The faithful subalgebra
|
|
- Ashley Campbell
- 5 years ago
- Views:
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!
92 A. an Huef and I. Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic Theory Dynam. Systems 17 (1997), J.H. Brown, G. Nagy, and S. Reznikoff, A generalized Cuntz-Krieger uniqueness theorem for higher-rank graphs, J. Funct. Anal. (2013), J.H. Brown, G. Nagy, S. Reznikoff, A. Sims, and D. Williams, Cartan subalgebras of groupoid C -algebras K.R. Davidson, S.C. Power, and D. Yang, Dilation theory for rank 2 graph algebras, J. Operator Theory.
93 P. Goldstein, On graph C*-algebras, J. Austral. Math. Soc. 72 (2002), A. Kumjian and D. Pask, Higher rank graph C*-algebras, New York J. Math. 6 (2000), A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998) A. Kumjian, D. Pask, I. Raeburn, and J. Renault, Graphs, groupoids and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), G. Nagy and S. Reznikoff, Abelian core of graph algebras, J. Lond. Math. Soc. (2) 85 (2012), no. 3,
94 G. Nagy and S. Reznikoff, Pseudo-diagonals and uniqueness theorems, (2013), to appear in Proc. AMS. D. Pask, I. Raeburn, M. Rørdam, A. Sims, Rank-two graphs whose C*-algebras are direct limits of circle algebras, J. Functional Anal. 144 (2006), I. Raeburn, A. Sims and T. Yeend, Higher-rank graphs and their C*-algebras, Proc. Edin. Math. Soc. 46 (2003) J. Renault, A groupoid approach to C -algebras, Irish Math. Soc. Bull. 61 (2008) D. Robertson and A. Sims, Simplicity of C*-algebras associated to higher-rank graphs, Bull. Lond. Math. Soc. 39 (2007), no. 2,
95 G. Robertson and T. Steger, Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, J. Reine Angew. Math. 513 (1999), A. Sims, Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs, Canad. J. Math. 58 (2006), no. 6, J. Spielberg, Graph-based models for Kirchberg algebras, J. Operator Theory 57 (2007), W. Szymański, General Cuntz-Krieger uniqueness theorem, Internat. J. Math. 13 (2002) D. Yang, Cycline subalgebras are Cartan, preprint.
96 T. Yeend, Groupoid models for the C*-algebras of topological higher-rank graphs, J. Operator Theory 57:1 (2007),
Uniqueness theorems for combinatorial C -algebras
Uniqueness theorems for combinatorial C -algebras joint work with Jonathan H. Brown, Gabriel Nagy, Aidan Sims, and Dana Williams funded in part by NSF DMS-1201564 Nebraska-Iowa Functional Analysis Seminar
More informationGroupoids and higher-rank graphs
Groupoids and higher-rank graphs James Rout Technion - Israel Institute of Technology 5/12/2017 (Ongoing work with Toke Meier Carlsen) James Rout Groupoids and higher-rank graphs 1 / 16 Higher-rank graphs
More informationUltragraph C -algebras via topological quivers
STUDIA MATHEMATICA 187 (2) (2008) Ultragraph C -algebras via topological quivers by Takeshi Katsura (Yokohama), Paul S. Muhly (Iowa City, IA), Aidan Sims (Wollongong) and Mark Tomforde (Houston, TX) Abstract.
More informationOn the simplicity of twisted k-graph C -algebras
Preliminary report of work in progress Alex Kumjian 1, David Pask 2, Aidan Sims 2 1 University of Nevada, Reno 2 University of Wollongong GPOTS14, Kansas State University, Manhattan, 27 May 2014 Introduction
More informationStable finiteness and pure infiniteness of the C -algebras of higher-rank graphs
Stable finiteness and pure infiniteness of the C -algebras of higher-rank graphs Astrid an Huef University of Houston, July 31 2017 Overview Let E be a directed graph such that the graph C -algebra C (E)
More informationTalk 5: Generalizations of Graph Algebras within the Cuntz-Pimsner Framework
Talk 5: Generalizations of Graph Algebras within the Cuntz-Pimsner Framework Mark Tomforde University of Houston, USA July, 2010 Mark Tomforde (University of Houston) Generalizations of Graph Algebras
More informationTwisted Higher Rank Graph C*-algebras
Alex Kumjian 1, David Pask 2, Aidan Sims 2 1 University of Nevada, Reno 2 University of Wollongong East China Normal University, Shanghai, 21 May 2012 Introduction. Preliminaries Introduction k-graphs
More informationValentin Deaconu, University of Nevada, Reno. Based on joint work with A. Kumjian and J. Quigg, Ergodic Theory and Dynamical Systems (2011)
Based on joint work with A. Kumjian and J. Quigg, Ergodic Theory and Dynamical Systems (2011) Texas Tech University, Lubbock, 19 April 2012 Outline We define directed graphs and operator representations
More informationGraphs and C -algebras
35 Graphs and C -algebras Aidan Sims Abstract We outline Adam Sørensen s recent characterisation of stable isomorphism of simple unital graph C -algebras in terms of moves on graphs. Along the way we touch
More informationCohomology of higher rank graphs, an interim report.
Cohomology of higher rank graphs, an interim report. Alex Kumjian 1, David Pask 2, Aidan Sims 2 1 University of Nevada, Reno 2 University of Wollongong GPOTS, Arizona State University Tempe, May 2011 Introduction
More informationTalk 1: An Introduction to Graph C -algebras
Talk 1: An Introduction to Graph C -algebras Mark Tomforde University of Houston, USA July, 2010 Mark Tomforde (University of Houston) Graph C -algebras July, 2010 1 / 39 Some Terminology A graph E = (E
More informationCUNTZ-KRIEGER ALGEBRAS OF DIRECTED GRAPHS. Alex Kumjian, David Pask and Iain Raeburn
pacific journal of mathematics Vol. 184, No. 1, 1998 CUNTZ-KRIEGER ALGEBRAS OF DIRECTED GRAPHS Alex Kumjian, David Pask and Iain Raeburn We associate to each row-finite directed graph E a universal Cuntz-Krieger
More informationPARTIAL DYNAMICAL SYSTEMS AND C -ALGEBRAS GENERATED BY PARTIAL ISOMETRIES
PARTIAL DYNAMICAL SYSTEMS AND C -ALGEBRAS GENERATED BY PARTIAL ISOMETRIES RUY EXEL 1, MARCELO LACA 2, AND JOHN QUIGG 3 November 10, 1997; revised May 99 Abstract. A collection of partial isometries whose
More informationNOTES ON CUNTZ KRIEGER UNIQUENESS THEOREMS AND C -ALGEBRAS OF LABELLED GRAPHS
NOTES ON CUNTZ KRIEGER UNIQUENESS THEOREMS AND C -ALGEBRAS OF LABELLED GRAPHS BERNHARD BURGSTALLER Abstract. In this note we deal with Cuntz-Krieger uniqueness theorems and extend the class of algebras
More informationKMS states for quasi-free actions on finite-graph algebras
KMS states for quasi-free actions on finite-graph algebras Christopher Chlebovec University of New Brunswick June 15, 2015 1 / 27 Throughout this talk, we will only consider finite graphs E, where E =
More informationEquivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 2017 Equivalence and stable isomorphism of groupoids,
More informationDecomposing the C -algebras of groupoid extensions
Decomposing the C -algebras of groupoid extensions Jonathan Brown, joint with Astrid an Huef Kansas State University Great Plains Operator Theory Symposium May 2013 J. Brown (KSU) Groupoid extensions GPOTS
More informationCartan sub-c*-algebras in C*-algebras
Plan Cartan sub-c*-algebras in C*-algebras Jean Renault Université d Orléans 22 July 2008 1 C*-algebra constructions. 2 Effective versus topologically principal. 3 Cartan subalgebras in C*-algebras. 4
More informationPartial semigroup actions and groupoids
Partial semigroup actions and groupoids Jean Renault University of Orléans May 12, 2014 PARS (joint work with Dana Williams) Introduction A few years ago, I was recruited by an Australian team to help
More informationGraphs of C*-correspondences and Fell bundles
University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 2010 Graphs of C*-correspondences and Fell bundles Valentin Deaconu Alexander
More informationExtensions of graph C -algebras
Extensions of graph C -algebras A Thesis Submitted to the Faculty in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics by Mark Tomforde DARTMOUTH COLLEGE Hanover,
More informationThe Structure of C -algebras Associated with Hyperbolic Dynamical Systems
The Structure of C -algebras Associated with Hyperbolic Dynamical Systems Ian F. Putnam* and Jack Spielberg** Dedicated to Marc Rieffel on the occasion of his sixtieth birthday. Abstract. We consider the
More informationSYMMETRIC IMPRIMITIVITY THEOREMS FOR GRAPH C -ALGEBRAS
SYMMETRIC IMPRIMITIVITY THEOREMS FOR GRAPH C -ALGEBRAS DAVID PASK AND IAIN RAEBURN The C -algebra C (E) of a directed graph E is generated by partial isometries satisfying relations which reflect the path
More informationSimplicity of C*-algebras associated to row-finite locally convex higher-rank graphs
University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 2009 Simplicity of C*-algebras associated to row-finite locally convex
More informationarxiv: v1 [math.oa] 13 Sep 2008
THE CO-UNIVERSAL C -ALGEBRA OF A ROW-FINITE GRAPH AIDAN SIMS arxiv:0809.2333v1 [math.oa] 13 Sep 2008 Abstract. Let E be a row-finite directed graph. We prove that there exists a C - algebra Cmin (E) with
More informationUniqueness Theorems For Topological Higherrank Graph C*-algebras
University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers: Part B Faculty of Engineering and Information Sciences 2018 Uniqueness Theorems For Topological Higherrank
More informationLeavitt Path Algebras
Leavitt Path Algebras At the crossroads of Algebra and Functional Analysis Mark Tomforde University of Houston USA In its beginnings, the study of Operator Algebras received a great deal of motivation
More informationHIGHER RANK GRAPH C -ALGEBRAS
HIGHER RANK GRAPH C -ALGEBRAS ALEX KUMJIAN AND DAVID PASK Abstract. Building on recent work of Robertson and Steger, we associate a C algebra to a combinatorial object which may be thought of as higher
More informationTalk 3: Graph C -algebras as Cuntz-Pimsner algebras
Talk 3: Graph C -algebras as Cuntz-Pimsner algebras Mark Tomforde University of Houston, USA July, 2010 Mark Tomforde (University of Houston) Graph C -algs. as Cuntz-Pimsner algs. July, 2010 1 / 29 Pimsner
More informationRadboud Universiteit Nijmegen
Radboud Universiteit Nijmegen Faculty of Science Properties of graph C -algebras in the Cuntz Krieger, Cuntz Pimsner and groupoid models Author: Student number: Program: Specialization: Supervisor: Second
More informationThe co-universal C*-algebra of a row-finite graph
University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 2010 The co-universal C*-algebra of a row-finite graph Aidan Sims University
More informationHaar Systems on Equivalent Groupoids
Dartmouth College Groupoidfest October 2015 Haar Systems In order to make a C -algebra from a groupoid G, we need a family of Radon measures { λ u : u G (0) } such that supp λ u = G u = { x G : r(x) =
More informationCONTINUOUS GRAPHS AND C*-ALGEBRAS
CONTINUOUS GRAPHS AND C*-ALGEBRAS by VALENTIN DEACONU 1 Timisoara, June 1998 Abstract We present the continuous graph approach for some generalizations of the Cuntz- Krieger algebras. These C*-algebras
More informationDiagonals in Fell algebras. an interim report.
, an interim report. Astrid an Huef 1, Alex Kumjian 2, Aidan Sims 3 1 University of New South Wales (as of 2010, Otago University) 2 University of Nevada, Reno 3 University of Wollongong AMS Sectional
More informationTHE C -ALGEBRAS OF ROW-FINITE GRAPHS
THE C -ALGEBRAS OF ROW-FINITE GRAPHS TERESA BATES, DAVID PASK, IAIN RAEBURN, AND WOJCIECH SZYMAŃSKI Abstract. We prove versions of the fundamental theorems about Cuntz-Krieger algebras for the C -algebras
More informationTopological full groups of étale groupoids
Topological full groups of étale groupoids Hiroki Matui Chiba University May 31, 2016 Geometric Analysis on Discrete Groups RIMS, Kyoto University 1 / 18 dynamics on Cantor set X Overview Cr (G) K i (Cr
More informationarxiv: v1 [math.oa] 6 May 2009
APERIODICITY AND COFINALITY FOR FINITELY ALIGNED HIGHER-RANK GRAPHS PETER LEWIN AND AIDAN SIMS arxiv:0905.0735v1 [math.oa] 6 May 2009 Abstract. We introduce new formulations of aperiodicity and cofinality
More informationGENERALIZED SOLENOIDS AND C*-ALGEBRAS
GENERALIZED SOLENOIDS AND C*-ALGEBRAS by VALENTIN DEACONU 1 Abstract We present the continuous graph approach for some generalizations of the Cuntz- Krieger algebras. These algebras are simple, nuclear,
More informationarxiv: v1 [math.oa] 13 Jul 2017
ON GRADED C -ALGEBRAS IAIN RAEBURN Abstract. WeshowthateverytopologicalgradingofaC -algebrabyadiscreteabelian group is implemented by an action of the compact dual group. arxiv:1707.04215v1 [math.oa] 13
More informationarxiv: v1 [math.oa] 21 Dec 2018
RECONSTRUCTING DIRECTED GRAPHS FROM GENERALISED GAUGE ACTIONS ON THEIR TOEPLITZ ALGEBRAS NATHAN BROWNLOWE, MARCELO LACA, DAVE ROBERTSON, AND AIDAN SIMS arxiv:1812.08903v1 [math.oa] 21 Dec 2018 Abstract.
More informationThe primitive ideals of some etale groupoid C*- algebras
University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 2016 The primitive ideals of some etale groupoid
More informationGeneralizations of Directed Graphs. Nura Patani
C -Correspondences and Topological Dynamical Systems Associated to Generalizations of Directed Graphs by Nura Patani A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor
More informationDiagonal preserving -isomorphisms of Leavitt Path Algebras
Diagonal preserving -isomorphisms of Leavitt Path Algebras Jonathan Brown The University of Dayton Joint with L. Clark and A. an Huef Groupoid Fest March 2017 Jonathan Brown (U. Dayton) Diagonal preserving
More informationarxiv:math/ v4 [math.oa] 28 Dec 2005
arxiv:math/0506151v4 [math.oa] 28 Dec 2005 TENSOR ALGEBRAS OF C -CORRESPONDENCES AND THEIR C -ENVELOPES. ELIAS G. KATSOULIS AND DAVID W. KRIBS Abstract. We show that the C -envelope of the tensor algebra
More informationC -algebras associated to dilation matrices
C -algebras associated to dilation matrices Iain Raeburn (University of Otago, NZ) This talk is about joint work with Ruy Exel and Astrid an Huef, and with Marcelo Laca and Jacqui Ramagge An Exel system
More informationOn k-morphs. A. Kumjian 1 D. Pask 2 A. Sims 2. WCOAS, 14 September 2008 Northern Arizona University Flagstaff AZ. University of Nevada, Reno
. A. Kumjian 1 D. Pask 2 A. Sims 2 1 Department of Mathematics and Statistics University of Nevada, Reno 2 School of Mathematics and Applied Statistics University of Wollongong, Australia WCOAS, 14 September
More informationRECONSTRUCTION OF GRADED GROUPOIDS FROM GRADED STEINBERG ALGEBRAS
RECONSTRUCTION OF GRADED GROUPOIDS FROM GRADED STEINBERG ALGEBRAS PERE ARA, JOAN BOSA, ROOZBEH HAZRAT, AND AIDAN SIMS Abstract. We show how to reconstruct a graded ample Hausdorff groupoid with topologically
More informationStrongly Self-Absorbing C -algebras which contain a nontrivial projection
Münster J. of Math. 1 (2008), 99999 99999 Münster Journal of Mathematics c Münster J. of Math. 2008 Strongly Self-Absorbing C -algebras which contain a nontrivial projection Marius Dadarlat and Mikael
More informationThe Abel Symposium 2015 Operator Algebras and Applications 7-11 August, 2015, Coastal Express, Norway Titles and abstracts of talks
The Abel Symposium 2015 Operator Algebras and Applications 7-11 August, 2015, Coastal Express, Norway Titles and abstracts of talks Erling Størmer (University of Oslo) Uffe Haagerup - In Memoriam Erik
More informationarxiv: v2 [math.ds] 28 Jun 2014
Symmetry, Integrability and Geometry: Methods and Applications Groupoid Actions on Fractafolds SIGMA 10 (2014), 068, 14 pages Marius IONESCU and Alex KUMJIAN arxiv:1311.3880v2 [math.ds] 28 Jun 2014 Department
More informationGraded K-theory and Graph Algebras
Alex Kumjian 1 David Pask 2 Aidan Sims 2 1 University of Nevada, Reno 2 University of Wollongong GPOTS, 1 June 2018 Miami University, Oxford Introduction A C -algebra A endowed with an automorphism of
More informationSEMICROSSED PRODUCTS OF THE DISK ALGEBRA
SEMICROSSED PRODUCTS OF THE DISK ALGEBRA KENNETH R. DAVIDSON AND ELIAS G. KATSOULIS Abstract. If α is the endomorphism of the disk algebra, A(D), induced by composition with a finite Blaschke product b,
More informationTHE CROSSED-PRODUCT OF A C*-ALGEBRA BY A SEMIGROUP OF ENDOMORPHISMS. Ruy Exel Florianópolis
THE CROSSED-PRODUCT OF A C*-ALGEBRA BY A SEMIGROUP OF ENDOMORPHISMS Ruy Exel Florianópolis This talk is based on: R. Exel, A new look at the crossed-product of a C*-algebra by an endomorphism, Ergodic
More informationarxiv: v4 [math.oa] 11 Jan 2013
COACTIONS AND SKEW PRODUCTS OF TOPOLOGICAL GRAPHS arxiv:1205.0523v4 [math.oa] 11 Jan 2013 S. KALISZEWSKI AND JOHN QUIGG Abstract. We show that the C -algebra of a skew-product topological graph E κ G is
More informationREALIZATION OF HYPERBOLIC GROUP C -ALGEBRAS AS DECREASING INTERSECTION OF CUNTZ ALGEBRAS O 2
REALIZATION OF HYPERBOLIC GROUP C -ALGEBRAS AS DECREASING INTERSECTION OF CUNTZ ALGEBRAS O 2 YUHEI SUZUKI Abstract. It is proved that for every ICC group which is embeddable into a hyperbolic group, the
More informationAlex Kumjian, David Pask, Aidan Sims
Documenta Math 161 C -Algebras Associated to Coverings of k-graphs Alex Kumjian, David Pask, Aidan Sims Received: March 15, 2007 Revised: May 24, 2008 Communicated by Joachim Cuntz Abstract A covering
More informationTENSOR ALGEBRAS OF PRODUCT SYSTEMS AND THEIR C -ENVELOPES
TENSOR ALGEBRAS OF PRODUCT SYSTEMS AND THEIR C -ENVELOPES ADAM DOR-ON AND ELIAS KATSOULIS Abstract. Let (G, P ) be an abelian, lattice ordered group and let X be a compactly aligned, φ-injective product
More informationRemarks on some fundamental results about higher-rank graphs and their C*-algebras
University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 2013 Remarks on some fundamental results about
More informationIterating the Cuntz-Nica-Pimsner construction for product systems
University of Wollongong Research Online University of Wollongong Thesis Collection 2017+ University of Wollongong Thesis Collections 2017 Iterating the Cuntz-Nica-Pimsner construction for product systems
More informationarxiv: v1 [math.oa] 20 Feb 2018
C*-ALGEBRAS FOR PARTIAL PRODUCT SYSTEMS OVER N RALF MEYER AND DEVARSHI MUKHERJEE arxiv:1802.07377v1 [math.oa] 20 Feb 2018 Abstract. We define partial product systems over N. They generalise product systems
More informationTOPOLOGICALLY FREE ACTIONS AND PURELY INFINITE C -CROSSED PRODUCTS
Bull. Korean Math. Soc. 31 (1994), No. 2, pp. 167 172 TOPOLOGICALLY FREE ACTIONS AND PURELY INFINITE C -CROSSED PRODUCTS JA AJEONG 1. Introduction For a given C -dynamical system (A, G,α) with a G-simple
More informationOn the Baum-Connes conjecture for Gromov monster groups
On the Baum-Connes conjecture for Gromov monster groups Martin Finn-Sell Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien martin.finn-sell@univie.ac.at June 2015 Abstract We present a geometric
More informationKMS states of graph algebras with a generalised gauge dynamics
KMS states of graph algebras with a generalised gauge dynamics Richard McNamara a thesis submitted for the degree of Doctor of Philosophy at the University of Otago, Dunedin, New Zealand. December 5, 2015
More informationInteractions between operator algebras and dynamical systems: Abstracts
Interactions between operator algebras and dynamical systems: Abstracts The composition series of ideals in the partial-isometric crossed product by a semigroup of extendible endomorphisms Sriwulan Adji
More informationarxiv: v1 [math.oa] 9 Jul 2018
AF-EMBEDDINGS OF RESIDUALLY FINITE DIMENSIONAL C*-ALGEBRAS arxiv:1807.03230v1 [math.oa] 9 Jul 2018 MARIUS DADARLAT Abstract. It is shown that a separable exact residually finite dimensional C*-algebra
More informationC*-algebras associated with interval maps
C*-algebras associated with interval maps Valentin Deaconu and Fred Shultz Brazilian Operator Algebras Conference Florianópolis, July 24-28, 2006 For each piecewise monotonic map τ of [0, 1], we associate
More informationPhase transitions in a number-theoretic dynamical system
Phase transitions in a number-theoretic dynamical system Iain Raeburn University of Wollongong GPOTS June 2009 This talk is about joint work with Marcelo Laca. In this talk, a dynamical system consists
More informationThe complexity of classification problem of nuclear C*-algebras
The complexity of classification problem of nuclear C*-algebras Ilijas Farah (joint work with Andrew Toms and Asger Törnquist) Nottingham, September 6, 2010 C*-algebras H: a complex Hilbert space (B(H),
More informationON THE KK-THEORY OF STRONGLY SELF-ABSORBING C -ALGEBRAS
MATH. SCAND. 104 (09), 95 107 ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C -ALGEBRAS MARIUS DADARLAT and WILHELM WINTER Abstract Let D and A be unital and separable C -algebras; let D be strongly self-absorbing.
More informationBoundary Quotients of Semigroup C*-algebras
Boundary Quotients of Semigroup C*-algebras Charles Starling uottawa May 14, 2015 Charles Starling (uottawa) Boundary Quotients of Semigroup C*-algebras May 14, 2015 1 / 23 Overview P - left cancellative
More informationRepresentations and the reduction theorem for ultragraph Leavitt path algebras
arxiv:1902.00013v1 [math.ra] 31 Jan 2019 Representations and the reduction theorem for ultragraph Leavitt path algebras Daniel Gonçalves and Danilo Royer February 4, 2019 Abstract In this paper we study
More information2 Alex Kumjian and David Pask of commuting 0 ; 1 matrices. In particular the associated C {algebras are simple, purely innite and generated by a famil
New York Journal of Mathematics New York J. Math. 6 (2000) 1{20. Higher Rank Graph C -Algebras Alex Kumjian and David Pask Abstract. Building on recent work of Robertson and Steger, we associate a C {algebra
More informationClassification of spatial L p AF algebras
Classification of spatial L p AF algebras Maria Grazia Viola Lakehead University joint work with N. C. Phillips Workshop on Recent Developments in Quantum Groups, Operator Algebras and Applications Ottawa,
More informationFlow equivalence of graph algebras
University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 2004 Flow equivalence of graph algebras David A Pask University of Wollongong,
More informationSpectrally Bounded Operators on Simple C*-Algebras, II
Irish Math. Soc. Bulletin 54 (2004), 33 40 33 Spectrally Bounded Operators on Simple C*-Algebras, II MARTIN MATHIEU Dedicated to Professor Gerd Wittstock on the Occasion of his Retirement. Abstract. A
More informationHigher rank graphs, k-subshifts and k-automata
Higher rank graphs, k-subshifts and k-automata R. Exel and B. Steinberg arxiv:1809.04932v1 [math.oa] 13 Sep 2018 Given a k-graph Λ we construct a Markov space M Λ, and a collection of k pairwise commuting
More informationarxiv: v2 [math.oa] 20 Dec 2016
GROUP ACTIONS ON GRAPHS AND C -CORRESPONDENCES arxiv:1410.3846v2 [math.oa] 20 Dec 2016 VALENTIN DEACONU Abstract. If G acts on a C -correspondence H over the C - algebra A (see Definition 2.4), then by
More informationHomology for higher-rank graphs and twisted C*- algebras
University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 2012 Homology for higher-rank graphs and twisted C*- algebras Alex Kumjian
More informationA class of C -algebras that are prime but not primitive
Münster J. of Math. 7 (2014), 489 514 DOI 10.17879/58269761901 urn:nbn:de:hbz:6-58269762205 Münster Journal of Mathematics c Münster J. of Math. 2014 A class of C -algebras that are prime but not primitive
More informationarxiv: v1 [math.oa] 11 Jul 2012
THE EQUIVALENCE RELATIONS OF LOCAL HOMEOMORPHISMS AND FELL ALGEBRAS LISA ORLOFF CLARK, ASTRID AN HUEF, AND IAIN RAEBURN arxiv:1207.2540v1 [math.oa] 11 Jul 2012 Abstract. We study the groupoid C -algebras
More informationOperator algebras and dynamical systems from number theory Nov 24 - Nov 29, 2013
Operator algebras and dynamical systems from number theory Nov 24 - Nov 29, 2013 MEALS *Breakfast (Bu et): 7:00 9:30 am, Sally Borden Building, Monday Friday *Lunch (Bu et): 11:30 am 1:30 pm, Sally Borden
More informationNon-separable AF-algebras
Non-separable AF-algebras Takeshi Katsura Department of Mathematics, Hokkaido University, Kita 1, Nishi 8, Kita-Ku, Sapporo, 6-81, JAPAN katsura@math.sci.hokudai.ac.jp Summary. We give two pathological
More informationINVERSE SEMIGROUPS AND COMBINATORIAL C*-ALGEBRAS. R. Exel*
INVERSE SEMIGROUPS AND COMBINATORIAL C*-ALGEBRAS R. Exel* We describe a special class of representations of an inverse semigroup S on Hilbert s space which we term tight. These representations are supported
More informationThe Spectrum of Groupoid C -algebras
Dartmouth College GPOTS 2008 Outline The Objects Groupoids Isotropy Group Bundles Induced Representations A Bicontinuous Bijection Contextualizing Groupoids Isotropy Group Bundles Groupoids Let G be a
More informationGroup actions on C -correspondences
David Robertson University of Wollongong AustMS 27th September 2012 AustMS 27th September 2012 1 / Outline 1 C -correspondences 2 Group actions 3 Cuntz-Pimsner algebras AustMS 27th September 2012 2 / Outline
More informationS-REGULARITY AND THE CORONA FACTORIZATION PROPERTY
MATH. SCAND. 99 (2006), 204 216 S-REGULARITY AND THE CORONA FACTORIZATION PROPERTY D. KUCEROVSKY and P. W. NG Abstract Stability is an important and fundamental property of C -algebras. Given a short exact
More informationarxiv: v2 [math.ra] 8 Sep 2016
RECONSTRUCTION OF GRADED GROUPOIDS FROM GRADED STEINBERG ALGEBRAS PERE ARA, JOAN BOSA, ROOZBEH HAZRAT, AND AIDAN SIMS arxiv:1601.02872v2 [math.ra] 8 Sep 2016 Abstract. We show how to reconstruct a graded
More informationarxiv:math/ v4 [math.oa] 22 Oct 2004
OPERATOR ALGEBRAS AND MAULDIN-WILLIAMS GRAPHS arxiv:math/0401408v4 [math.oa] 22 Oct 2004 MARIUS IONESCU Abstract. We describe a method for associating a C -correspondence to a Mauldin-Williams graph and
More informationLECTURE NOTES ON HIGHER-RANK GRAPHS AND THEIR C -ALGEBRAS
LECTURE NOTES ON HIGHER-RANK GRAPHS AND THEIR C -ALGEBRAS AIDAN SIMS Abstract. These are notes for a short lecture course on k-graph C -algebras to be delivered at the Summer School on C -algebras and
More informationKMS states on the C-algebras of reducible graphs
University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 2014 KMS states on the C-algebras of reducible
More informationthe fiber of not being finitely generated. On the other extreme, even if the K-theory of the fiber vanishes,
J. Bosa and M. Dadarlat. () Local triviality for continuous field C -algebras, International Mathematics Research Notices, Vol., Article ID, 10 pages. doi:10.1093/imrn/ Local triviality for continuous
More informationarxiv: v1 [math.oa] 30 Oct 2017
Hausdorff étale groupoids and their C -algebras arxiv:1710.10897v1 [math.oa] 30 Oct 2017 Aidan Sims October 31, 2017 School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW
More informationSemigroup C*-algebras
Semigroup C*-algebras Xin Li Queen Mary University of London (QMUL) Plan for the talks Plan for my talks: Constructions and examples Nuclearity and amenability K-theory Classification, and some questions
More informationCrossed products of k-graph C*-algebras by Zl
University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 2009 Crossed products of k-graph C*-algebras by Zl Cyntyhia Farthing David
More informationCO-UNIVERSAL ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS, AND GAUGE-INVARIANT UNIQUENESS THEOREMS
CO-UNIVERSL LGEBRS SSOCITED TO PRODUCT SYSTEMS, ND GUGE-INVRINT UNIQUENESS THEOREMS TOKE M. CRLSEN, NDI S. LRSEN, IDN SIMS, ND SEN T. VITTDELLO bstract. Let (G, P ) be a quasi-lattice ordered group, and
More informationPurely infinite partial crossed products
University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers Faculty of Engineering and Information Sciences 2014 Purely infinite partial crossed products Thierry Giordano
More informationarxiv: v1 [math.oa] 22 Jan 2018
SIMPLE EQUIVARIANT C -ALGEBRAS WHOSE FULL AND REDUCED CROSSED PRODUCTS COINCIDE YUHEI SUZUKI arxiv:1801.06949v1 [math.oa] 22 Jan 2018 Abstract. For any second countable locally compact group G, we construct
More informationequivalence in expansive dynamics
On the C -algebras of homoclinic and heteroclinic equivalence in expansive dynamics matkt@imf.au.dk Institut for Matematiske Fag Det Naturvidenskabelige Fakultet Aarhus Universitet November 2007 Based
More informationTrees, Ultrametrics, and Noncommutative Geometry
Pure and Applied Mathematics Quarterly Volume 8, Number 1 (Special Issue: In honor of F. Thomas Farrell and Lowell E. Jones, Part 1 of 2 ) 221 312, 2012 Trees, Ultrametrics, and Noncommutative Geometry
More information