What becomes of continuous dimension in a Subfactor?

Abstract: In a II_1 factor the trace gives projections a continuous dimension: 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 relative to N," now taking values in N. Which positive contractions of N arise this way? In finite index, the Pimsner–Popa inequality rules out most 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 that interval has remained open since. I will explain two results. First, in joint work with Dietmar Bisch, we show that if N ⊂ M has uniform infinite index, then every positive contraction of N is the expectation of a projection. In particular this holds for crossed products by infinite groups and for L(H) ⊂ L(G) with [G:H] = ∞. Second, for every extremal inclusion 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.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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Noncommutative geometry in NYC

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