Coarse geometry and quantum groups

Transcription

1 Coarse geometry and quantum groups University of Glasgow Sheffield March 27, 2013

2 Noncommutative discrete spaces

3 Noncommutative discrete spaces Definition A noncommutative discrete space X is a triple (Irr(X ), C c (X ), φ) where Irr(X ) is a set, C c (X ) is a complex -algebra of the form C c (X ) = alg- where n x N for all x Irr(X ), x Irr(X ) M nx (C) φ : C c (X ) C is a faithful positive linear functional.

4 Noncommutative discrete spaces

5 Noncommutative discrete spaces We use the notation in the sequel. C 0 (X ) = C - l (X ) = l - C(X ) = x Irr(X ) x Irr(X ) x Irr(X ) M nx (C) M nx (C) M nx (C) We denote by l 2 (X ) the Hilbert space completion of C c (X ) with respect to φ.

6 Noncommutative discrete spaces We will be interested in certain bounded operators on l 2 (X ).

7 Noncommutative discrete spaces We will be interested in certain bounded operators on l 2 (X ). Notice first that any bounded operator T L(l 2 (X )) determines an (operator-valued) matrix (T x,y ) x,y Irr(X ) where T x,y = p x Tp y L(M ny (C), M nx (C)). Here p z L(l 2 (X )) for z Irr(X ) is the orthogonal projection onto M nx (C) l 2 (X ).

8 Noncommutative discrete spaces Using the identification ι φ : M ny (C) = M ny (C) given by ι φ (f )(g) = φ(fg) we shall sometimes identify T = (T x,y ) x,y Irr(X ) with its kernel, that is, with the corresponding element K T C(X X ) = M nx (C) M ny (C) in the sequel. x,y Irr(X )

9 Noncommutative discrete spaces We say that K = (K x,y ) C(X X ) is a finite kernel if K defines a bounded operator on l 2 (X ), that is, K = K T for some T L(l 2 (X )) K is row-finite and column-finite, that is, for every x Irr(X ) K x,y 0, K z,x 0 for only finitely many y, z Irr(X ).

10 Noncommutative discrete spaces Let K, L C(X X ) be finite kernels. We write (K L) x,z = K x,y L y,z, σ(k) x,y = Ky,x y Irr(X ) for their composition and for the adjoint, respectively.

11 Noncommutative discrete spaces Let K, L C(X X ) be finite kernels. We write (K L) x,z = y Irr(X ) K x,y L y,z, σ(k) x,y = K y,x for their composition and for the adjoint, respectively. These operations correspond to composition and taking adjoints of operators. That is, K R T = K R K T, K T = σ(k T ), for R, T L(l 2 (X )).

12 Coarse structures

13 Coarse structures Definition Let X = (Irr(X ), C c (X ), φ) be a noncommutative discrete space. A coarse structure for X is a collection E of linear subspaces of C(X X ), called controlled subspaces, consisting of finite kernels such that If E E and F E then F E. If E 1, E 2 E then E 1 + E 2 E. If E E then σ(e) E. If E 1, E 2 E then E 1 E 2 E. The space C c (X X ) is contained in E and all kernels corresponding to operators in Z(l (X )) are contained in E. A noncommutative coarse space is a noncommutative set X equipped with a coarse structure.

14 The uniform Roe algebra By the definition of coarse structures, the collection of all operators associated to kernels in E for some E E forms a -subalgebra C u (X ) of L(l 2 (X )). Definition Let (X, E) be a noncommutative coarse space. The uniform Roe algebra C u (X ) L(l 2 (X )) is the C -algebra obtained as the norm closure of C u (X ). All these definitions obviously recover the standard definitions in the case that all matrix blocks in X have size one.

15 Discrete quantum groups

16 Discrete quantum groups Definition A discrete quantum group G is given by a unital C -algebra S = Cred (G) together with a unital -homomorphism : S S S such that S S S id S S id S S S is commutative and (S)(1 S) and (S 1) (S) are dense subspaces of S S.

17 Example: Discrete groups If G is a discrete group then S = Cred (G) defines a discrete quantum group.

18 Example: Discrete groups If G is a discrete group then S = Cred (G) defines a discrete quantum group. The comultiplication : Cred (G) C red (G) C red (G) is given by (s) = s s for s G C[G] C red (G).

19 Example: Compact groups If G is a compact group then S = C(G) determines a discrete quantum group.

20 Example: Compact groups If G is a compact group then S = C(G) determines a discrete quantum group. The comultiplication : C(G) C(G) C(G) = C(G G) is given by (f )(s, t) = f (st)

21 Example: Compact groups If G is a compact group then S = C(G) determines a discrete quantum group. The comultiplication : C(G) C(G) C(G) = C(G G) is given by (f )(s, t) = f (st) Every commutative discrete quantum group is of this form.

22 Example: The quantum group SU q (2)

23 Example: The quantum group SU q (2) Definition Fix q [ 1, 1] \ {0}.

24 Example: The quantum group SU q (2) Definition Fix q [ 1, 1] \ {0}. The C -algebra C(SU q (2)) is the universal C -algebra generated by elements α and γ satisfying the relations αγ = qγα, αγ = qγ α, γγ = γ γ, α α + γ γ = 1, αα + q 2 γγ = 1.

25 Example: The quantum group SU q (2) Definition Fix q [ 1, 1] \ {0}. The C -algebra C(SU q (2)) is the universal C -algebra generated by elements α and γ satisfying the relations αγ = qγα, αγ = qγ α, γγ = γ γ, α α + γ γ = 1, αα + q 2 γγ = 1. These relations are equivalent to saying that the fundamental matrix ( ) α qγ is unitary. γ α

26 Example: The quantum group SU q (2) If we write ( ) ( ) u11 u 12 α qγ = u 21 u 22 γ α then : C(SU q (2)) C(SU q (2)) C(SU q (2)) is given by (u ij ) = u i1 u 1j + u i2 u 2j.

27 Example: The quantum group SU q (2) If we write ( ) ( ) u11 u 12 α qγ = u 21 u 22 γ α then : C(SU q (2)) C(SU q (2)) C(SU q (2)) is given by (u ij ) = u i1 u 1j + u i2 u 2j. For q = 1 one obtains in this way the C -algebra C(SU(2)) of functions on SU(2) together with the group structure of SU(2).

28 Peter-Weyl theory

29 Peter-Weyl theory Let G be a discrete quantum group. A finite dimensional corepresentation of G is a unitary u π = (uij π) C red (G) M dim(π)(c) = M n (Cred (G)) such that (u π ij ) = n uik π uπ kj. k=1

30 Peter-Weyl theory Let G be a discrete quantum group. A finite dimensional corepresentation of G is a unitary u π = (uij π) C red (G) M dim(π)(c) = M n (Cred (G)) such that (u π ij ) = n uik π uπ kj. k=1 A corepresentation is called irreducible if (id T )u π = u π (id T ) for T M dim(π) (C) implies T C id.

31 Peter-Weyl theory Let G be a discrete quantum group. A finite dimensional corepresentation of G is a unitary u π = (uij π) C red (G) M dim(π)(c) = M n (Cred (G)) such that (u π ij ) = n uik π uπ kj. k=1 A corepresentation is called irreducible if (id T )u π = u π (id T ) for T M dim(π) (C) implies T C id. We write Irr(G) for the set of all isomorphism classes of irreducible corepresentations of G.

32 Peter-Weyl theory Theorem (Peter-Weyl) Let G be a discrete quantum group. There exists a canonical dense Hopf -subalgebra C[G] inside Cred (G) such that C c (G) = C[G] = M dim(π) (C) π Irr(G) Moreover C c (G) is a multiplier Hopf -algebra equipped with a faithful positive left invariant linear functional φ, uniquely determined up to a scalar. We may therefore view (Irr(G), C c (G), φ) as a noncommutative discrete space.

33 The standard coarse structure

34 The standard coarse structure Let G be a discrete quantum group.

35 The standard coarse structure Let G be a discrete quantum group. If F Irr(G) is a subset we write C[F ] C[G] for the linear span of all matrix coefficients of corepresentations in F.

36 The standard coarse structure Let G be a discrete quantum group. If F Irr(G) is a subset we write C[F ] C[G] for the linear span of all matrix coefficients of corepresentations in F. Let π : C 0 (G) L(l 2 (G)) and ˆπ : C red (G) L(l 2 (G)) be the left regular representations.

37 The standard coarse structure Let G be a discrete quantum group. If F Irr(G) is a subset we write C[F ] C[G] for the linear span of all matrix coefficients of corepresentations in F. Let π : C 0 (G) L(l 2 (G)) and ˆπ : C red (G) L(l 2 (G)) be the left regular representations. We consider as basic controlled subspaces in C(G G) the spaces of kernels of operators on l 2 (G) of the form ˆπ(x)π(f ) L(l 2 (G)) where x C[F ] for some finite set F Irr(G) and f Z(l (G)), the center of l (G).

38 The standard coarse structure Definition The standard coarse structure on G is the coarse structure generated by all basic controlled subspaces. We write C u (G) for the uniform Roe algebra associated to the standard coarse structure on G.

39 The standard coarse structure Definition The standard coarse structure on G is the coarse structure generated by all basic controlled subspaces. We write C u (G) for the uniform Roe algebra associated to the standard coarse structure on G. We always have G red C 0 (G) C u (G) G red l (G), and these inclusions are typically strict.

40 The Roe algebra and exactness

41 The Roe algebra and exactness Theorem Let G be a discrete quantum group. Then the following conditions are equivalent. G is exact. The uniform Roe algebra C u (G) is nuclear. We point out again that we have C 0 (G) red G C u (G) l (G) red G, and these inclusions are typically strict.