Aspects of unconditionality of bases in spaces of compact operators

E. Tutaj has introduced classes of Schauder bases termed "unconditional-like" (UL) and "unconditional-like*" (UL*) whose intersection is the class of unconditional bases. In view of this association with unconditional bases, it is interesting to note that there exist Banach spaces which have no unconditional basis and yet have a basis of one of these two types (e.g., the space [0,1]). In the same spirit, we show in this paper that the space of all compact operators on a reflexive Banach space with an unconditional basis has a basis of type UL*, even though it is well-known that this space has no unconditional basis.