The Independence Of Operator Algebras And The Joint Spectrum Of Operator Tuples

Several characterization of the C^* algebras to be C^*-independent are given by use of the joint spectrum of operators,and equivalent conditions for C^* algebra to be prime are given.In 1970,Hang and Kastler.D introduce the concept of C^*independent in order to study the relations between two subsystem of a system.Let A and B be two algebras generated by the observables of systems,then the C^* independent can be viewed as:the observable of system A would not influence system B,and respectively,the observable of system B would not influnce system A.We give characterizations of C^* independent as follows:Theorem 1 Let A and B be mutually C^* algebras of B(H),then the following statements are equivalent:1.A and B are C^* independent.2.For all operator A∈A and all operator B∈B.then there is: Sp(A,B)=σ(A)×σ(B).3.For all positive operator A∈A and all positive operator B∈B,then there is: Sp(a,b)=Sp(a)×Sp(b). 4.For all operator tuples a=(a₁,…,a_n) of A and all operator tuples b=(b₁,…,b_n) of B,then there is: Sp(a,b)=Sp(a)×Sp(b).5.For all operator A∈A and for all operator B∈B,then there is: r(AB)=r(A)r(B). where r(A) denotes the spectral radius of operator A.6.For all positive operator A of A and all positive operator B of B,there is: W(a,b)=W(A)×W(B).7.For all operator A of A and all operator B of B,there is: W(A,B)=W(A)×W(B).8.For all positive operator A of A and all positive operator B of B,there is:‖A+B‖=‖A‖+‖B‖.9.For all normal operator A of A and all positive operator B of B,there is:‖A+B‖=max{|λ-μ;λ∈σ(A),μ∈σ(B).}Since the joint spectrum does not change under the transform of similarity,then: Corollary 2 Let A and B be C^* independent in B(H),then for any unitary operator U∈B(H),U AU^-1 and UBU^-1 are C^* independent.In order to study quantum system consisting of n sub-systems,we introduce the concept of C^* independent for n quantum systems,because of the close relationship between C^* independent and joint spectrum,we give the definition of the independence of joint spectrum for n Banach algebras.Definition 3 Let A be a unital C^* algebra,A_i,i=1,2,…,n be mutually commutative C^* subalgebras of A,we call they are C^* independent if for any disjoint subset {i₁,…,i_k} and {j₁,…,j_l} of {1,2,…,n},the C^* algebra C^*(A_i1,…,A_ik) and A_jl,…,A_jl are C^* independent.Theorem 4 Let a_i,i=1,2,…,n be double commuting operators in B(H),then the following statements are equivalent:1.(A_i)_iⁿ=1 are C^* independent.2.For any disjoint subset {i₁,…,ik} and {j1,…,jl} of {1,2,…,n},for any operator w₁∈C^*(a_il,…,a_ik) and any operator w₂∈C^*(a_j1,…,a_jl),then: Sp(w₁,w₂)=σ(w₁)×σ(w₂).3.For any disjoint subset {i₁,…,ik} and {j₁,…,j_l} of {1,2,…,n},for any positive operator w₁∈C^*(a_i1,…,a_ik) and any positive operator w₂∈C^*(a_j1,…,a_jl), then: Sp(w₁,w₂)=σ(w1)×σ(w₂).4.There is a faithful stateφon C^*(a₁,…,a_n),such that(C^*(a_i))_iⁿ=1 are independent in(C^*(a₁,…,a_n),φ).Definition 5 Let B be unital Banach algebra:A_i,i=1,2,…,n be commutative Banach subalgebras,we call they are independence of joint spectrum,if for any operator a_i∈A_i,i=1,2,…,n,there is: Sp(a₁,…,a_n)=σ(a₁)×…×σ(a_n). We have the following equivalent statements about independence of joint spectrum, Theorem 6 Let A_i,i=1,2,…,n be mutually commutative in B(H),then the following statements are equivalent:1.(A_i)_i=1ⁿ are independence of joint spectrum.2.For any positive operator A_i∈A_i,there is:Sp(A₁,…,A_n)=σ(A₁)×…×σ(A_n).3.For any element a_i∈A_i,i=1,2,…,n,then:4.Let A_i∈A_i,if A_i≠0,there is:Corollary 7 For any sub-C^* algebras,C^* independent is not weaker than independence of joint spectrum.Corollary 8 Let(M_n(C),φ) be a C^* probability space,whereφis faithful,and (A_i)_i=1^k are mutually commutative normal operators in M_n(C),If(C^*(A_i))_i=1^k are independent in(M_n(C),φ),then k≤[log₂n],where[x]is the least integer less than x.In chapter 4,we give characterizations of primeness of C^* algebra by use of joint spectrum.Theorem 9 Let A be a unital C^* algebra,then the following statements are equivalent:1.A is prime. 2.For any positive element of A,there is:Sp(L_A,R_B)=σ(A)×σ(B).3.For any positive operator tuples a=(a₁,…,a_n),b=(b₁,…,b_n) of A,there is:Sp(L_a,R_b)=Sp(L_a)×Sp(R_b).Corollary 10 Let A be a unital C^* algebra,then the following statements are equivalent:1.For any commutative positive element a=(a₁,…,a_n),b=(b₁,…,b_n) of A,there is:σ_B(L_a,R_b)=σ_B(L_a)×σ_B(R_b).where B is the Banach algebra generated by L_a,R_b,I.2.For any commutative positive element a=(a₁,…,a_n),b=(b₁,…,b_n) of A,there is:Sp(L_a,R_b)=Sp(L_a)×Sp(R_b).In the second part of chapter 3,we consider the problem of distance of the independent normal operator.Theorem 11 Let(C^*(A_i))_i=1ⁿ be C^* independent,then for any normal operator C_k∈C^*(A_k),D_l∈C^*(A_l),k,l=1,2,…,n,there is:In the last chapter,we consider the problem of the property of the direct of C^* independent algebras. Theorem 12 Let A_i and B_i be independent C^* algebras,then:A₁⊕…⊕A_n and B₁⊕…⊕B_n are conditional independent.