Quantum Metric Space Structures Of Twisted Group C~*-algebras

In this dissertation,we investigate the metric geometry of noncommutative compact spaces,and discuss the quantum metric space structures of twisted group C*-algebras.This dissertation contains four chapters,and the contents of the chapters of this disser-tation are as follows.In Chapter 1,we focus on the introduction to the backgrounds and some relevant basic concepts and preliminaries of this dissertation.In Chapter 2 we construct a new class of compact quantum metric spaces,using the twisted rapid decay property of discrete groups.We extend the classical rapid decay property and the twisted rapid decay property of finitely generated groups to a more general setting of the twisted rapid decay property of discrete groups,and give several equivalent characterizations.For any 2-cocycle ? on a discrete group ? using the operator Ml on l2(?)of the pointwise multiplication induced by a length function l on ?,we obtain a sequence of derivation {?k}k=1? on the twisted reduced group C*-algebra Cr*((?,?),and then we have a Lipschitz seminorm sequence {LDk}k=1? on the twisted reduced group C*-algebra Cr*(?,?).We show that when a discrete group ? has the ?-twisted rapid decay-property with respect to a proper length function l on ? for a 2-cocycle on ?,there exists a positive integer k0 such that for any integer k?k0,the seminorm LDk is a Lip-norn on Cr*(?,?),that is,(Cr*(?,?),LDk)is a compact quantum metric space.Consequently,from this theorem we can give the compact quantum metric space structures on the twisted reduced group C*-algebra Cr*(?,?)generated by hyperbolic groups,finitely generated free groups,finitely generated groups with polynomial growth and discrete Heisenberg groups and so on,respectively.In particular,we can endow the noncommutative tori with the compact quantum metric space structures.Moreover,we prove that the compact quantum metric space structures(Cr*(?,?),LDk)on the twisted reduced group C*-algebras Cr*(?,?)do not depend the choice of representative on the 2-cocycle cohomology class in the Lipschitz isometric sense,and hence they depend only on the 2-cocycle cohomology class.In Chapter 3,we construct a new class of compact quantum metric spaces,using the bounded ?-dilation property of the length function on discrete groups.We formulate a key inequality by the means of considering the twisted convolution operators from the left regular projective representation of Cc(?,?)on l2(?)as integral operators.Then we define a new seminorm JD,? on,(?,?)from this inequality,and prove that it depends only on the cohomology class of the 2-cocycle ?.Moreover,we truncate the twisted convo-lution operators from the left regular projective representation,and control the truncated operators with respect to the operator norm and the seminorm JD,?,respectively.We extend the classical polynomial growth conditions of the length function on a finitely gen-erated group to a more general setting of the bounded ?-dilation property of the length function on a discrete group.For a discrete group with a proper length function with the bounded ?-dilation property,we decompose the functions with finite support on this group into "three" parts,and dominate them by the operator norm and the seminorm JD,?,re-spectively.Using these results,we prove that for any 2-cocycle ? on a discrete group ?endowed with a length function l with the bounded ?-dilation property,(Cr*(?,?),LD)is a compact quantum metric space,and its compact quantum metric space structures depend only on the cohomology class of the 2-cocycle a in the Lipschitz isometric sense.In particular,for a finitely generated discrete group ? with polynomial growth(Namely,it is a nilpotent-by-finite group.)and any 2-cocycle ? on ?,the pair(Cr*(?,?),LD)is a compact quantum metric space.In Chapter 4 we discuss the isometry group of twisted reduced group C*-algebras with respect to the standard spectral triple from the point of view of topological groups.We give a sufficient and necessary condition that the triple(Cr*(?,?),l2(?),Ml)is a spectral triple,that is,l is a proper length function on ? with a discrete and unbounded range.For the spectral triple(Cr*(?,?),l2(?),Ml),we prove that the isometry group Iso(Cr*(?,?),l2((?),Ml)equipped with the point-norm topology is a topological group,and that it depends only on the cohomology class of the 2-cocycle ? in the sense of topological group isomorphism.Endowing the set Map(?,T)of all functions from a discrete group ? to the unit circle T and the automorphical group Autl(r)preserving a length function l on ? with the pointwise convergence topology,we prove that the isometry group Iso(Cr*(?,?),l2(?),Ml)and the semi-direct product topological group Map(r,T)×? Autl(r)are isomorphic for some special case of l.Moreover,we show that the isometry group Iso(Cr*(?,?),l2(?),Ml)is a compact topological group under the point-norm topology by embedding this isometry group into the direct product of the unitary operator groups of the eigenspace of l topologically.In particular,when ? is a finitely generated discrete group,the isometry group Iso(Cr*(?,?),l2(?),Ml)is a compact Lie group in the point-norm topology.More generally,we extend this result to a unital C*-algebra with a*-filtration,and prove that its isometry group with respect to the spectral triple from this*-filtration is a compact topology group under the point-norm topology.