Extensions Of Locally Compact Quantum Metric Spaces

In this thesis, we study the extension and inverse problems of quantum metric spaces in the framework of locally compact quantum metric space. We divide this thesis into four chapters.In the first chapter we introduce the background and some basics on compact quantum metric spaces.In the second chapter we introduce the motivation, the definition and some properties of locally compact quantum metric spaces. Particularly we introduce the relationship between locally compact quantum metric spaces and compact quantum metric spaces.In the third chapter we generalize some results on quotients and ideals of compact quantum metric spaces to the case of locally compact quantum metric spaces:（1） Let A be a separable C*-algebra without identity. Assume that I is a closed two-sided ideal and Ï€ is the quotient map from A to A/I. When （A, L. B） is a locally compact quantum metric space, we define a seminorm LA/I on A/I and prove that （A/I,LA/I,π（B）） is locally compact quantum metric space.（2） Let A be separable unital C*-algebra, and let （A, L） be a compact quantum metric space. Assume that I is an ideal of A and has not identity as C*-algebra. Let LI be the restriction of L in I. Suppose that B is an abelian C*-subalgebra of I and contains an approximation unit element of I. We prove that （I,LI,B） is a locally compact quantum metric space.In the fourth chapter, we consider the expansion problem of locally compact quantum metric spaces. For a C*-algebra exact sequence 0 →A₀→lA₁→πA₂→₀ which is positive linear split exact at A₁, if there are Lipschit triples （A₀,L₀, B₀） and （A₂,L₂, B₂） on A₀ and A₂, respectively, we can construct a Lipschitz triple （A₁, L₁, B₁） on A₁. When （A₀,L₀） is a compact quantum metric space and （A₂, L₂, B₂） is a locally compact quantum metric space, we prove that （A₁, L₁,B₁） is a locally compact quantum metric space.