Tensor Products of C*-Algebras with the Ideal Property

Combining a construction of Dadarlat of a unital, simple, non-exact C*-algebra C of real rank zero and stable rank one, which is shape equivalent to a UHF-algebra, with results of Kirchberg and a result obtained by Dadarlat and the first-named author, we show that B(H)⊗C contains an ideal that is not generated by its projections. We also find a unital, separable sub C*-algebra A of B(H) such that A is of real rank zero and A⊗C has an ideal that is not generated by its projections.