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Cornel Pasnicu - One of the best experts on this subject based on the ideXlab platform.
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The weak ideal property and Topological Dimension zero
Canadian Journal of Mathematics, 2017Co-Authors: Cornel Pasnicu, N. Christopher PhillipsAbstract:Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or Topological Dimension zero, and some results for C*-algebras with related properties. Some of the more important results include: The weak ideal property implies Topological Dimension zero. For a separable C*-algebra~A, Topological Dimension zero is equivalent to RR (O_2 \otimes A) = 0, to D \otimes A having the ideal property for some (or any) Kirchberg algebra~D, and to A being residually hereditarily in the class of all C*-algebras B such that O_{\infty} \otimes B contains a nonzero projection. Extending the known result for Z_2, the classes of C*-algebras with Topological Dimension zero, with the weak ideal property, and with residual (SP) are closed under crossed products by arbitrary actions of abelian 2-groups. If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A \otimes_{min} B has the weak ideal property. If X is a totally disconnected locally compact Hausdorff space and A is a C_0 (X)-algebra all of whose fibers have one of the weak ideal property, Topological Dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for Topological Dimension zero, provided A is separable). Topological Dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH algebras. The weak ideal property does not imply the ideal property for separable Z-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for.
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the weak ideal property and Topological Dimension zero
Canadian Journal of Mathematics, 2017Co-Authors: Cornel Pasnicu, Christopher N PhillipsAbstract:Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or Topological Dimension zero, and some results for C*-algebras with related properties. Some of the more important results include: • The weak ideal property implies Topological Dimension zero. • For a separable C*-algebra A, Topological Dimension zero is equivalent to RR(O2 ⊗ A) = 0, to D ⊗ A having the ideal property for some (or any) Kirchberg algebra D, and to A being residually hereditarily in the class of all C*-algebras B such that O∞ ⊗B contains a nonzero projection. • Extending the known result for Z2, the classes of C*-algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian 2-groups. • If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A ⊗min B has the weak ideal property. • If X is a totally disconnected locally compact Hausdorff space and A is a C0(X)-algebra all of whose fibers have one of the weak ideal property, Topological Dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for Topological Dimension zero, provided A is separable). • Topological Dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH algebras. • The weak ideal property does not imply the ideal property for separable Z-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for. The weak ideal property (recalled in Definition 1.3 below) was introduced in [25]; it is the property for which there are good permanence results (see Section 8 of [25]) which seems to be closest to the ideal property. (The ideal property fails to pass to extensions, by Theorem 5.1 of [18], to corners, by Example 2.8 of [24], and to fixed point algebras under actions of Z2, by Example 2.7 of [24]. The weak ideal property does all of these.) Topological Dimension zero was introduced in [4]; it is a non-Hausdorff version of total disconnectedness of the primitive ideal space of Date: 10 February 2016. 2010 Mathematics Subject Classification. Primary 46L05.
N. Christopher Phillips - One of the best experts on this subject based on the ideXlab platform.
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The weak ideal property and Topological Dimension zero
Canadian Journal of Mathematics, 2017Co-Authors: Cornel Pasnicu, N. Christopher PhillipsAbstract:Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or Topological Dimension zero, and some results for C*-algebras with related properties. Some of the more important results include: The weak ideal property implies Topological Dimension zero. For a separable C*-algebra~A, Topological Dimension zero is equivalent to RR (O_2 \otimes A) = 0, to D \otimes A having the ideal property for some (or any) Kirchberg algebra~D, and to A being residually hereditarily in the class of all C*-algebras B such that O_{\infty} \otimes B contains a nonzero projection. Extending the known result for Z_2, the classes of C*-algebras with Topological Dimension zero, with the weak ideal property, and with residual (SP) are closed under crossed products by arbitrary actions of abelian 2-groups. If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A \otimes_{min} B has the weak ideal property. If X is a totally disconnected locally compact Hausdorff space and A is a C_0 (X)-algebra all of whose fibers have one of the weak ideal property, Topological Dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for Topological Dimension zero, provided A is separable). Topological Dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH algebras. The weak ideal property does not imply the ideal property for separable Z-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for.
Christopher N Phillips - One of the best experts on this subject based on the ideXlab platform.
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the weak ideal property and Topological Dimension zero
Canadian Journal of Mathematics, 2017Co-Authors: Cornel Pasnicu, Christopher N PhillipsAbstract:Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or Topological Dimension zero, and some results for C*-algebras with related properties. Some of the more important results include: • The weak ideal property implies Topological Dimension zero. • For a separable C*-algebra A, Topological Dimension zero is equivalent to RR(O2 ⊗ A) = 0, to D ⊗ A having the ideal property for some (or any) Kirchberg algebra D, and to A being residually hereditarily in the class of all C*-algebras B such that O∞ ⊗B contains a nonzero projection. • Extending the known result for Z2, the classes of C*-algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian 2-groups. • If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A ⊗min B has the weak ideal property. • If X is a totally disconnected locally compact Hausdorff space and A is a C0(X)-algebra all of whose fibers have one of the weak ideal property, Topological Dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for Topological Dimension zero, provided A is separable). • Topological Dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH algebras. • The weak ideal property does not imply the ideal property for separable Z-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for. The weak ideal property (recalled in Definition 1.3 below) was introduced in [25]; it is the property for which there are good permanence results (see Section 8 of [25]) which seems to be closest to the ideal property. (The ideal property fails to pass to extensions, by Theorem 5.1 of [18], to corners, by Example 2.8 of [24], and to fixed point algebras under actions of Z2, by Example 2.7 of [24]. The weak ideal property does all of these.) Topological Dimension zero was introduced in [4]; it is a non-Hausdorff version of total disconnectedness of the primitive ideal space of Date: 10 February 2016. 2010 Mathematics Subject Classification. Primary 46L05.
Stacy Patterson - One of the best experts on this subject based on the ideXlab platform.
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effect of Topological Dimension on rigidity of vehicle formations fundamental limitations of local feedback
Conference on Decision and Control, 2008Co-Authors: Bassam Bamieh, Mihailo R Jovanovic, Partha P Mitra, Stacy PattersonAbstract:We consider the role of Topological Dimension in problems of network consensus and vehicular formations where only local feedback is available. In particular, we consider the simple network topologies of regular lattices in 1, 2 and higher Dimensions. Performance measures for consensus and formation problems are proposed that measure the deviation from average and rigidity or tightness of formations respectively. A common phenomenon appears where in Dimensions 1 and 2, consensus is impossible in the presence of any amount of additive stochastic perturbations, and in the limit of large formations. In Dimensions 3 and higher, consensus is indeed possible. We show that microscopic error measures that involve only neighboring sites do not suffer from this effect. This phenomenon reflects the fact that in Dimensions 1 and 2, local stabilizing feedbacks can not suppress long spatial wavelength ?meandering? motions. These effects are significantly more pronounced in vehicular problems than in consensus, and yet they are unrelated to string stability issues.
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CDC - Effect of Topological Dimension on rigidity of vehicle formations: Fundamental limitations of local feedback
2008 47th IEEE Conference on Decision and Control, 2008Co-Authors: Bassam Bamieh, Mihailo R Jovanovic, Partha P Mitra, Stacy PattersonAbstract:We consider the role of Topological Dimension in problems of network consensus and vehicular formations where only local feedback is available. In particular, we consider the simple network topologies of regular lattices in 1, 2 and higher Dimensions. Performance measures for consensus and formation problems are proposed that measure the deviation from average and rigidity or tightness of formations respectively. A common phenomenon appears where in Dimensions 1 and 2, consensus is impossible in the presence of any amount of additive stochastic perturbations, and in the limit of large formations. In Dimensions 3 and higher, consensus is indeed possible. We show that microscopic error measures that involve only neighboring sites do not suffer from this effect. This phenomenon reflects the fact that in Dimensions 1 and 2, local stabilizing feedbacks can not suppress long spatial wavelength ?meandering? motions. These effects are significantly more pronounced in vehicular problems than in consensus, and yet they are unrelated to string stability issues.
Bassam Bamieh - One of the best experts on this subject based on the ideXlab platform.
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effect of Topological Dimension on rigidity of vehicle formations fundamental limitations of local feedback
Conference on Decision and Control, 2008Co-Authors: Bassam Bamieh, Mihailo R Jovanovic, Partha P Mitra, Stacy PattersonAbstract:We consider the role of Topological Dimension in problems of network consensus and vehicular formations where only local feedback is available. In particular, we consider the simple network topologies of regular lattices in 1, 2 and higher Dimensions. Performance measures for consensus and formation problems are proposed that measure the deviation from average and rigidity or tightness of formations respectively. A common phenomenon appears where in Dimensions 1 and 2, consensus is impossible in the presence of any amount of additive stochastic perturbations, and in the limit of large formations. In Dimensions 3 and higher, consensus is indeed possible. We show that microscopic error measures that involve only neighboring sites do not suffer from this effect. This phenomenon reflects the fact that in Dimensions 1 and 2, local stabilizing feedbacks can not suppress long spatial wavelength ?meandering? motions. These effects are significantly more pronounced in vehicular problems than in consensus, and yet they are unrelated to string stability issues.
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CDC - Effect of Topological Dimension on rigidity of vehicle formations: Fundamental limitations of local feedback
2008 47th IEEE Conference on Decision and Control, 2008Co-Authors: Bassam Bamieh, Mihailo R Jovanovic, Partha P Mitra, Stacy PattersonAbstract:We consider the role of Topological Dimension in problems of network consensus and vehicular formations where only local feedback is available. In particular, we consider the simple network topologies of regular lattices in 1, 2 and higher Dimensions. Performance measures for consensus and formation problems are proposed that measure the deviation from average and rigidity or tightness of formations respectively. A common phenomenon appears where in Dimensions 1 and 2, consensus is impossible in the presence of any amount of additive stochastic perturbations, and in the limit of large formations. In Dimensions 3 and higher, consensus is indeed possible. We show that microscopic error measures that involve only neighboring sites do not suffer from this effect. This phenomenon reflects the fact that in Dimensions 1 and 2, local stabilizing feedbacks can not suppress long spatial wavelength ?meandering? motions. These effects are significantly more pronounced in vehicular problems than in consensus, and yet they are unrelated to string stability issues.