Conditions Equivalent To The Independence Of Operator Algebras

In quantum field theory, the term “two observables are independent” is of great significance. So in operator algebras, the concept of C* independence is introduced by R. Haag. And the independence of operator algebras is an important topic in operator algebra. In recent years, a lot of researchers obtain excellent works on the study of the independence of operator algebras. One of them is Roos Theorem and it indicates that twoC* algebras areC* independent if they satisfy the S property.In this paper, l is a C* algebra and has a unit e contained in subalgebra A and B.In the paper, we study the independence of operator algebras.First, we give the characterization of C* independence of twoC* subalgebras which have the unit and do not commute with each other through uncoupled product states: AbelianC* subalgebra A and C* subalgebra B that are not commuting with each other areC* independent if and only if l has theC* uncoupled product property;secondly, we solve the problem of faithfully independently commute of twoC* subalgebras which have the unit and do not commute with each other via unique common extensions of pairs of states; thirdly, we introduce the concept ofC*P-C independence of two C* subalgebras which have the unit and commute with each other and give the equivalent description of C*P-C independence: l is isomorphic to the maximal tensor product of twoC* subalgebras which commute with each other if and only if they are C*P-C independent.Last, we introduce Hahn-Banach independence,at the same time we proof that two C* subalgebras that commute with each other are Hahn-Banach independent if and only if they are C* independent Furthermore, we give the conditions of equivalence of C* algebras that are Hahn-Banach independent in a special case.