The faithful subalgebra

Transcription

1 joint work with Jonathan H. Brown, Gabriel Nagy, Aidan Sims, and Dana Williams funded in part by NSF DMS Classification of C -Algebras, etc. University of Louisiana at Lafayette May 11 15, 2015

2 Let G be a graph, k-graph, or groupoid, and C (G ) the universal C*-algebra defined from it. 1 an-huef, 97 2 Fowler-Kumjian-Pask-Raeburn, 97

3 Let G be a graph, k-graph, or groupoid, and C (G ) the universal C*-algebra defined from it. Question: Under what circumstances is a -homomorphism φ : C (G ) B(H) injective? 1 an-huef, 97 2 Fowler-Kumjian-Pask-Raeburn, 97

4 Let G be a graph, k-graph, or groupoid, and C (G ) the universal C*-algebra defined from it. Question: Under what circumstances is a -homomorphism φ : C (G ) B(H) injective? Classical theorems addressing this question assume either 1 an-huef, 97 2 Fowler-Kumjian-Pask-Raeburn, 97

5 Let G be a graph, k-graph, or groupoid, and C (G ) the universal C*-algebra defined from it. Question: Under what circumstances is a -homomorphism φ : C (G ) B(H) injective? Classical theorems addressing this question assume either (a) the existence of intertwining gauge actions" on the algebras (Gauge Invariant Uniqueness Theorem 1 ), or 1 an-huef, 97 2 Fowler-Kumjian-Pask-Raeburn, 97

6 Let G be a graph, k-graph, or groupoid, and C (G ) the universal C*-algebra defined from it. Question: Under what circumstances is a -homomorphism φ : C (G ) B(H) injective? Classical theorems addressing this question assume either (a) the existence of intertwining gauge actions" on the algebras (Gauge Invariant Uniqueness Theorem 1 ), or (b) an aperiodicity condition on the graph itself (Cuntz-Krieger Uniqueness Theorem 2 ), 1 an-huef, 97 2 Fowler-Kumjian-Pask-Raeburn, 97

7 Let G be a graph, k-graph, or groupoid, and C (G ) the universal C*-algebra defined from it. Question: Under what circumstances is a -homomorphism φ : C (G ) B(H) injective? Classical theorems addressing this question assume either (a) the existence of intertwining gauge actions" on the algebras (Gauge Invariant Uniqueness Theorem 1 ), or (b) an aperiodicity condition on the graph itself (Cuntz-Krieger Uniqueness Theorem 2 ), and conclude that φ is injective iff it is nondegenerate, i.e., injective on the diagonal subalgebra D. 1 an-huef, 97 2 Fowler-Kumjian-Pask-Raeburn, 97

8 Theorem (Brown-Nagy-R-Sims-Williams)

9 Theorem (Brown-Nagy-R-Sims-Williams) There is a canonical subalgebra M C (G ) such that a -homomorphism φ : C (G ) B(H) is injective iff φ M is injective.

10 Theorem (Brown-Nagy-R-Sims-Williams) There is a canonical subalgebra M C (G ) such that a -homomorphism φ : C (G ) B(H) is injective iff φ M is injective. [NR1] Nagy and Reznikoff, Abelian core of graph algebras, J. Lond. Math. Soc. (2) 85 (2012), no. 3, [NR2] Nagy and Reznikoff, Pseudo-diagonals and uniqueness theorems, Proc. AMS (2013). [BNR] Brown, Nagy, Reznikoff A generalized Cuntz-Krieger uniqueness theorem for higher-rank graphs, JFA (2013). [BNRSW] Brown, Nagy, Reznikoff, Sims, and Williams, Cartan subalgebras in C -algebras of Hausdorff étale groupoids (2015).

11 Drinen (1999): Every AF algebra is Morita equivalent to a graph algebra.

12 Drinen (1999): Every AF algebra is Morita equivalent to a graph algebra. Spielberg k-graphs can be used to construct any UCT Kirchberg algebra.

13 Drinen (1999): Every AF algebra is Morita equivalent to a graph algebra. Spielberg k-graphs can be used to construct any UCT Kirchberg algebra. Hong-Szymański (2004): the ideal structure of the algebra can be completely described from the graph.

14 Drinen (1999): Every AF algebra is Morita equivalent to a graph algebra. Spielberg k-graphs can be used to construct any UCT Kirchberg algebra. Hong-Szymański (2004): the ideal structure of the algebra can be completely described from the graph. Generalizations: Exel crossed product algebras, Leavitt path algebras (Abrams, Ruiz, Tomforde), topological graph algebras (Katsura), Ruelle algebras (Putnam, Spielberg), Exel-Laca algebras, ultragraphs (Tomforde), Steinberg algebras (Brown, Clark, Farthing, Sims, etal.) Cuntz-Pimsner algebras, higher-rank Cuntz-Krieger algebras (Robertson-Steger), etc.

15 Let k N +. We regard N k as a category with a single object, 0, and with composition of morphisms given by addition.

16 Let k N +. We regard N k as a category with a single object, 0, and with composition of morphisms given by addition. A k-graph is a countable category Λ along with a degree functor d : Λ N k satisfying the unique factorization property: For all λ Λ, and m, n N k, if d(λ) = m + n then there are unique µ d 1 (m) and ν d 1 (n) such that λ = µν.

17 Let k N +. We regard N k as a category with a single object, 0, and with composition of morphisms given by addition. A k-graph is a countable category Λ along with a degree functor d : Λ N k satisfying the unique factorization property: For all λ Λ, and m, n N k, if d(λ) = m + n then there are unique µ d 1 (m) and ν d 1 (n) such that λ = µν. Denote the range and source maps r, s : Λ Λ.

18 Let k N +. We regard N k as a category with a single object, 0, and with composition of morphisms given by addition. A k-graph is a countable category Λ along with a degree functor d : Λ N k satisfying the unique factorization property: For all λ Λ, and m, n N k, if d(λ) = m + n then there are unique µ d 1 (m) and ν d 1 (n) such that λ = µν. Denote the range and source maps r, s : Λ Λ. Refer to objects as vertices and morphisms as paths.

19 Let k N +. We regard N k as a category with a single object, 0, and with composition of morphisms given by addition. A k-graph is a countable category Λ along with a degree functor d : Λ N k satisfying the unique factorization property: For all λ Λ, and m, n N k, if d(λ) = m + n then there are unique µ d 1 (m) and ν d 1 (n) such that λ = µν. Denote the range and source maps r, s : Λ Λ. Refer to objects as vertices and morphisms as paths. Denote Λ n = d 1 ({n}) = {morphisms of degree n}.

20 Let k N +. We regard N k as a category with a single object, 0, and with composition of morphisms given by addition. A k-graph is a countable category Λ along with a degree functor d : Λ N k satisfying the unique factorization property: For all λ Λ, and m, n N k, if d(λ) = m + n then there are unique µ d 1 (m) and ν d 1 (n) such that λ = µν. Denote the range and source maps r, s : Λ Λ. Refer to objects as vertices and morphisms as paths. Denote Λ n = d 1 ({n}) = {morphisms of degree n}. We assume: for all v Λ 0, n N k, 0 < r 1 ({v}) Λ n <.

21 Let k N +. We regard N k as a category with a single object, 0, and with composition of morphisms given by addition. A k-graph is a countable category Λ along with a degree functor d : Λ N k satisfying the unique factorization property: For all λ Λ, and m, n N k, if d(λ) = m + n then there are unique µ d 1 (m) and ν d 1 (n) such that λ = µν. Denote the range and source maps r, s : Λ Λ. Refer to objects as vertices and morphisms as paths. Denote Λ n = d 1 ({n}) = {morphisms of degree n}. We assume: for all v Λ 0, n N k, 0 < r 1 ({v}) Λ n <. Example The set of finite paths in a directed graph, with d(α) = the length of α, forms a 1-graph.

22 Example: Standard rectangles in N k

23 Example: Standard rectangles in N k Let Ω k := {(l, n) N k N k l n} with d(l, n) = n l, s(m, l) = l = r(l, n), and (m, l)(l, n) = (m, n).

24 Example: Standard rectangles in N k Let Ω k := {(l, n) N k N k l n} with d(l, n) = n l, s(m, l) = l = r(l, n), and (m, l)(l, n) = (m, n). 0

25 Example: Standard rectangles in N k Let Ω k := {(l, n) N k N k l n} with d(l, n) = n l, s(m, l) = l = r(l, n), and (m, l)(l, n) = (m, n). l n 0

26 Example: Standard rectangles in N k Let Ω k := {(l, n) N k N k l n} with d(l, n) = n l, s(m, l) = l = r(l, n), and (m, l)(l, n) = (m, n). l n m 0

27 Example: Standard rectangles in N k Let Ω k := {(l, n) N k N k l n} with d(l, n) = n l, s(m, l) = l = r(l, n), and (m, l)(l, n) = (m, n). n m 0

28 A Cuntz-Krieger Λ-family in a C*-algebra A is a set {T λ, λ Λ} of partial isometries in A satisfying

29 A Cuntz-Krieger Λ-family in a C*-algebra A is a set {T λ, λ Λ} of partial isometries in A satisfying (i) {T v v Λ 0 }) is a family of mutually orthogonal projections,

30 A Cuntz-Krieger Λ-family in a C*-algebra A is a set {T λ, λ Λ} of partial isometries in A satisfying (i) {T v v Λ 0 }) is a family of mutually orthogonal projections, (ii) T λµ = T λ T µ for all λ, µ Λ s.t. s(λ) = r(µ),

31 A Cuntz-Krieger Λ-family in a C*-algebra A is a set {T λ, λ Λ} of partial isometries in A satisfying (i) {T v v Λ 0 }) is a family of mutually orthogonal projections, (ii) T λµ = T λ T µ for all λ, µ Λ s.t. s(λ) = r(µ), (iii) T λ T λ = T s(λ) for all λ Λ, and

32 A Cuntz-Krieger Λ-family in a C*-algebra A is a set {T λ, λ Λ} of partial isometries in A satisfying (i) {T v v Λ 0 }) is a family of mutually orthogonal projections, (ii) T λµ = T λ T µ for all λ, µ Λ s.t. s(λ) = r(µ), (iii) T λ T λ = T s(λ) for all λ Λ, and (iv) for all v Λ 0 and n N k, T v = T λ Tλ. λ Λ n r(λ)=v

33 A Cuntz-Krieger Λ-family in a C*-algebra A is a set {T λ, λ Λ} of partial isometries in A satisfying (i) {T v v Λ 0 }) is a family of mutually orthogonal projections, (ii) T λµ = T λ T µ for all λ, µ Λ s.t. s(λ) = r(µ), (iii) T λ T λ = T s(λ) for all λ Λ, and (iv) for all v Λ 0 and n N k, T v = T λ Tλ. λ Λ n r(λ)=v C (Λ) will denote the C*-algebra generated by a universal Cuntz-Krieger Λ-family, (S λ, λ Λ).

34 A Cuntz-Krieger Λ-family in a C*-algebra A is a set {T λ, λ Λ} of partial isometries in A satisfying (i) {T v v Λ 0 }) is a family of mutually orthogonal projections, (ii) T λµ = T λ T µ for all λ, µ Λ s.t. s(λ) = r(µ), (iii) T λ T λ = T s(λ) for all λ Λ, and (iv) for all v Λ 0 and n N k, T v = T λ Tλ. λ Λ n r(λ)=v C (Λ) will denote the C*-algebra generated by a universal Cuntz-Krieger Λ-family, (S λ, λ Λ). Prop: C (Λ) = span{s α Sβ α, β Λ, s(α) = s(β)}

35 A Cuntz-Krieger Λ-family in a C*-algebra A is a set {T λ, λ Λ} of partial isometries in A satisfying (i) {T v v Λ 0 }) is a family of mutually orthogonal projections, (ii) T λµ = T λ T µ for all λ, µ Λ s.t. s(λ) = r(µ), (iii) T λ T λ = T s(λ) for all λ Λ, and (iv) for all v Λ 0 and n N k, T v = T λ Tλ. λ Λ n r(λ)=v C (Λ) will denote the C*-algebra generated by a universal Cuntz-Krieger Λ-family, (S λ, λ Λ). Prop: C (Λ) = span{s α Sβ α, β Λ, s(α) = s(β)} The diagonal D := span{s α S α α Λ}.

36 Classic uniqueness theorems

37 Classic uniqueness theorems Assume nondegeneracy.

38 Classic uniqueness theorems Assume nondegeneracy. Coburn s Theorem ( 67) C (T e, T f ) = T, the Toeplitz algebra. e f

39 Classic uniqueness theorems Assume nondegeneracy. Coburn s Theorem ( 67) C (T e, T f ) = T, the Toeplitz algebra. e f Cuntz ( 77) C (T ei 1 i n) = O n, the Cuntz algebra. n loops

40 Classic uniqueness theorems Assume nondegeneracy. Coburn s Theorem ( 67) C (T e, T f ) = T, the Toeplitz algebra. e f Cuntz ( 77) C (T ei 1 i n) = O n, the Cuntz algebra. n loops Cuntz-Krieger ( 80) When the adjacency matrix A of G satisfies a fullness condition (I), C (T e e G) = O A, the Cuntz-Krieger algebra.

41 Is non-degeneracy enough?

42 Is non-degeneracy enough? No! Consider the cycle of length three, E. The map φ : C (E) M 3 (C) given by S vi ε i,i S ei,j ε j,i. e 1,2 v 2 E e 2,3 v 3 e 3,1 is a non-injective -homomorphism. v 1

43 Is non-degeneracy enough? No! Consider the cycle of length three, E. The map φ : C (E) M 3 (C) given by S vi ε i,i S ei,j ε j,i. e 1,2 v 2 E e 2,3 v 3 e 3,1 is a non-injective -homomorphism. v 1 Cuntz-Krieger Uniqueness Theorem:

44 Is non-degeneracy enough? No! Consider the cycle of length three, E. The map φ : C (E) M 3 (C) given by S vi ε i,i S ei,j ε j,i. e 1,2 v 2 E e 2,3 v 3 e 3,1 is a non-injective -homomorphism. v 1 Cuntz-Krieger Uniqueness Theorem: When φ is nondegenerate and the graph satisfies then φ is injective. (L) every cycle has an entry

45 Is non-degeneracy enough? No! Consider the cycle of length three, E. The map φ : C (E) M 3 (C) given by S vi ε i,i S ei,j ε j,i. e 1,2 v 2 E e 2,3 v 3 e 3,1 is a non-injective -homomorphism. v 1 Cuntz-Krieger Uniqueness Theorem: When φ is nondegenerate and the graph satisfies then φ is injective. (L) every cycle has an entry Theorem Szymański (2001), Nagy-R (2010): Condition (L) can be replaced with a condition on the spectrum of φ(s λ ) for cycles λ without entry.

46 Aperiodicity defined via the infinite path space Λ

47 Aperiodicity defined via the infinite path space Λ An infinite path in a k-graph Λ is a degree-preserving covariant functor x : Ω k Λ.

48 Aperiodicity defined via the infinite path space Λ An infinite path in a k-graph Λ is a degree-preserving covariant functor x : Ω k Λ. k = 1 picture e 1 e 2 x(1, 3) = e 1 e 2 d(e 1 e 2 ) = 2

49 Aperiodicity defined via the infinite path space Λ An infinite path in a k-graph Λ is a degree-preserving covariant functor x : Ω k Λ. k = 1 picture e 1 e 2 x(1, 3) = e 1 e 2 d(e 1 e 2 ) = 2 k = 2 picture

50 Aperiodicity defined via the infinite path space Λ An infinite path in a k-graph Λ is a degree-preserving covariant functor x : Ω k Λ. k = 1 picture e 1 e 2 x(1, 3) = e 1 e 2 d(e 1 e 2 ) = 2 k = 2 picture s(α) x((1, 2), (3, 3)) = α d(α) = (2, 1) r(α)

51 An infinite path x in a k-graph is eventually periodic if there are α β in Λ and y Λ such that x = αy = βy; otherwise x is aperiodic.

52 An infinite path x in a k-graph is eventually periodic if there are α β in Λ and y Λ such that x = αy = βy; otherwise x is aperiodic. x Λ

53 An infinite path x in a k-graph is eventually periodic if there are α β in Λ and y Λ such that x = αy = βy; otherwise x is aperiodic. x Λ α

54 An infinite path x in a k-graph is eventually periodic if there are α β in Λ and y Λ such that x = αy = βy; otherwise x is aperiodic. x Λ α β

55 An infinite path x in a k-graph is eventually periodic if there are α β in Λ and y Λ such that x = αy = βy; otherwise x is aperiodic. x Λ y α y β

56 Λ is aperiodic if every vertex is the range vertex of an aperiodic infinite path.

57 Λ is aperiodic if every vertex is the range vertex of an aperiodic infinite path. In a directed graph, cycles without entry reveal failure of aperiodicity.

58 Λ is aperiodic if every vertex is the range vertex of an aperiodic infinite path. In a directed graph, cycles without entry reveal failure of aperiodicity. v α λ Clearly the only infinite path with range v, αλλλ, is eventually periodic.

59 Λ is aperiodic if every vertex is the range vertex of an aperiodic infinite path. In a directed graph, cycles without entry reveal failure of aperiodicity. v α λ Clearly the only infinite path with range v, αλλλ, is eventually periodic. Uniqueness theorems of Raeburn-Sims-Yeend and Kumjian-Pask assume aperiodicity of the k-graph.

60 Λ is aperiodic if every vertex is the range vertex of an aperiodic infinite path. In a directed graph, cycles without entry reveal failure of aperiodicity. v α λ Clearly the only infinite path with range v, αλλλ, is eventually periodic. Uniqueness theorems of Raeburn-Sims-Yeend and Kumjian-Pask assume aperiodicity of the k-graph. Theorem Nagy-R (2010), Nagy-Brown-R (2013) A -homomorphism φ : C (Λ) A is injective iff it is injective on the subalgebra M := C (S α S β γ Λ αγ = βγ}.

61 Properties of the subalgebra

62 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if

63 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if (i) a faithful conditional expectation A B,

64 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if (i) a faithful conditional expectation A B, (ii) The normalizer of B in A generates A,

65 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if (i) a faithful conditional expectation A B, (ii) The normalizer of B in A generates A, and (iii) B contains an approximate unit of A.

66 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if (i) a faithful conditional expectation A B, (ii) The normalizer of B in A generates A, and (iii) B contains an approximate unit of A. Extension properties for pure states on masa B A:

67 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if (i) a faithful conditional expectation A B, (ii) The normalizer of B in A generates A, and (iii) B contains an approximate unit of A. Extension properties for pure states on masa B A: (UEP) Every pure state extends uniquely to A.

68 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if (i) a faithful conditional expectation A B, (ii) The normalizer of B in A generates A, and (iii) B contains an approximate unit of A. Extension properties for pure states on masa B A: (UEP) Every pure state extends uniquely to A. A Cartan subalgebra with the UEP is a Kumjian C -diagonal.

69 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if (i) a faithful conditional expectation A B, (ii) The normalizer of B in A generates A, and (iii) B contains an approximate unit of A. Extension properties for pure states on masa B A: (UEP) Every pure state extends uniquely to A. A Cartan subalgebra with the UEP is a Kumjian C -diagonal.

70 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if (i) a faithful conditional expectation A B, (ii) The normalizer of B in A generates A, and (iii) B contains an approximate unit of A. Extension properties for pure states on masa B A: (UEP) Every pure state extends uniquely to A. A Cartan subalgebra with the UEP is a Kumjian C -diagonal. (AEP) Densely many pure states extend uniquely.

71 Properties of the subalgebra (Renault, 80) A masa C*-subalgebra B A is Cartan if (i) a faithful conditional expectation A B, (ii) The normalizer of B in A generates A, and (iii) B contains an approximate unit of A. Extension properties for pure states on masa B A: (UEP) Every pure state extends uniquely to A. A Cartan subalgebra with the UEP is a Kumjian C -diagonal. (AEP) Densely many pure states extend uniquely. Thm (Nagy-R, 2011) When G is a directed graph, M C (G) is Cartan and satisfies AEP; i.e, it is a pseudo-diagonal.

72 A groupoid is a small category in which every element has an inverse. A topological groupoid is one in which multiplication and inversion are continuous. It is étale if the range and source are local homeomorphisms.

73 A groupoid is a small category in which every element has an inverse. A topological groupoid is one in which multiplication and inversion are continuous. It is étale if the range and source are local homeomorphisms. C (G) is defined to be a completion of C c (G).

74 A groupoid is a small category in which every element has an inverse. A topological groupoid is one in which multiplication and inversion are continuous. It is étale if the range and source are local homeomorphisms. C (G) is defined to be a completion of C c (G). Iso(G) := {g G r(g) = s(g)}, the isotropy subgroupoid of G.

75 A groupoid is a small category in which every element has an inverse. A topological groupoid is one in which multiplication and inversion are continuous. It is étale if the range and source are local homeomorphisms. C (G) is defined to be a completion of C c (G). Iso(G) := {g G r(g) = s(g)}, the isotropy subgroupoid of G. Theorem (Brown-Nagy-R-Sims-Williams, 2014) Let G be a locally compact, amenable, Hausdorff, étale groupoid. If φ : C (G) A is a C -homomorphism, then the following are equivalent. (i) φ is injective. (ii) φ is injective on C ((Iso(G)) ).

76 To a k-graph Λ, we associate the groupoid

77 To a k-graph Λ, we associate the groupoid with G Λ = {(αy, d, βy) y Λ, α, β Λ, d = d Λ (β) d Λ (α)}

78 To a k-graph Λ, we associate the groupoid G Λ = {(αy, d, βy) y Λ, α, β Λ, d = d Λ (β) d Λ (α)} with s(x, d, y) = y = r(y, d, z) (x, d, y) 1 = (y, d, x) (x, d, y)(y, d, w) = (x, d + d, w)

79 To a k-graph Λ, we associate the groupoid G Λ = {(αy, d, βy) y Λ, α, β Λ, d = d Λ (β) d Λ (α)} with s(x, d, y) = y = r(y, d, z) (x, d, y) 1 = (y, d, x) (x, d, y)(y, d, w) = (x, d + d, w) The cylinder sets Z (α, β) = {(αy, d, βy)} form a basis for an étale topology.

80 To a k-graph Λ, we associate the groupoid G Λ = {(αy, d, βy) y Λ, α, β Λ, d = d Λ (β) d Λ (α)} with s(x, d, y) = y = r(y, d, z) (x, d, y) 1 = (y, d, x) (x, d, y)(y, d, w) = (x, d + d, w) The cylinder sets Z (α, β) = {(αy, d, βy)} form a basis for an étale topology. Iso(G Λ ) = {(αy, d, βy) G Λ αy = βy}

81 To a k-graph Λ, we associate the groupoid G Λ = {(αy, d, βy) y Λ, α, β Λ, d = d Λ (β) d Λ (α)} with s(x, d, y) = y = r(y, d, z) (x, d, y) 1 = (y, d, x) (x, d, y)(y, d, w) = (x, d + d, w) The cylinder sets Z (α, β) = {(αy, d, βy)} form a basis for an étale topology. Iso(G Λ ) = {(αy, d, βy) G Λ αy = βy} Recall: C (G Λ ) is a completion of C c (G Λ ).

82 To a k-graph Λ, we associate the groupoid G Λ = {(αy, d, βy) y Λ, α, β Λ, d = d Λ (β) d Λ (α)} with s(x, d, y) = y = r(y, d, z) (x, d, y) 1 = (y, d, x) (x, d, y)(y, d, w) = (x, d + d, w) The cylinder sets Z (α, β) = {(αy, d, βy)} form a basis for an étale topology. Iso(G Λ ) = {(αy, d, βy) G Λ αy = βy} Recall: C (G Λ ) is a completion of C c (G Λ ). The map S α S β χ Z (α,β) implements an isomorphism C (Λ) = C (G Λ ) that restricts to an iso M = C (Iso(G Λ ) )..

83 Thm (BNRSW, 2015) Let G be a Hausdorff, étale groupoid. (a) If (Iso(G)) is closed, then the restriction map f f Iso(G) extends to a faithful conditional expectation E : C (G) M = C ((Iso(G)) ). (b) If (Iso(G)) is not closed, then there is no conditional expectation onto the subalgebra. (c) If (Iso(G)) is closed and abelian, then M is a masa.

84 Thm (BNRSW, 2015) Let G be a Hausdorff, étale groupoid. (a) If (Iso(G)) is closed, then the restriction map f f Iso(G) extends to a faithful conditional expectation E : C (G) M = C ((Iso(G)) ). (b) If (Iso(G)) is not closed, then there is no conditional expectation onto the subalgebra. (c) If (Iso(G)) is closed and abelian, then M is a masa. Thm (BNRSW, 2015; Yang, 2014) Let Λ be a k-graph. (a) M is always a masa in C (Λ). (b) There are examples of 2-graph C -algebras with (Iso(G)) not closed, and hence M not Cartan.

85 Thm (BNRSW, 2015) Let G be a Hausdorff, étale groupoid. (a) If (Iso(G)) is closed, then the restriction map f f Iso(G) extends to a faithful conditional expectation E : C (G) M = C ((Iso(G)) ). (b) If (Iso(G)) is not closed, then there is no conditional expectation onto the subalgebra. (c) If (Iso(G)) is closed and abelian, then M is a masa. Thm (BNRSW, 2015; Yang, 2014) Let Λ be a k-graph. (a) M is always a masa in C (Λ). (b) There are examples of 2-graph C -algebras with (Iso(G)) not closed, and hence M not Cartan. Thm (NRBSW, 2014) All Cartan subalgebras satisfy the AEP.

86 Abstract Uniqueness Theorem (Brown-Nagy-R)

87 Abstract Uniqueness Theorem (Brown-Nagy-R) Let A be a C*-algebra and M A a C -subalgebra. Suppose there is a set S of pure states on M satisfying

88 Abstract Uniqueness Theorem (Brown-Nagy-R) Let A be a C*-algebra and M A a C -subalgebra. Suppose there is a set S of pure states on M satisfying (i) each ψ S extends uniquely to a state ψ on A, and

89 Abstract Uniqueness Theorem (Brown-Nagy-R) Let A be a C*-algebra and M A a C -subalgebra. Suppose there is a set S of pure states on M satisfying (i) each ψ S extends uniquely to a state ψ on A, and (ii) the direct sum ψ S π ψ of the GNS representations associated to the extensions to A of elements in S is faithful on A.

90 Abstract Uniqueness Theorem (Brown-Nagy-R) Let A be a C*-algebra and M A a C -subalgebra. Suppose there is a set S of pure states on M satisfying (i) each ψ S extends uniquely to a state ψ on A, and (ii) the direct sum ψ S π ψ of the GNS representations associated to the extensions to A of elements in S is faithful on A. Then a -homomorphism Φ : A B is injective iff Φ M is injective.

91 Thank you!

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