  We investigate C*-algebras whose central sequence algebra has no characters,
and we raise the question if such C*-algebras necessarily must absorb the
Jiang-Su algebra (provided that they also are separable). We relate this
question to a question of Dadarlat and Toms if the Jiang-Su algebra always
embeds into the infinite tensor power of any unital C*-algebra without
characters. We show that absence of characters of the central sequence algebra
implies that the C*-algebra has the so-called strong Corona Factorization
Property, and we use this result to exhibit simple nuclear separable unital
C*-algebras whose central sequence algebra does admit a character. We show how
stronger divisibility properties on the central sequence algebra imply stronger
regularity properties of the underlying C*-algebra.
