Tensor decompositions of II 1 factors arising Introduction - PowerPoint PPT Presentation

Tensor decompositions of II1 factors arising from AFP Groups R. de Santiago Tensor decompositions of II 1 factors arising Introduction Motivation Results from AFP Groups Product Rigidity Classification of Tensor Decomposition New Directions Rolando de Santiago Joint with Ionut Chifan and Wanchalerm Sukpicarnon UC Los Angeles West Coast Operator Algebras Symposium Seattle 5 Oct 2018

Tensor decompositions von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Definition Introduction M ⊆ B ( H ) is a von Neumann algebra if M is a unital, WOT Motivation Results Product Rigidity closed, ∗ -subalgebra. Classification of Tensor { x n } → x in WOT iff |� ( x n − x ) η, ξ �| → 0 for every η, ξ ∈ H Decomposition New Directions

Tensor decompositions von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Definition Introduction M ⊆ B ( H ) is a von Neumann algebra if M is a unital, WOT Motivation Results Product Rigidity closed, ∗ -subalgebra. Classification of Tensor { x n } → x in WOT iff |� ( x n − x ) η, ξ �| → 0 for every η, ξ ∈ H Decomposition New Directions Let M ⊆ B ( H ) be a von Neumann algebra. ◮ M a factor if Z ( M ) = M ′ ∩ M ∼ = C , where M ′ = { y ∈ B ( H ) : xy = yx ∀ x ∈ M } .

Tensor decompositions von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Definition Introduction M ⊆ B ( H ) is a von Neumann algebra if M is a unital, WOT Motivation Results Product Rigidity closed, ∗ -subalgebra. Classification of Tensor { x n } → x in WOT iff |� ( x n − x ) η, ξ �| → 0 for every η, ξ ∈ H Decomposition New Directions Let M ⊆ B ( H ) be a von Neumann algebra. ◮ M a factor if Z ( M ) = M ′ ∩ M ∼ = C , where M ′ = { y ∈ B ( H ) : xy = yx ∀ x ∈ M } . ◮ M is type II 1 if M is infinite dimensional and admits a normal faithful tracial state τ : M → C .

Tensor decompositions von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Definition Introduction M ⊆ B ( H ) is a von Neumann algebra if M is a unital, WOT Motivation Results Product Rigidity closed, ∗ -subalgebra. Classification of Tensor { x n } → x in WOT iff |� ( x n − x ) η, ξ �| → 0 for every η, ξ ∈ H Decomposition New Directions Let M ⊆ B ( H ) be a von Neumann algebra. ◮ M a factor if Z ( M ) = M ′ ∩ M ∼ = C , where M ′ = { y ∈ B ( H ) : xy = yx ∀ x ∈ M } . ◮ M is type II 1 if M is infinite dimensional and admits a normal faithful tracial state τ : M → C . ◮ M is a von Neumann algebra iff M ′′ = M (Murray von Neumann 36).

Tensor decompositions von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Definition Introduction M ⊆ B ( H ) is a von Neumann algebra if M is a unital, WOT Motivation Results Product Rigidity closed, ∗ -subalgebra. Classification of Tensor { x n } → x in WOT iff |� ( x n − x ) η, ξ �| → 0 for every η, ξ ∈ H Decomposition New Directions Let M ⊆ B ( H ) be a von Neumann algebra. ◮ M a factor if Z ( M ) = M ′ ∩ M ∼ = C , where M ′ = { y ∈ B ( H ) : xy = yx ∀ x ∈ M } . ◮ M is type II 1 if M is infinite dimensional and admits a normal faithful tracial state τ : M → C . ◮ M is a von Neumann algebra iff M ′′ = M (Murray von Neumann 36). Remark ◮ If M is type II 1 , then τ ( P ( M )) = [0 , 1] .

Tensor decompositions von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Definition Introduction M ⊆ B ( H ) is a von Neumann algebra if M is a unital, WOT Motivation Results Product Rigidity closed, ∗ -subalgebra. Classification of Tensor { x n } → x in WOT iff |� ( x n − x ) η, ξ �| → 0 for every η, ξ ∈ H Decomposition New Directions Let M ⊆ B ( H ) be a von Neumann algebra. ◮ M a factor if Z ( M ) = M ′ ∩ M ∼ = C , where M ′ = { y ∈ B ( H ) : xy = yx ∀ x ∈ M } . ◮ M is type II 1 if M is infinite dimensional and admits a normal faithful tracial state τ : M → C . ◮ M is a von Neumann algebra iff M ′′ = M (Murray von Neumann 36). Remark ◮ If M is type II 1 , then τ ( P ( M )) = [0 , 1] . ◮ If t > 0 then M t := pMp where p ∈ P ( M n ( C )¯ ⊗ M ) with ( τ n ⊗ τ )( p ) = t / n for n “large enough.”

Tensor decompositions The Hyperfinite II 1 Factor of II1 factors arising from AFP Groups R. de Santiago Introduction Let τ n : M n ( C ) → C be the normalize trace. Motivation Results Product Rigidity Classification of Tensor Decomposition New Directions

Tensor decompositions The Hyperfinite II 1 Factor of II1 factors arising from AFP Groups R. de Santiago Introduction Let τ n : M n ( C ) → C be the normalize trace. Motivation Results Product Rigidity ( M 2 ( C ) , τ 2 ) ֒ → ( M 4 ( C ) , τ 4 ) ֒ → ( M 8 ( C ) , τ 8 ) ֒ → · · · Classification of Tensor Decomposition � � New Directions x 0 x �→ 0 x

Tensor decompositions The Hyperfinite II 1 Factor of II1 factors arising from AFP Groups R. de Santiago Introduction Let τ n : M n ( C ) → C be the normalize trace. Motivation Results Product Rigidity ( M 2 ( C ) , τ 2 ) ֒ → ( M 4 ( C ) , τ 4 ) ֒ → ( M 8 ( C ) , τ 8 ) ֒ → · · · Classification of Tensor Decomposition � � New Directions x 0 x �→ 0 x WOT � R := ( M 2 ( C ) , τ 2 ) n ∈ N WOT � = ( M k ( C ) , τ k ) ∀ k ∈ N \ { 1 } n ∈ N WOT � ∀ { k n } ∈ ( N \ { 1 } ) N = ( M k n ( C ) , τ k n ) n ∈ N

Tensor decompositions Group von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Introduction Motivation Results Product Rigidity Γ discrete countable group � L (Γ) group von Neumann algebra Classification of Tensor Decomposition New Directions

Tensor decompositions Group von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Introduction Motivation Results Product Rigidity Γ discrete countable group � L (Γ) group von Neumann algebra Classification of Tensor ◮ Γ ֒ → U ( ℓ 2 (Γ)) by Decomposition γ · ( η )( λ ) := η ( γ − 1 λ ) ∀ η ∈ ℓ 2 (Γ) , γ, λ ∈ Γ. New Directions

Tensor decompositions Group von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Introduction Motivation Results Product Rigidity Γ discrete countable group � L (Γ) group von Neumann algebra Classification of Tensor ◮ Γ ֒ → U ( ℓ 2 (Γ)) by Decomposition γ · ( η )( λ ) := η ( γ − 1 λ ) ∀ η ∈ ℓ 2 (Γ) , γ, λ ∈ Γ. New Directions WOT = { γ } ′′ ◮ L (Γ) := C [Γ] γ ∈ Γ ⊆ B ( ℓ 2 (Γ))

Tensor decompositions Group von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Introduction Motivation Results Product Rigidity Γ discrete countable group � L (Γ) group von Neumann algebra Classification of Tensor ◮ Γ ֒ → U ( ℓ 2 (Γ)) by Decomposition γ · ( η )( λ ) := η ( γ − 1 λ ) ∀ η ∈ ℓ 2 (Γ) , γ, λ ∈ Γ. New Directions WOT = { γ } ′′ ◮ L (Γ) := C [Γ] γ ∈ Γ ⊆ B ( ℓ 2 (Γ)) ◮ τ : L (Γ) → C by τ ( x ) = � x δ e , δ e � .

Tensor decompositions Group von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Introduction Motivation Results Product Rigidity Γ discrete countable group � L (Γ) group von Neumann algebra Classification of Tensor ◮ Γ ֒ → U ( ℓ 2 (Γ)) by Decomposition γ · ( η )( λ ) := η ( γ − 1 λ ) ∀ η ∈ ℓ 2 (Γ) , γ, λ ∈ Γ. New Directions WOT = { γ } ′′ ◮ L (Γ) := C [Γ] γ ∈ Γ ⊆ B ( ℓ 2 (Γ)) ◮ τ : L (Γ) → C by τ ( x ) = � x δ e , δ e � . Remark ◮ For Σ ⊆ Γ, γ Σ = σγσ − 1 : σ ∈ Σ � � .

Tensor decompositions Group von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Introduction Motivation Results Product Rigidity Γ discrete countable group � L (Γ) group von Neumann algebra Classification of Tensor ◮ Γ ֒ → U ( ℓ 2 (Γ)) by Decomposition γ · ( η )( λ ) := η ( γ − 1 λ ) ∀ η ∈ ℓ 2 (Γ) , γ, λ ∈ Γ. New Directions WOT = { γ } ′′ ◮ L (Γ) := C [Γ] γ ∈ Γ ⊆ B ( ℓ 2 (Γ)) ◮ τ : L (Γ) → C by τ ( x ) = � x δ e , δ e � . Remark ◮ For Σ ⊆ Γ, γ Σ = σγσ − 1 : σ ∈ Σ � � . ◮ Γ is icc if | γ Γ | = ∞ for all γ ∈ Γ \ { e } .

Tensor decompositions Group von Neumann Algebras of II1 factors arising from AFP Groups R. de Santiago Introduction Motivation Results Product Rigidity Γ discrete countable group � L (Γ) group von Neumann algebra Classification of Tensor ◮ Γ ֒ → U ( ℓ 2 (Γ)) by Decomposition γ · ( η )( λ ) := η ( γ − 1 λ ) ∀ η ∈ ℓ 2 (Γ) , γ, λ ∈ Γ. New Directions WOT = { γ } ′′ ◮ L (Γ) := C [Γ] γ ∈ Γ ⊆ B ( ℓ 2 (Γ)) ◮ τ : L (Γ) → C by τ ( x ) = � x δ e , δ e � . Remark ◮ For Σ ⊆ Γ, γ Σ = σγσ − 1 : σ ∈ Σ � � . ◮ Γ is icc if | γ Γ | = ∞ for all γ ∈ Γ \ { e } . ◮ Γ icc iff L (Γ) is a II 1 factor.