Classifying spaces of compact Lie groups that are p–compact for all prime numbers

A p–compact group (see Dwyer–Wilkerson [8]) is a loop space X such that X is Fp –finite and that its classifying space BX is Fp –complete (see Andersen–Grodal– Møller–Viruel [2] and Dwyer–Wilkerson [11]). We recall that the p–completion of a compact Lie group G is a p–compact group if π0(G) is a p–group. Next, if C(ρ) denotes the centralizer of a group homomorphism ρ from a p–toral group to a compact Lie group, according to [8, Theorem 6.1], the loop space of the p–completion Ω(BC(ρ))p is a p–compact group.