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Paulo Varandas - One of the best experts on this subject based on the ideXlab platform.
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A variational principle for the metric mean dimension of Free Semigroup actions
Ergodic Theory and Dynamical Systems, 2021Co-Authors: Maria Carvalho, Fagner B. Rodrigues, Paulo VarandasAbstract:Abstract We consider continuous Free Semigroup actions generated by a family $(g_y)_{y \,\in \, Y}$ of continuous endomorphisms of a compact metric space $(X,d)$ , subject to a random walk $\mathbb P_\nu =\nu ^{\mathbb N}$ defined on a shift space $Y^{\mathbb N}$ , where $(Y, d_Y)$ is a compact metric space with finite upper box dimension and $\nu $ is a Borel probability measure on Y. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the Semigroup action with the corresponding notions for the associated skew product and the shift map $\sigma $ on $Y^{\mathbb {N}}$ , and compare them with the upper box dimension of Y. In particular, we obtain exact formulas whenever $\nu $ is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure $\nu $ , and to test the scope of our results.
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Quantitative recurrence for Free Semigroup actions
Nonlinearity, 2018Co-Authors: Maria Carvalho, Fagner B. Rodrigues, Paulo VarandasAbstract:We consider finitely generated Free Semigroup actions on a compact metric space and obtain quantitative information on Poincare recurrence, average first return time and hitting frequency for the random orbits induced by the Semigroup action. Besides, we relate the recurrence to balls with the rates of expansion of the Semigroup generators and the topological entropy of the Semigroup action. Finally, we establish a partial variational principle and prove an ergodic optimization for this kind of dynamical action.
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A variational principle for Free Semigroup actions
Advances in Mathematics, 2018Co-Authors: Maria Carvalho, Fagner B. Rodrigues, Paulo VarandasAbstract:Abstract In this paper we introduce a notion of measure theoretical entropy for a finitely generated Free Semigroup action and establish a variational principle when the Semigroup is generated by continuous self maps on a compact metric space and has finite topological entropy. In the case of Semigroups generated by Ruelle-expanding maps we prove the existence of equilibrium states and describe some of their properties. Of independent interest are the different ways we will present to compute the metric entropy and a characterization of the stationary measures.
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Semigroup Actions of Expanding Maps
Journal of Statistical Physics, 2017Co-Authors: Maria Carvalho, Fagner B. Rodrigues, Paulo VarandasAbstract:We consider Semigroups of Ruelle-expanding maps, parameterized by random walks on the Free Semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the Semigroup action and the thermodynamic formalism of the associated skew-product. In particular, we clarify the connection between the topological entropy of the Semigroup action and the growth rate of the periodic points, establish the main properties of the dynamical zeta function of the Semigroup action and relate these notions to recent research on annealed and quenched thermodynamic formalism. Meanwhile, we examine how the choice of the random walk in the Semigroup unsettles the ergodic properties of the action.
Maria Carvalho - One of the best experts on this subject based on the ideXlab platform.
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A variational principle for the metric mean dimension of Free Semigroup actions
Ergodic Theory and Dynamical Systems, 2021Co-Authors: Maria Carvalho, Fagner B. Rodrigues, Paulo VarandasAbstract:Abstract We consider continuous Free Semigroup actions generated by a family $(g_y)_{y \,\in \, Y}$ of continuous endomorphisms of a compact metric space $(X,d)$ , subject to a random walk $\mathbb P_\nu =\nu ^{\mathbb N}$ defined on a shift space $Y^{\mathbb N}$ , where $(Y, d_Y)$ is a compact metric space with finite upper box dimension and $\nu $ is a Borel probability measure on Y. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the Semigroup action with the corresponding notions for the associated skew product and the shift map $\sigma $ on $Y^{\mathbb {N}}$ , and compare them with the upper box dimension of Y. In particular, we obtain exact formulas whenever $\nu $ is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure $\nu $ , and to test the scope of our results.
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Quantitative recurrence for Free Semigroup actions
Nonlinearity, 2018Co-Authors: Maria Carvalho, Fagner B. Rodrigues, Paulo VarandasAbstract:We consider finitely generated Free Semigroup actions on a compact metric space and obtain quantitative information on Poincare recurrence, average first return time and hitting frequency for the random orbits induced by the Semigroup action. Besides, we relate the recurrence to balls with the rates of expansion of the Semigroup generators and the topological entropy of the Semigroup action. Finally, we establish a partial variational principle and prove an ergodic optimization for this kind of dynamical action.
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A variational principle for Free Semigroup actions
Advances in Mathematics, 2018Co-Authors: Maria Carvalho, Fagner B. Rodrigues, Paulo VarandasAbstract:Abstract In this paper we introduce a notion of measure theoretical entropy for a finitely generated Free Semigroup action and establish a variational principle when the Semigroup is generated by continuous self maps on a compact metric space and has finite topological entropy. In the case of Semigroups generated by Ruelle-expanding maps we prove the existence of equilibrium states and describe some of their properties. Of independent interest are the different ways we will present to compute the metric entropy and a characterization of the stationary measures.
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Semigroup Actions of Expanding Maps
Journal of Statistical Physics, 2017Co-Authors: Maria Carvalho, Fagner B. Rodrigues, Paulo VarandasAbstract:We consider Semigroups of Ruelle-expanding maps, parameterized by random walks on the Free Semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the Semigroup action and the thermodynamic formalism of the associated skew-product. In particular, we clarify the connection between the topological entropy of the Semigroup action and the growth rate of the periodic points, establish the main properties of the dynamical zeta function of the Semigroup action and relate these notions to recent research on annealed and quenched thermodynamic formalism. Meanwhile, we examine how the choice of the random walk in the Semigroup unsettles the ergodic properties of the action.
Kenneth R. Davidson - One of the best experts on this subject based on the ideXlab platform.
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A note on absolute continuity in Free Semigroup algebras
Houston Journal of Mathematics, 2008Co-Authors: Kenneth R. Davidson, Dilian YangAbstract:A Free Semigroup algebra is the weak operator topology closed (nonself-adjoint, unital) algebra generated by n isometries with pairwise orthogonal ranges. The prototype is the algebra generated by the left regular representation of the Free Semigroup on n letters. A Free Semigroup algebra which is isomorphic to the left regular algebra is called type L. If the infinite ampliation of the isometries generates a type L algebra, it is called weak-* type L. A Free Semigroup algebra is absolutely continuous if the vector functionals on it are equivalent to (some) vector functionals on the left regular representation. The purpose of this note is to show that absolutely continuous Free Semigroup algebras are weak-* type L
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B(H) is a Free Semigroup algebra
Proceedings of the American Mathematical Society, 2005Co-Authors: Kenneth R. DavidsonAbstract:We provide a simplified version of a construction of Charles Read. For any n > 2, there are n isometries with orthogonal ranges with the property that the nonselfadjoint weak-*-closed algebra that they generate is all of B(H).
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ABSOLUTELY CONTINUOUS REPRESENTATIONS AND A KAPLANSKY DENSITY THEOREM FOR Free Semigroup ALGEBRAS
Journal of Functional Analysis, 2005Co-Authors: Kenneth R. Davidson, David R. PittsAbstract:We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra An. Absolutely continuous functionals are used to help identify the type L part of the Free Semigroup algebra associated to a *-extendible representation σ. A *-extendible representation of An is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given *-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of σ(An) is weak-* dense in the unit ball of the associated Free Semigroup algebra if and only if σ is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts.
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Free Semigroup Algebras A Survey
Systems Approximation Singular Integral Operators and Related Topics, 2001Co-Authors: Kenneth R. DavidsonAbstract:This is a survey of recent results on Free Semigroup algebras, which are the WOT-closed algebras generated by n isometries with pairwise orthogonal ranges.
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Invariant Subspaces and Hyper-Reflexivity for Free Semigroup Algebras
Proceedings of the London Mathematical Society, 1999Co-Authors: Kenneth R. Davidson, David R. PittsAbstract:In this paper, we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call Free Semigroup algebras. This enables us to prove that they are reflexive, and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces. Such algebras are called hyper-reflexive. This property is very strong, but it has been established in only a very few cases. Moreover the prototypes of this class of algebras are the natural candidate for a non-commutative analytic Toeplitz algebra on n variables. The case we make for this analogy is very compelling. In particular, in this paper, the key to the invariant subspace analysis is a good analogue of the Beurling theorem for invariant subspaces of the unilateral shift. This leads to a notion of inner–outer factorization in these algebras. In a sequel to this paper [13], we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in Cn. A Free Semigroup algebra is the weak operator topology closed algebra generated by a set S1, . . . , Sn of isometries with pairwise orthogonal ranges. These conditions are described algebraically by S∗ i Sj = δijI for 1 ≤ i, j ≤ n; (F)
David R. Pitts - One of the best experts on this subject based on the ideXlab platform.
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ABSOLUTELY CONTINUOUS REPRESENTATIONS AND A KAPLANSKY DENSITY THEOREM FOR Free Semigroup ALGEBRAS
Journal of Functional Analysis, 2005Co-Authors: Kenneth R. Davidson, David R. PittsAbstract:We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra An. Absolutely continuous functionals are used to help identify the type L part of the Free Semigroup algebra associated to a *-extendible representation σ. A *-extendible representation of An is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given *-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of σ(An) is weak-* dense in the unit ball of the associated Free Semigroup algebra if and only if σ is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts.
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Invariant Subspaces and Hyper-Reflexivity for Free Semigroup Algebras
Proceedings of the London Mathematical Society, 1999Co-Authors: Kenneth R. Davidson, David R. PittsAbstract:In this paper, we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call Free Semigroup algebras. This enables us to prove that they are reflexive, and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces. Such algebras are called hyper-reflexive. This property is very strong, but it has been established in only a very few cases. Moreover the prototypes of this class of algebras are the natural candidate for a non-commutative analytic Toeplitz algebra on n variables. The case we make for this analogy is very compelling. In particular, in this paper, the key to the invariant subspace analysis is a good analogue of the Beurling theorem for invariant subspaces of the unilateral shift. This leads to a notion of inner–outer factorization in these algebras. In a sequel to this paper [13], we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in Cn. A Free Semigroup algebra is the weak operator topology closed algebra generated by a set S1, . . . , Sn of isometries with pairwise orthogonal ranges. These conditions are described algebraically by S∗ i Sj = δijI for 1 ≤ i, j ≤ n; (F)
Gelu Popescu - One of the best experts on this subject based on the ideXlab platform.
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Multivariable moment problems
Positivity, 2004Co-Authors: Gelu PopescuAbstract:In this paper we solve moment problems for Poisson transforms and, more generally, for completely positive linear maps on unital C ^*-algebras generated by “universal” row contractions associated with $$\mathbb{F}_{n}^{+}$$ , the Free Semigroup with n generators. This class of C*-algebras includes the Cuntz-Toeplitz algebra $$C^{*} (S_{1}, \ldots, S_{n})$$ (resp. $$C^{*} (B_{1}, \ldots, B_{n})$$ ) generated by the creation operators on the full (resp. symmetric, or anti-symmetric)) Fock space with n generators. As consequences, we obtain characterizations for the orbits of contractive Hilbert modules over complex Free Semigroup algebras such as $$\mathbb{CF}_{n}^{+}, \mathbb{C}[z_{1}, \ldots, z_{n}]$$ ,and, more generally, the quotient algebra $$\mathbb{CF}_{n}^{+}$$ , where J is an arbitrary two-sided ideal of $$\mathbb{CF}_{n}^{+}$$ . All these results are extended to the generalized Cuntz algebra $$\cal{O} (x_{i=1}^{n} G_{i}^{+})$$ , where G _ i ^+ are the positive cones ofdiscrete subgroups G _ i ^+ of the real line $$\mathbb{R}$$ . Moreover, we characterize the orbits of Hilbert modules over the quotient algebra $$\mathbb{C}_{i=1}^{*n} G_{i}^{+}/J$$ , where J is an arbitrary two-sided ideal ofthe Free Semigroup algebra $$\mathbb{C}_{i=1}^{*n} G_{i}^{+}$$ .
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similarity and ergodic theory of positive linear maps
arXiv: Operator Algebras, 2003Co-Authors: Gelu PopescuAbstract:In this paper we study the operator inequality \phi(X)\leq X and the operator equation \phi(X)= X, where \phi is a w^*-continuous positive (resp. completely positive) linear map on B(H). We show that their solutions are in one-to-one correspondence with a class of Poisson transforms on Cuntz-Toeplitz C^*-algebras, if \phi is completely positive. Canonical decompositions, ergodic type theorems, and lifting theorems are obtained and used to provide a complete description of all solutions, when \phi(I)\leq I. We show that the above-mentioned inequality (resp. equation) and the structure of its solutions have strong implications in connection with representations of Cuntz-Toeplitz C^*-algebras, common invariant subspaces for n-tuples of operators, similarity of positive linear maps, and numerical invariants associated with Hilbert modules over \CF_n^+, the complex Free Semigroup algebra generated by the Free Semigroup on n generators.
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Curvature Invariant for Hilbert Modules over Free Semigroup Algebras
Advances in Mathematics, 2001Co-Authors: Gelu PopescuAbstract:Abstract We introduce a notion of relative curvature (resp. Euler characteristic) for finite rank contractive Hilbert modules over C F +n, the complex Free Semigroup algebra generated by the Free Semigroup F +n on n generators. Asymptotic formulas and basic properties for both the curvature and the Euler characteristic are established. In particular, it is shown that the standard relative curvature invariant (resp. Euler characteristic) of a Hilbert module H is a nonnegative number less than or equal to the rank of H , and it depends only on the properties of the completely positive map φT(X)≔∑ni=1 TiXT*i, where [T1, …, Tn] is the row contraction of (not necessarily commuting) operators uniquely determined by the C F +n-module structure of H . Moreover, we prove that for every t⩾0 there is a Hilbert module H such that curv( H )=χ( H )=t. The module structure defined by the left creation operators on the full Fock space F2(Hn) on n generators occupies the position of the rank-one Free module in the algebraic theory. We obtain a complete description of the closed submodules (resp. quotients) of the Free Hilbert module F2(Hn) and calculate their curvature invariant. It is shown that the curvature is a complete invariant for the finite rank submodules of the Free Hilbert module F2(Hn)⊗ K , where K is a finite dimensional Hilbert space. A noncommutative version of the Gauss–Bonnet–Chern theorem from Riemannian geometry is obtained for graded Hilbert modules over C F +n. In particular, it is proved that the curvature and Euler characteristic coincide for certain classes of pure Hilbert modules. Our investigation is based on noncommutative Poisson transforms, noncommutative dilation theory, and harmonic analysis on Fock spaces.
