A CONCEPTUAL PROOF OF A BASIC RESULT IN THE COMBINATORIAL APPROACH TO FREENESS
Roland Speicher · Infinite Dimensional Analysis Quantum Probability and Related Topics · 2000
We present a conceptual proof for the theorem describing free cumulants with products as arguments. We reduce that theorem to a statement about the behavior of Möbius inversion under the embedding of lattices. As direct consequence of this insight we get generalizations of the theorem to the operator-valued case as well as analogous statements for classical and Boolean cumulants.