Real rank and exponential length of tensor products with O1

Abstract. Let D be any C∗-algebra. We prove that O∞⊗D has real rank at most 1, exponential length at most 2pi, exponential rank at most 2 + ε, and C ∗ projective length at most pi. The algebra O ∞ can be replaced with any separable nuclear purely infinite simple C∗-algebra