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SUMMARY:Julio Cáceres (Vanderbilt University)
DTSTART:20261118T200000Z
DTEND:20261118T210000Z
DTSTAMP:20261111T024539Z
UID:NYC-NCG/231
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NYC-NCG/231/
 ">What becomes of continuous dimension in a Subfactor?</a>\nby Julio Cáce
 res (Vanderbilt University) as part of Noncommutative geometry in NYC\n\n\
 nAbstract\nIn a II_1 factor the trace gives projections a continuous dimen
 sion: every value in [0\,1] occurs. Replace the scalars by a subfactor N 
 ⊂ M\, and the conditional expectation E_N(p) becomes a "dimension of p r
 elative to N\," now taking values in N. Which positive contractions of N a
 rise this way? In finite index\, the Pimsner–Popa inequality rules out m
 ost of them\, and the question reduces to Popa's relative dimension set Λ
 (M\,N)\, the scalars α with E_N(p) = α1. Popa computed Λ(M\,N) in 1989 
 except on a middle interval that opens up once the index exceeds 4\, and t
 hat interval has remained open since.\nI will explain two results. First\,
  in joint work with Dietmar Bisch\, we show that if N ⊂ M has uniform in
 finite index\, then every positive contraction of N is the expectation of 
 a projection. In particular this holds for crossed products by infinite gr
 oups and for L(H) ⊂ L(G) with [G:H] = ∞. Second\, for every extremal i
 nclusion of finite index d > 4\, the whole middle interval lies in Λ(M\,N
 ). Together\, these show that Λ(M\,N) depends only on the index for every
  irreducible inclusion of II₁ factors.\n
LOCATION:https://researchseminars.org/talk/NYC-NCG/231/
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