Direct sums of tensor products and non-commutative independence
The classical probability theory corresponds to commutative *-algebras and in this case two sub-*-algebras are independent in the state \(\Phi\) if in some sense the value of the state on the tensor product of restrictions is equal to the tensor product of the values of the state on each of its restrictions. These facts can be generalized into the non-commutative case in replacing tensor products by free products of *-algebras. The author has outlined the close ``connection between three known fundamental notions of non-commutative independence in quantum probability and the three basic choices for forming an associative direct sum of alternating tensor products of two vector spaces''.
associative direct sum of alternating tensor products
scientific article; zbMATH DE number 850597