Completely Correlation Preserving Functions

In this chapter, we introduce and investigate one of the central topics of our monograph: the class of completely correlation preserving functions (CCP), i.e., non-linear functions which map a classical correlation matrix of any size—entrywise—into a correlation matrix of the same size, such as Grothendieck’s function \(f(\rho ) : = \frac {2}{\pi }\arcsin (\rho ), \rho \in [-1,1]\) (in the real case). Important connections between CCP functions, an entrywise functional calculus for positive semidefinite matrices and the famous theorem of Schoenberg are pointed out.