The study of Fourier multipliers has been the central topic of classical harmonic analysis. The main task there is to find criteria for the boundedness of Fourier multipliers on various function spaces, especially on Lp-spaces. A basic example of Fourier multipliers is the well-known Hilbert transform.

Noncommutative analysis can be viewed as quantised analysis, where functions become operators (Noncommutative functions). Noncommutative analysis finds one of its origins in the investigation of approximation properties of operator algebras initiated by Haagerup’s pioneering work (Invent. Math. 1979) on Fourier multipliers on free groups in which he solved the longstanding problem on Grothendieck’s AP for the reduced C*-algebra of a free group: this algebra has the CBAP.

In this talk, we will discuss some recent progress in the study of Fourier multipliers on noncommutative function spaces.