  We prove that the infinite tensor power of a unital separable C*-algebra
absorbs the Jiang-Su algebra Z tensorially if and only if it contains,
unitally, a subhomogeneous algebra without characters. This yields a succinct
universal property for Z in a category so large that there are no unital
separable C*-algebras without characters known to lie outside it. This category
moreover contains the vast majority of our stock-in-trade separable amenable
C*-algebras, and is closed under passage to separable superalgebras and
quotients, and hence to unital tensor products, unital direct limits, and
crossed products by countable discrete groups.
  One consequence of our main result is that strongly self-absorbing ASH
algebras are Z-stable, and therefore satisfy the hypotheses of a recent
classification theorem of W. Winter. One concludes that Z is the only
projectionless strongly self-absorbing ASH algebra, completing the
classification of strongly self-absorbing ASH algebras.
