Topology Graded C * - Algebra Diagonal Invariant Ideals And Inductive Limits,

Let A be a C^* — algebra ( not necessarily unital or commutative ), let {E_i}i∈I be an infinite set of Hilbert A-modules, then their direct sum E_i is still Hilbert A-modules. We defineL(E,E) = {t|t: E→E,(?)t^* : E→ E, such that (tx,y) = (x,t^*y), (?)x,y 6 E}, then L{E,E) is a C^*-algebra.Let J be a closed, two-sided ideal in a topologically graded C^*-algebra B = (?)B_t. DefineJ₁ = (J∩B_e), that is, the ideal generated by J∩B_e;J₂ = {b ∈ B|F_t(b) ∈ J,(?)t ∈ Γ}; J₃ = Ind J = {6 ∈B|F_e{b^*b) ∈ J}, then J₁ (?) J₂ = J₃, and J₃ is an ideal of B. Define B^∞ = Span{ b_t |b_t(?) B_t, t ∈ Γ}, we can draw a conclusion as follows: An ideal I of a topologically graded C^*— algebra B is diagonal invariant if and only if (?) I(?) IndJ, for certain ideal J of B.Let (G₁,E₁), (G₂,E₂) be two quasi-lattice ordered groups, denote T^E1,T^E2 are Toepiitz operator algebras correspondingly. Suppose (?) : G₁ → G₂ is a unital group homomorphism, such that (?){E₁) (?) E₂, then the natural homomorphism r^E2,E1 from Toepiitz algebra T^E1 onto T^E2 becomes a unital homomorphism of C^*—algebras if and only if the following conditions are all satisfied : (1) (?)(x) = (?)(y) => x = y, for any x,y ∈ E₁. (2) x(?) y E₁<=>(?)(x) (?) (?)(y) ∈ E₂, for any x,y ∈ E₁; and (?)(x (?)y) = (?)(x) (?) (?)(y), for any x,y ∈ E₁, with a common upper bound in E\. (3) (?)(G₁) ∩ E₂ = (?)(E₁). An application is given to the inductive limits of Toepiitz algebras.