Unitarizability, Maurey-Nikishin factorization, and Polish groups of finite type

TY - JOUR

T1 - Unitarizability, Maurey-Nikishin factorization, and Polish groups of finite type

AU - Ando, Hiroshi

AU - Matsuzawa, Yasumichi

AU - Thom, Andreas

AU - Törnquist, Asger Dag

PY - 2021

Y1 - 2021

N2 - Let Γ be a countable discrete group, and let πW Γ → GL(H) be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group G = H ⋈π Γ is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group U(ℓ2(ℕ))) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a II1-factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey-Nikishin factorization theorem for continuous maps from a Hilbert space to the space L0(X, m) of all measurable maps on a probability space.

AB - Let Γ be a countable discrete group, and let πW Γ → GL(H) be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group G = H ⋈π Γ is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group U(ℓ2(ℕ))) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a II1-factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey-Nikishin factorization theorem for continuous maps from a Hilbert space to the space L0(X, m) of all measurable maps on a probability space.

UR - http://www.scopus.com/inward/record.url?scp=85037690404&partnerID=8YFLogxK

U2 - 10.1515/crelle-2017-0047

DO - 10.1515/crelle-2017-0047

M3 - Journal article

SN - 0075-4102

JO - Journal fuer die Reine und Angewandte Mathematik

JF - Journal fuer die Reine und Angewandte Mathematik

ER -