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DTSTART;TZID=America/Toronto:20250729T100000
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URL:https://uwaterloo.ca/pure-mathematics/events/phd-thesis-defence-37
SUMMARY:PhD Thesis Defence
CLASS:PUBLIC
DESCRIPTION:LARISSA KROELL\, UNIVERSITY OF WATERLOO\n\n_Partial C*-dynamica
 l systems and the ideal structure of partial\nreduced crossed products_\n\
 nWe study C*-algebras stemming from partial C*-dynamical systems. We\ndeve
 lop equivariant injective envelopes associated to such systems\,\nwhich al
 low us to obtain canonical connections to enveloping actions\nas well as r
 esults on the ideal structure of partial crossed products.\n\nWe extend th
 e theory of equivariant injective envelopes pioneered by\nHamana in the 19
 80s to partial C*-dynamical systems. To do so\, we\nintroduce the category
  of generalized unital partial actions by\nallowing for *-automorphisms ac
 ting on families of special hereditary\nsubalgebras. Utilizing properties 
 of injective envelopes and the\nnotion of an injective unitization of part
 ial C*-dynamical systems\, we\nargue that it suffices to consider unital o
 bjects in our category.\nThis also allows us to connect our theory to Abad
 ie's notion of\nenveloping actions leading to a canonical relationship of 
 their\nG-injective envelopes. \n\nUtilizing properties of injective envelo
 pes\, we introduce novel\nnon-triviality conditions for partial *-automorp
 hisms inspired by\nglobal C*-dynamics. We contrast this notion with existi
 ng conditions\nin the literature. Lastly\, we study a non-commutative gene
 ralization\nof stabilizer subgroups for pseudo-Glimm ideals. In particular
 \, we\nshow that for Glimm ideals in the C*-injective envelope\, these\nst
 abilizer subgroups are in fact amenable --- a result which is\ncrucial for
  our main theorems regarding the ideal structure of partial\nreduced cross
 ed products.\n\nFinally\, our main application of the theory of G-injectiv
 e envelopes\nis a characterization of the ideal intersection property for 
 partial\nC*-dynamical systems subject to a cohomological condition as a\ng
 eneralization of the result for global group actions. To state this\ngener
 alization\, we utilize the dynamical conditions introduced\npreviously and
  generalize the notion of equivariant\npseudo-expectations to partial C*-d
 ynamical systems. We also give a\nsufficient intrinsic condition in terms 
 non-commutative uniformly\nrecurrent partial subsystems utilizing pseudo-G
 limm ideals. As a\nconsequence of our results\, we obtain a full character
 ization of the\nideal intersection property for partial actions on commuta
 tive\nC*-algebras in terms of freeness of the partial action on the spectr
 um\nof the G-injective envelope. \n\nMC5403
DTSTAMP:20260504T211825Z
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