We study properties of the central sequence algebra A ω ∩ A of a C *algebra A, 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

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... lso 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.