Rokhlin Dimension for C*-correspondences

Open Access
Author:
Zelenberg, Aleksey
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
April 23, 2015
Committee Members:
  • Nathanial Patrick Brown, Dissertation Advisor
  • Nathanial Patrick Brown, Committee Chair
  • Nigel David Higson, Committee Member
  • John Roe, Committee Member
  • Martin Bojowald, Special Member
Keywords:
  • nuclear dimension
  • Pimsner algebra
  • Rokhlin dimension
  • crossed product
Abstract:
The notion of nuclear dimension for C*-algebras was defined by Winter and Zacharias as a noncommutative analog of covering dimension for topological spaces. In recent years nuclear dimension has generated a great deal of interest, not only due to its connection to other important structural properties of C*-algebras such as Jiang-Su stability and strict comparison, but also because it seems to be a unifying principle in the classification program of nuclear C*-algebras using K-theory. As such, much work as been done to understand how nuclear dimension behaves for various C*-constructions. Along these lines, Hirshberg, Winter, and Zacharias proved that if A is a C*-algebra having finite nuclear dimension, and if A is being acted on by an automorphism having finite Rokhlin dimension, then the associated crossed product has finite nuclear dimension. This thesis substantially generalizes this result. Indeed, since a crossed product by the integers can be regarded as a Cuntz-Pimsner algebra associated to a singly-generated C*-correspondence, we propose a definition of Rokhlin dimension for arbitrary C*-correspondences that agrees with the traditional one in the singly-generated case. We then show that in many cases (such as for finitely generated projective correspondences), finiteness of nuclear dimension for Pimsner algebras is preserved in the presence of finite Rokhlin dimension. We conclude by using these results to prove that certain types of amalgamated free products have finite nuclear dimension.