Conditional and unconditional decompositions of noncommutative L^p spaces

In a number of problems in different areas of analysis it has been shown that the condition that one needs is not that a certain set be bounded, but rather that it be `R bounded'. This concept has been used, for example, in recent important work of Kalton and Weis and of le Merdy, concerning H^\infty functional calculus and maximal regularity, and of Celement, de Pagter, Sukochev and Witvliet on decompositions of Banach spaces.

In this talk I will show how this concept is used to prove some theorems that can be thought of, either as concerning particular types of decomposition of Banach spaces, or else concerning particular types of functional calculus. As particular applications of these theorems we get analogues of classical multiplier theorems of harmonic analysis, but now acting on the von Neumann-Schatten spaces S_p rather than on the Lebesgue spaces L^p. This is joint work with T.A. Gillespie.