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Higher symmetries in operator algebras

Chen, Quan
http://orcid.org/0000-0001-5853-5487
http://rave.ohiolink.edu/etdc/view?acc_num=osu1681856056654948

Abstract Details

2023, Doctor of Philosophy, Ohio State University, Mathematics.
This dissertation consists of two self-contained papers from my graduate work at Ohio State University. In Chapter 2, we review the definition of C*-algebras, von Neumann algebras, C*/W* categories/ 2-categories. Some of this background material is taken from [GLR85, CHPJP22]. We also review the 2-categories C*Alg/W*Alg of C*/W*-correspondences and Q-system realization construction taken directly from my articles [CHPJP22, CP22]. Chapter 3 is joint work with Corey Jones and David Penneys [CJP21]. First We discuss the construction of a unitary braided tensor category End_loc(C) from a given W*- category C. When C is the category of finitely generated projective modules over a type II1 factor M, the underlying tensor category of its dualizable part of End_loc(Mod_fgp(M)) is Connes’ bimodule version χ ̃(M) due to Popa. Second, for each unitary fusion category C, we construct a II1-factor M such that χ ̃(M) \cong Z(C). Chapter 4 is joint work with Roberto Hern ́andez Palomares and Corey Jones [CPJ22]. We introduce a K-theoretic invariant for actions of unitary fusion categories on unital C*-algebras. We show that for inductive limits of finite dimensional actions of fusion categories on AF-algebras, this is a complete invariant. In particular, this gives a complete invariant for inductive limit actions of finite groups on unital AF-algebras. We apply our results to obtain a classification of finite depth, strongly AF-inclusions of unital AF- algebras.
David Penneys (Advisor)
183 p.
Mathematics
C*-algebra, von Neumann algebra, subfactor, tensor category, 2-category, braided tensor category

Recommended Citations

Citations

  • Chen, Q. (2023). Higher symmetries in operator algebras [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1681856056654948

    APA Style (7th edition)

  • Chen, Quan. Higher symmetries in operator algebras. 2023. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1681856056654948.

    MLA Style (8th edition)

  • Chen, Quan. "Higher symmetries in operator algebras." Doctoral dissertation, Ohio State University, 2023. http://rave.ohiolink.edu/etdc/view?acc_num=osu1681856056654948

    Chicago Manual of Style (17th edition)

osu1681856056654948
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This open access ETD is published by The Ohio State University and OhioLINK.