C*-Independence, Product States and Commutation

Abstract

Let D be a unital C * -algebra generated by C * -subalgebras A and B possessing the unit of D. Motivated by the commutation problem of C * -independent algebras arising in quantum field theory, the interplay between commutation phenomena, product type extensions of pairs of states and tensor product structure is studied. Roos's theorem [11] is generalized in showing that the following conditions are equivalent: (i) every pair of states on A and B extends to an uncoupled product state on D; (ii) there is a representation π of D such that π(A) and π(B) commute and π is faithful on both A and B; (iii) A ⊗ min B is canonically isomorphic to a quotient of D.The main results involve unique common extensions of pairs of states. One consequence of a general theorem proved is that, in conjunction with the unique product state extension property, the existence of a faithful family of product states forces commutation. Another is that if D is simple and has the unique product extension property across A and B then the latter C * -algebras must commute and D be their minimal tensor product.

States on Operator Algebras and Axiomatic System of Quantum Theory

“…All the result presented here have been obtained jointly by L. J. Bunce and the author in Bunce and Hamhalter (2004). We characterize tensor products of C * -algebras in quite simple terms of state extensions and, in particular, without assuming their mutual commutation.…”

“…(Bunce and Hamhalter, 2004) If D is simple and there is at least one uncoupled product state across A and B, then D is canonically * -isomorphic to A ⊗ min B. Now we shall deal with the uniqueness of common state extensions.…”

“…Another contribution to the commutation problem is the following theorem: Theorem 4.3. (Bunce and Hamhalter, 2004) D is canonically * -isomorphic to the minimal tensor product A ⊗ min B if and only if there is a set S of uncoupled product states across A and B such that the set of the corresponding G.N.S. representations {π ϕ | ϕ ∈ S} is faithful on D. (We say that a system of representations is faithful if the intersection of their kernels is zero.…”

“…(Bunce and Hamhalter, 2004) Suppose that for each pair of states and τ of A and B, respectively, there is a unique state ϕ ∈ S (D) extending and τ . Let D have a faithful family of G.N.S.…”

Independence of group algebras

“…We say that C * -algebras A and B are C * -independent in the product sense if the C * -algebra, C * (A, B), generated by A and B is isomorphic to the minimal tensor product A ⊗ min B via the natural map a b → a ⊗ b, a ∈ A, b ∈ B. This condition is forced by the existence of a faithful product state over the pair A and B provided that A and B commute [12], or by the existence of a separating family of states with the product property without assuming mutual commutativity of A and B (see [5] for more detailed analysis). In the category of von Neumann algebras the counterpart of the C * -independence is called the W * -independence and it is defined as follows: von Neumann subalgebras M and N in a von Neumann algebra F are called W * -independent if for each pair of normal states ϕ and ψ on M and N , respectively, there is a normal state on F extending both ϕ and ψ. W * -independence always implies C * -independence, the reverse implication is not true in general (see [15] for more details).…”