Topology Graded C * - Algebra Of The Same Ideal

This paper consists of three parts: Introductions of Hilbert C^*-modules, Topolog-ically graded Toeplitz cross sectional algebras over Fell bundles, The Primitive idealsand Invariant ideals of Toeplitz algebras.In the first charpter, we introduce the fundamental of Hilbert C^*-modules and theconcepts and basic terms which is ready for this paper.In the second charpter, we firstly introduce and study the Toeplitz cross sectionalalgebra on discrete groups, which is the natural generalization of group C^*-algebrasand Fell bundles. Given a Fell bundle B=(B_t)_t∈Γ over a discrete groupΓ. For anynon-empty subset E ofΓ, a Toeplitz cross sectional algebra T^E is introduced, and weprove that the Toeplitz cross sectional algebra T^E is topologically graded if the Fellbundle B=(B_t)_t∈Γ is regular with respect to the given subset E.In the third charper, we study two kinds of ideals of Toeplitz algebras: primitiveideals and invariant ideals. Let (G, G₊) be a discrete abelian ordered group, denote byT^G+ the associated Toeplitz algebra. The primitive ideals of T^G+ were firstly studiedby Adji and Raeburn in. In this paper, we'll give a much simpler proof of theirresult (Corollary 3.4). Another important aspect of the study about Toeplitzalgebras is their ideal structures. In this paper, we firstly introduce the concept ofΦ-invariant ideal and discuss the relationship among diagonal invariant ideals, invariantideals andΦ-invariant ideals of Toeplitz algebras. Specially, we give an example of aninvariant ideal of certain Toeplitz algebra, which is not diagonal invariant.