The concept of conditional freeness generalizes in some aspects that of freeness. It was first studied by M. Bozejko, M. Leinert and R. Speicher (around the 90's) in the context of Quantum Stochastic Processes.

Since then many tools and concepts of free probability have been shown to have their counterparts in the conditionally free picture.

We present some results of Mihai Popa and Jiun-Chau Wang about the generalization of the T-transform and the multiplicative free convolution in the conditionally free case.