Semigroup C*-algebras

Transcription

1 Semigroup C*-algebras Xin Li Queen Mary University of London (QMUL)

2 Plan for the talks Plan for my talks: Constructions and examples Nuclearity and amenability K-theory Classification, and some questions

3 Plan for the talks Plan for my talks: Constructions and examples Nuclearity and amenability K-theory Classification, and some questions Reference: Survey paper Semigroup C*-algebras (arxiv).

4 Semigroup C*-algebras. The construction Let P be a left cancellative semigroup. Left cancellation means that for all p, x, y P, px = py implies x = y.

5 Semigroup C*-algebras. The construction Let P be a left cancellative semigroup. Left cancellation means that for all p, x, y P, px = py implies x = y. For p P, define V p : l 2 P l 2 P, δ x δ px.

6 Semigroup C*-algebras. The construction Let P be a left cancellative semigroup. Left cancellation means that for all p, x, y P, px = py implies x = y. For p P, define V p : l 2 P l 2 P, δ x δ px. A C*-algebra is an algebra of bounded linear operators on a Hilbert space, invariant under adjoints, closed in the norm topology.

7 Semigroup C*-algebras. The construction Let P be a left cancellative semigroup. Left cancellation means that for all p, x, y P, px = py implies x = y. For p P, define V p : l 2 P l 2 P, δ x δ px. A C*-algebra is an algebra of bounded linear operators on a Hilbert space, invariant under adjoints, closed in the norm topology. Definition C λ (P) := C ({V p : p P}) L(l 2 P)

8 Semigroup C*-algebras. Examples Positive cones: P = {x G: x e} in a left-ordered group (G, ), where is a total order on G with y x gy gx for all y, x, g G.

9 Semigroup C*-algebras. Examples Positive cones: P = {x G: x e} in a left-ordered group (G, ), where is a total order on G with y x gy gx for all y, x, g G. First example N Z: Cλ (N) is the Toeplitz algebra [Coburn].

10 Semigroup C*-algebras. Examples Positive cones: P = {x G: x e} in a left-ordered group (G, ), where is a total order on G with y x gy gx for all y, x, g G. First example N Z: Cλ (N) is the Toeplitz algebra [Coburn]. For G (R, +), P = G [0, ): Cλ (P) determines P [Douglas].

11 Semigroup C*-algebras. Examples Positive cones: P = {x G: x e} in a left-ordered group (G, ), where is a total order on G with y x gy gx for all y, x, g G. First example N Z: Cλ (N) is the Toeplitz algebra [Coburn]. For G (R, +), P = G [0, ): Cλ (P) determines P [Douglas]. What about positive cones in general left-ordered groups?

12 Semigroup C*-algebras. Examples Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V, E) be a graph with E V V. A + Γ := {σ v : v V } σ v σ w = σ w σ v (v, w) E +.

13 Semigroup C*-algebras. Examples Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V, E) be a graph with E V V. A + Γ := {σ v : v V } σ v σ w = σ w σ v (v, w) E +. e.g. N N: C λ (N N) = C λ (N) C λ (N).

14 Semigroup C*-algebras. Examples Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V, E) be a graph with E V V. A + Γ := {σ v : v V } σ v σ w = σ w σ v (v, w) E +. e.g. N N: C λ (N N) = C λ (N) C λ (N). e.g. N N: C λ (N N) = T 2, where T 2 is the Toeplitz extension of the Cuntz algebra O 2, 0 K T 2 O 2 0.

15 Semigroup C*-algebras. Examples Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V, E) be a graph with E V V. A + Γ := {σ v : v V } σ v σ w = σ w σ v (v, w) E +. e.g. N N: C λ (N N) = C λ (N) C λ (N). e.g. N N: C λ (N N) = T 2, where T 2 is the Toeplitz extension of the Cuntz algebra O 2, 0 K T 2 O 2 0. What about general graph products?

16 Semigroup C*-algebras. Examples Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V, E) be a graph with E V V. A + Γ := {σ v : v V } σ v σ w = σ w σ v (v, w) E +. e.g. N N: C λ (N N) = C λ (N) C λ (N). e.g. N N: C λ (N N) = T 2, where T 2 is the Toeplitz extension of the Cuntz algebra O 2, 0 K T 2 O 2 0. What about general graph products? What about general Artin monoids? e.g. Braid monoids, e.g. B + 3 = σ 1, σ 2 σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 +

17 Semigroup C*-algebras. Examples Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V, E) be a graph with E V V. A + Γ := {σ v : v V } σ v σ w = σ w σ v (v, w) E +. e.g. N N: C λ (N N) = C λ (N) C λ (N). e.g. N N: C λ (N N) = T 2, where T 2 is the Toeplitz extension of the Cuntz algebra O 2, 0 K T 2 O 2 0. What about general graph products? What about general Artin monoids? e.g. Braid monoids, e.g. B + 3 = σ 1, σ 2 σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 + Baumslag-Solitar monoid [Spielberg]: a, b ab c = b d a + or a, b a = b d ab c + What about general graphs of semigroups?

18 Semigroup C*-algebras. Examples Right-angled Artin monoids [Crisp-Laca]: Let Γ = (V, E) be a graph with E V V. A + Γ := {σ v : v V } σ v σ w = σ w σ v (v, w) E +. e.g. N N: C λ (N N) = C λ (N) C λ (N). e.g. N N: C λ (N N) = T 2, where T 2 is the Toeplitz extension of the Cuntz algebra O 2, 0 K T 2 O 2 0. What about general graph products? What about general Artin monoids? e.g. Braid monoids, e.g. B + 3 = σ 1, σ 2 σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 + Baumslag-Solitar monoid [Spielberg]: a, b ab c = b d a + or a, b a = b d ab c + What about general graphs of semigroups? Thompson monoid x 0, x 1, x 2,... x n x k = x k x n+1 for k < n +

19 Semigroup C*-algebras. Examples R or R R for an integral domain R. R = R \ {0}, R R = R R as sets, multiplication given by (d, c)(b, a) = (d + cb, ca).

20 Semigroup C*-algebras. Examples R or R R for an integral domain R. R = R \ {0}, R R = R R as sets, multiplication given by (d, c)(b, a) = (d + cb, ca). e.g. R ring of algebraic integers in number field, e.g. Z[i] or Z[ζ] (ζ: root of unity)

21 Semigroup C*-algebras. Examples R or R R for an integral domain R. R = R \ {0}, R R = R R as sets, multiplication given by (d, c)(b, a) = (d + cb, ca). e.g. R ring of algebraic integers in number field, e.g. Z[i] or Z[ζ] (ζ: root of unity) P Z n finitely generated.

22 Semigroup C*-algebras. Examples R or R R for an integral domain R. R = R \ {0}, R R = R R as sets, multiplication given by (d, c)(b, a) = (d + cb, ca). e.g. R ring of algebraic integers in number field, e.g. Z[i] or Z[ζ] (ζ: root of unity) P Z n finitely generated. e.g. numerical semigroups: P Z finitely generated, e.g. P = N \ {1} = {0, 2, 3, 4,...}.

23 Semigroup C*-algebras. Examples R or R R for an integral domain R. R = R \ {0}, R R = R R as sets, multiplication given by (d, c)(b, a) = (d + cb, ca). e.g. R ring of algebraic integers in number field, e.g. Z[i] or Z[ζ] (ζ: root of unity) P Z n finitely generated. e.g. numerical semigroups: P Z finitely generated, e.g. P = N \ {1} = {0, 2, 3, 4,...}. P G, e.g. Zappa-Szép products, e.g. from self-similar groups [Nekrashevych, Brownlowe-Ramagge-Robertson-Whittaker]

24 Inverse semigroups To construct full semigroup C*-algebras, we first need to isolate a class of isometric representations. We need the notion of inverse semigroups.

25 Inverse semigroups To construct full semigroup C*-algebras, we first need to isolate a class of isometric representations. We need the notion of inverse semigroups. Definition An inverse semigroup is a semigroup S with the property that for every x S, there is a unique y S with x = xyx and y = yxy. We write y = x 1 (or y = x ).

26 Inverse semigroups To construct full semigroup C*-algebras, we first need to isolate a class of isometric representations. We need the notion of inverse semigroups. Definition An inverse semigroup is a semigroup S with the property that for every x S, there is a unique y S with x = xyx and y = yxy. We write y = x 1 (or y = x ). Every inverse semigroup can be realized as partial bijections on a fixed set, where multiplication is given by composition (wherever it makes sense);

27 Inverse semigroups To construct full semigroup C*-algebras, we first need to isolate a class of isometric representations. We need the notion of inverse semigroups. Definition An inverse semigroup is a semigroup S with the property that for every x S, there is a unique y S with x = xyx and y = yxy. We write y = x 1 (or y = x ). Every inverse semigroup can be realized as partial bijections on a fixed set, where multiplication is given by composition (wherever it makes sense);... as partial isometries on a Hilbert space whose source and range projections commute, where multiplication is composition.

28 The left inverse hull Let P be a left cancellative semigroup with identity e.

29 The left inverse hull Let P be a left cancellative semigroup with identity e. Definition The left inverse hull I l (P) of P is the inverse semigroup generated by the partial bijections P pp, x px.

30 The left inverse hull Let P be a left cancellative semigroup with identity e. Definition The left inverse hull I l (P) of P is the inverse semigroup generated by the partial bijections P pp, x px. So I l (P) is the smallest semigroup of partial bijections on P which is closed under inverses and contains {P pp, x px: p P}.

31 The left inverse hull Let P be a left cancellative semigroup with identity e. Definition The left inverse hull I l (P) of P is the inverse semigroup generated by the partial bijections P pp, x px. So I l (P) is the smallest semigroup of partial bijections on P which is closed under inverses and contains {P pp, x px: p P}. Alternatively, I l (P) is the smallest semigroup of partial isometries on l 2 P which is closed under adjoints and contains {V p : p P}.

32 Idempotents in inverse semigroups Let S be an inverse semigroup. Definition E := { x 1 x: x S } = { xx 1 : x S } = { e S: e = e 2} is the semilattice of idempotents of S. For e, f E, set e f if e = ef.

33 Idempotents in inverse semigroups Let S be an inverse semigroup. Definition E := { x 1 x: x S } = { xx 1 : x S } = { e S: e = e 2} is the semilattice of idempotents of S. For e, f E, set e f if e = ef. For I l (P), E = { qn 1 p n q 1 1 p 1P: q i, p i P } =: J P. J P always contains all principal right ideals pp, p P, and if we have J P \ { } = {pp: p P}, then we say that P is right LCM.

34 Idempotents in inverse semigroups Let S be an inverse semigroup. Definition E := { x 1 x: x S } = { xx 1 : x S } = { e S: e = e 2} is the semilattice of idempotents of S. For e, f E, set e f if e = ef. For I l (P), E = { qn 1 p n q 1 1 p 1P: q i, p i P } =: J P. J P always contains all principal right ideals pp, p P, and if we have J P \ { } = {pp: p P}, then we say that P is right LCM. Ê = {χ : E {0, 1} non-zero semigroup homomorphism}, with the topology of pointwise convergence, is the spectrum of E. Ê can be identified with the space of filters via χ χ 1 (1).

35 Idempotents in inverse semigroups Let S be an inverse semigroup. Definition E := { x 1 x: x S } = { xx 1 : x S } = { e S: e = e 2} is the semilattice of idempotents of S. For e, f E, set e f if e = ef. For I l (P), E = { qn 1 p n q 1 1 p 1P: q i, p i P } =: J P. J P always contains all principal right ideals pp, p P, and if we have J P \ { } = {pp: p P}, then we say that P is right LCM. Ê = {χ : E {0, 1} non-zero semigroup homomorphism}, with the topology of pointwise convergence, is the spectrum of E. Ê can be identified with the space of filters via χ χ 1 (1). Exercise: Work out J P and ĴP for P = N N.

36 C*-algebras of inverse semigroups

37 C*-algebras of inverse semigroups Let S be an inverse semigroup. For s S, define λ s : l 2 S l 2 S by δ x δ sx if s 1 s xx 1 and δ x 0 otherwise.

38 C*-algebras of inverse semigroups Let S be an inverse semigroup. For s S, define λ s : l 2 S l 2 S by δ x δ sx if s 1 s xx 1 and δ x 0 otherwise. Definition C λ (S) := C ({λ s : s S}) L(l 2 S).

39 C*-algebras of inverse semigroups Let S be an inverse semigroup. For s S, define λ s : l 2 S l 2 S by δ x δ sx if s 1 s xx 1 and δ x 0 otherwise. Definition C λ (S) := C ({λ s : s S}) L(l 2 S). (If 0 S, replace l 2 S by l 2 S, where S = S \ {0}.)

40 C*-algebras of inverse semigroups Let S be an inverse semigroup. For s S, define λ s : l 2 S l 2 S by δ x δ sx if s 1 s xx 1 and δ x 0 otherwise. Definition C λ (S) := C ({λ s : s S}) L(l 2 S). (If 0 S, replace l 2 S by l 2 S, where S = S \ {0}.) The full inverse semigroup C*-algebra is given by Definition C (S) := C ( {v s } s S v s v t = v st, vs = v s 1 (and v 0 = 0 if 0 S) ).

41 C*-algebras of inverse semigroups Let S be an inverse semigroup. For s S, define λ s : l 2 S l 2 S by δ x δ sx if s 1 s xx 1 and δ x 0 otherwise. Definition C λ (S) := C ({λ s : s S}) L(l 2 S). (If 0 S, replace l 2 S by l 2 S, where S = S \ {0}.) The full inverse semigroup C*-algebra is given by Definition C (S) := C ( {v s } s S v s v t = v st, vs = v s 1 (and v 0 = 0 if 0 S) ). By construction, there is a homomorphism C (S) C λ (S), v s λ s.

42 Full semigroup C*-algebras

43 Full semigroup C*-algebras Let P be a left cancellative semigroup. Its full semigroup C*-algebra is Definition C (P) := C (I l (P)).

44 Full semigroup C*-algebras Let P be a left cancellative semigroup. Its full semigroup C*-algebra is Definition C (P) := C (I l (P)). Note: We say that 0 I l (P) if lies in I l (P), i.e., J P.

45 Full semigroup C*-algebras Let P be a left cancellative semigroup. Its full semigroup C*-algebra is Definition C (P) := C (I l (P)). Note: We say that 0 I l (P) if lies in I l (P), i.e., J P. By universal property, there is a canonical homomorphism C (P) Cλ (P) called the left regular representation.

46 Full semigroup C*-algebras Let P be a left cancellative semigroup. Its full semigroup C*-algebra is Definition C (P) := C (I l (P)). Note: We say that 0 I l (P) if lies in I l (P), i.e., J P. By universal property, there is a canonical homomorphism C (P) Cλ (P) called the left regular representation. Examples: C (N) = C (v v v = 1).

47 Full semigroup C*-algebras Let P be a left cancellative semigroup. Its full semigroup C*-algebra is Definition C (P) := C (I l (P)). Note: We say that 0 I l (P) if lies in I l (P), i.e., J P. By universal property, there is a canonical homomorphism C (P) Cλ (P) called the left regular representation. Examples: C (N) = C (v v v = 1). C (N N) = C (v a, v b v a v a = 1 = v b v b, v a v b = v b v a, v a v b = v b v a ).

48 Full semigroup C*-algebras Let P be a left cancellative semigroup. Its full semigroup C*-algebra is Definition C (P) := C (I l (P)). Note: We say that 0 I l (P) if lies in I l (P), i.e., J P. By universal property, there is a canonical homomorphism C (P) Cλ (P) called the left regular representation. Examples: C (N) = C (v v v = 1). C (N N) = C (v a, v b v a v a = 1 = v b v b, v a v b = v b v a, v a v b = v b v a ). C (N N) = C (v a, v b v a v a = 1 = v b v b, v a v a v b v b = 0).

49 Full semigroup C*-algebras

50 Full semigroup C*-algebras If P is right reversible, i.e., Pp Pq for all p, q P, or if P is right LCM, then C (P) = C {e X : X J P } {v p : p P} e X = e X = e 2 X ; v p v p = 1; e = 0 if J P, e P = 1, e X Y = e X e Y ; v pq = v p v q ; v p e X v p = e px

51 Full semigroup C*-algebras If P is right reversible, i.e., Pp Pq for all p, q P, or if P is right LCM, then C (P) = C {e X : X J P } {v p : p P} e X = e X = e 2 X ; v p v p = 1; e = 0 if J P, e P = 1, e X Y = e X e Y ; v pq = v p v q ; v p e X v p = e px So C (P) = D(P) α P, where D(P) = C ({e X : X J P }) C (P), and α p : D(P) D(P), e X e px.