Semigroup C*-algebras Xin Li Queen Mary University of London (QMUL) Reference: Survey paper \Semigroup C*-algebras" (arXiv). Plan for the talks Plan for my talks: I Constructions and examples I Nuclearity and amenability I K-theory I Classification, and some questions Plan for the talks Plan for my talks: I Constructions and examples I Nuclearity and amenability I K-theory I Classification, and some questions Reference: Survey paper \Semigroup C*-algebras" (arXiv). 2 2 For p 2 P, define Vp : ` P ! ` P; δx 7! δpx . A C*-algebra is an algebra of bounded linear operators on a Hilbert space, invariant under adjoints, closed in the norm topology. Definition ∗ ∗ 2 Cλ (P) := C (fVp: p 2 Pg) ⊆ L(` P) Semigroup C*-algebras. The construction Let P be a left cancellative semigroup. Left cancellation means that for all p; x; y 2 P, px = py implies x = y. A C*-algebra is an algebra of bounded linear operators on a Hilbert space, invariant under adjoints, closed in the norm topology. Definition ∗ ∗ 2 Cλ (P) := C (fVp: p 2 Pg) ⊆ L(` P) Semigroup C*-algebras. The construction Let P be a left cancellative semigroup. Left cancellation means that for all p; x; y 2 P, px = py implies x = y. 2 2 For p 2 P, define Vp : ` P ! ` P; δx 7! δpx . Definition ∗ ∗ 2 Cλ (P) := C (fVp: p 2 Pg) ⊆ L(` P) Semigroup C*-algebras. The construction Let P be a left cancellative semigroup. Left cancellation means that for all p; x; y 2 P, px = py implies x = y. 2 2 For p 2 P, define Vp : ` P ! ` P; δx 7! δpx . A C*-algebra is an algebra of bounded linear operators on a Hilbert space, invariant under adjoints, closed in the norm topology. Semigroup C*-algebras. The construction Let P be a left cancellative semigroup. Left cancellation means that for all p; x; y 2 P, px = py implies x = y. 2 2 For p 2 P, define Vp : ` P ! ` P; δx 7! δpx . A C*-algebra is an algebra of bounded linear operators on a Hilbert space, invariant under adjoints, closed in the norm topology. Definition ∗ ∗ 2 Cλ (P) := C (fVp: p 2 Pg) ⊆ L(` P) ∗ First example N ⊆ Z: Cλ (N) is the Toeplitz algebra [Coburn]. ∗ For G ⊆ (R; +), P = G \ [0; 1): Cλ (P) determines P [Douglas]. What about positive cones in general left-ordered groups? Semigroup C*-algebras. Examples I Positive cones: P = fx 2 G: x ≥ eg in a left-ordered group (G; ≥), where ≥ is a total order on G with y ≥ x ) gy ≥ gx for all y; x; g 2 G. ∗ For G ⊆ (R; +), P = G \ [0; 1): Cλ (P) determines P [Douglas]. What about positive cones in general left-ordered groups? Semigroup C*-algebras. Examples I Positive cones: P = fx 2 G: x ≥ eg in a left-ordered group (G; ≥), where ≥ is a total order on G with y ≥ x ) gy ≥ gx for all y; x; g 2 G. ∗ First example N ⊆ Z: Cλ (N) is the Toeplitz algebra [Coburn]. What about positive cones in general left-ordered groups? Semigroup C*-algebras. Examples I Positive cones: P = fx 2 G: x ≥ eg in a left-ordered group (G; ≥), where ≥ is a total order on G with y ≥ x ) gy ≥ gx for all y; x; g 2 G. ∗ First example N ⊆ Z: Cλ (N) is the Toeplitz algebra [Coburn]. ∗ For G ⊆ (R; +), P = G \ [0; 1): Cλ (P) determines P [Douglas]. Semigroup C*-algebras. Examples I Positive cones: P = fx 2 G: x ≥ eg in a left-ordered group (G; ≥), where ≥ is a total order on G with y ≥ x ) gy ≥ gx for all y; x; g 2 G. ∗ First example N ⊆ Z: Cλ (N) is the Toeplitz algebra [Coburn]. ∗ For G ⊆ (R; +), P = G \ [0; 1): Cλ (P) determines P [Douglas]. What about positive cones in general left-ordered groups? ∗ ∼ ∗ ∗ e.g. N × N: Cλ (N × N) = Cλ (N) ⊗ Cλ (N). ∗ ∼ e.g. N ∗ N: Cλ (N ∗ N) = T2, where T2 is the Toeplitz extension of the Cuntz algebra O2, 0 !K!T2 !O2 ! 0. What about general graph products? What about general Artin monoids? + + e.g. Braid monoids, e.g. B3 = hσ1; σ2 j σ1σ2σ1 = σ2σ1σ2i I Baumslag-Solitar monoid [Spielberg]: + + a; b j abc = bd a or a; b j a = bd abc What about general graphs of semigroups? + I Thompson monoid hx0; x1; x2;::: j xnxk = xk xn+1 for k < ni Semigroup C*-algebras. Examples I Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V ; E) be a graph with E ⊆ V × V . + + AΓ := hfσv : v 2 V g j σv σw = σw σv 8 (v; w) 2 Ei . ∗ ∼ e.g. N ∗ N: Cλ (N ∗ N) = T2, where T2 is the Toeplitz extension of the Cuntz algebra O2, 0 !K!T2 !O2 ! 0. What about general graph products? What about general Artin monoids? + + e.g. Braid monoids, e.g. B3 = hσ1; σ2 j σ1σ2σ1 = σ2σ1σ2i I Baumslag-Solitar monoid [Spielberg]: + + a; b j abc = bd a or a; b j a = bd abc What about general graphs of semigroups? + I Thompson monoid hx0; x1; x2;::: j xnxk = xk xn+1 for k < ni Semigroup C*-algebras. Examples I Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V ; E) be a graph with E ⊆ V × V . + + AΓ := hfσv : v 2 V g j σv σw = σw σv 8 (v; w) 2 Ei . ∗ ∼ ∗ ∗ e.g. N × N: Cλ (N × N) = Cλ (N) ⊗ Cλ (N). What about general graph products? What about general Artin monoids? + + e.g. Braid monoids, e.g. B3 = hσ1; σ2 j σ1σ2σ1 = σ2σ1σ2i I Baumslag-Solitar monoid [Spielberg]: + + a; b j abc = bd a or a; b j a = bd abc What about general graphs of semigroups? + I Thompson monoid hx0; x1; x2;::: j xnxk = xk xn+1 for k < ni Semigroup C*-algebras. Examples I Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V ; E) be a graph with E ⊆ V × V . + + AΓ := hfσv : v 2 V g j σv σw = σw σv 8 (v; w) 2 Ei . ∗ ∼ ∗ ∗ e.g. N × N: Cλ (N × N) = Cλ (N) ⊗ Cλ (N). ∗ ∼ e.g. N ∗ N: Cλ (N ∗ N) = T2, where T2 is the Toeplitz extension of the Cuntz algebra O2, 0 !K!T2 !O2 ! 0. What about general Artin monoids? + + e.g. Braid monoids, e.g. B3 = hσ1; σ2 j σ1σ2σ1 = σ2σ1σ2i I Baumslag-Solitar monoid [Spielberg]: + + a; b j abc = bd a or a; b j a = bd abc What about general graphs of semigroups? + I Thompson monoid hx0; x1; x2;::: j xnxk = xk xn+1 for k < ni Semigroup C*-algebras. Examples I Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V ; E) be a graph with E ⊆ V × V . + + AΓ := hfσv : v 2 V g j σv σw = σw σv 8 (v; w) 2 Ei . ∗ ∼ ∗ ∗ e.g. N × N: Cλ (N × N) = Cλ (N) ⊗ Cλ (N). ∗ ∼ e.g. N ∗ N: Cλ (N ∗ N) = T2, where T2 is the Toeplitz extension of the Cuntz algebra O2, 0 !K!T2 !O2 ! 0. What about general graph products? I Baumslag-Solitar monoid [Spielberg]: + + a; b j abc = bd a or a; b j a = bd abc What about general graphs of semigroups? + I Thompson monoid hx0; x1; x2;::: j xnxk = xk xn+1 for k < ni Semigroup C*-algebras. Examples I Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V ; E) be a graph with E ⊆ V × V . + + AΓ := hfσv : v 2 V g j σv σw = σw σv 8 (v; w) 2 Ei . ∗ ∼ ∗ ∗ e.g. N × N: Cλ (N × N) = Cλ (N) ⊗ Cλ (N). ∗ ∼ e.g. N ∗ N: Cλ (N ∗ N) = T2, where T2 is the Toeplitz extension of the Cuntz algebra O2, 0 !K!T2 !O2 ! 0. What about general graph products? What about general Artin monoids? + + e.g. Braid monoids, e.g. B3 = hσ1; σ2 j σ1σ2σ1 = σ2σ1σ2i + I Thompson monoid hx0; x1; x2;::: j xnxk = xk xn+1 for k < ni Semigroup C*-algebras. Examples I Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V ; E) be a graph with E ⊆ V × V . + + AΓ := hfσv : v 2 V g j σv σw = σw σv 8 (v; w) 2 Ei . ∗ ∼ ∗ ∗ e.g. N × N: Cλ (N × N) = Cλ (N) ⊗ Cλ (N). ∗ ∼ e.g. N ∗ N: Cλ (N ∗ N) = T2, where T2 is the Toeplitz extension of the Cuntz algebra O2, 0 !K!T2 !O2 ! 0. What about general graph products? What about general Artin monoids? + + e.g. Braid monoids, e.g. B3 = hσ1; σ2 j σ1σ2σ1 = σ2σ1σ2i I Baumslag-Solitar monoid [Spielberg]: + + a; b j abc = bd a or a; b j a = bd abc What about general graphs of semigroups? Semigroup C*-algebras. Examples I Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V ; E) be a graph with E ⊆ V × V . + + AΓ := hfσv : v 2 V g j σv σw = σw σv 8 (v; w) 2 Ei . ∗ ∼ ∗ ∗ e.g. N × N: Cλ (N × N) = Cλ (N) ⊗ Cλ (N). ∗ ∼ e.g. N ∗ N: Cλ (N ∗ N) = T2, where T2 is the Toeplitz extension of the Cuntz algebra O2, 0 !K!T2 !O2 ! 0. What about general graph products? What about general Artin monoids? + + e.g. Braid monoids, e.g. B3 = hσ1; σ2 j σ1σ2σ1 = σ2σ1σ2i I Baumslag-Solitar monoid [Spielberg]: + + a; b j abc = bd a or a; b j a = bd abc What about general graphs of semigroups? + I Thompson monoid hx0; x1; x2;::: j xnxk = xk xn+1 for k < ni e.g. R ring of algebraic integers in number field, e.g. Z[i] or Z[ζ](ζ: root of unity) n I P ⊆ Z finitely generated.