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

Hiroshi Ando; Yasumichi Matsuzawa; Andreas Thom; Asger Dag Törnquist

doi:10.1515/crelle-2017-0047

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

Hiroshi Ando, Yasumichi Matsuzawa, Andreas Thom, Asger Dag Törnquist

Department of Mathematical Sciences

1 Citation (Scopus)

Abstract

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.

Original language	English
Journal	Journal fuer die Reine und Angewandte Mathematik
Number of pages	29
ISSN	0075-4102
DOIs	https://doi.org/10.1515/crelle-2017-0047
Publication status	Published - 2021

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

abstract = "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.",

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AU - Ando, Hiroshi

AU - Matsuzawa, Yasumichi

AU - Thom, Andreas

AU - Törnquist, Asger Dag

PY - 2021

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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.

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

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