Summary: Let be a separable unital -algebra and let be the canonical contraction from the Haagerup tensor product of with itself to the space of completely bounded maps on . In our previous paper [I. Gogić, Proc. Edinb. Math. Soc., II. Ser. 54, No. 1, 99–111 (2011; Zbl 1213.46046)] we showed that if satisfies that (a) the lengths of elementary operators on are uniformly bounded, or (b) the image of equals the set of all elementary operators on , then is necessarily SFT (subhomogeneous of finite type). In this paper, we extend this result; we show that if satisfies (a) or (b), then the codimensions of 2-primal ideals of are also finite and uniformly bounded. Using this, we provide an example of a unital separable SFT algebra which satisfies neither (a) nor (b).
However, if the primitive spectrum of a unital SFT algebra is Hausdorff, we show that such an satisfies both (a) and (b).