuclear C*-algebra, whose trace simplex has finite dimensional closed extreme
boundary, tensorially absorbs the Jiang-Su algebra if and only if it has the
local weak comparison property.
  We can also eliminate the nuclearity assumption, if instead we assume the
(SI) property of Matui and Sato, and, moreover, that each II_1-factor
representation of the C*-algebra is a McDuff factor.
