The Induced Ideals Of Toeplitz Algebras

In this paper, we'll study Toeplitz algebras on quasi-latticed ordered groups. The paperconsists of four chapters. In chapter one, we'll recall some basic concepts, such as quasi-lattice ordered group, directed and hereditary set and the inductive limits of groups. Wemainly focus on the study of the inductive limits of quasi-latticed ordered groups. As shownby Xu and Zhang in [18], the inductive limits of quasi-latticed ordered groups will also bequasi-latticed ordered provided that certain conditions are satisfied. One important aspectof this topic is the study of the directed and hereditary sets, and this is carried out in Section1.3. Our main results in this chapter are Propositions 1.3.3 and 1.3.6.Ordered groups, free groups, and (Z~n, Z_+~n) all are quasi-lattice ordered, however exceptfor these well-known examples, it seems quite difficult to find other (non-abelian) quasi-latticed ordered groups. In chapter two, we'll construct a concrete quasi-lattice orderedgroup by choosing certain 2 by 2 upper triangular matrices. The detailed structure of S(H)is clarified, where S(H) stands for the closedθ-invariant subset generated by H, whichamount to eight cases. Based on this result, we then focus on the study of the induced idealsof certain Toeplitz algebras in Chapter three, where all induced ideals of the underlyingToeplitz algebras are given.In [18], Xu and Zhang studied on quasi-latticed ordered groups. Among other things, weproved that the inductive limits of Toeplitz algebras will be amenable if every element in thedirected system is. Replacing the single point set {e} by a general directed and hereditary setH, we have get a generalized version of the main results of [18]. For the details, see Theorem4.2.2 and Theorem4.2.3.