Constructing Well-bounded Operators not of type (B) on a Class of Inductive Limits

Well-bounded operators are linear operators on a Banach space $X$ that have an $AC[a,b]$ functional calculus for some interval $[a,b]$. A well-bounded operator is of type (B) if it can be written as an integral against a spectral family of projections, and this is always the case when $X$ is reflexive. There are many examples of well-bounded operators on non-reflexive spaces that are not of type (B), and it is open whether there is a non-reflexive Banach space upon which every well-bounded operator is of type (B). The spaces constructed by Pisier, which answered a conjecture of Grothendieck in the negative, have been suggested by Cheng and Doust as a candidate to answer this open problem. In this paper, it will be shown that on a class of Banach spaces containing these spaces, there is always a well-bounded operator not of type (B).