Results 1  10
of
23
The Entropy Theory of Symbolic Extensions
, 2002
"... Fix a topological system (X; T ), with its space K(X;T ) of T  invariant Borel probabilities. If (Y; S) is a symbolic system (subshift) and ' : (Y; S) ! (X; T ) is a topological extension (factor map), then the function ext on K(X;T ) which assigns to each the maximal entropy of a measur ..."
Abstract

Cited by 30 (4 self)
 Add to MetaCart
Fix a topological system (X; T ), with its space K(X;T ) of T  invariant Borel probabilities. If (Y; S) is a symbolic system (subshift) and ' : (Y; S) ! (X; T ) is a topological extension (factor map), then the function ext on K(X;T ) which assigns to each the maximal entropy of a measure on Y mapping to is called the extension entropy function of '. The in mum of such functions over all symbolic extensions is called the symbolic extension entropy function and is denoted by hsex . In this paper we completely characterize these functions in terms of functional analytic properties of an entropy structure on (X; T ). The entropy structure H is a sequence of entropy functions h k de ned with respect to a re ning sequence of partitions of X (or of X Z, for some auxiliary system (Z; R) with simple dynamics) whose boundaries have measure zero for all the invariant Borel probabilities.
On the cone of lower semicontinuous traces on a C*algebra
, 2009
"... The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2quasitraces on a (nonexact) C* ..."
Abstract

Cited by 17 (8 self)
 Add to MetaCart
The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2quasitraces on a (nonexact) C*algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It is shown that if a C*algebra absorbs the JiangSu algebra, then the subsemigroup of its Cuntz semigroup consisting of the purely noncompact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2quasitraces. This yields a computation of the Cuntz semigroup for the following two classes of C*algebras: C*algebras that absorb the JiangSu algebra and have no nonzero simple subquotients, and simple C*algebras that absorb the JiangSu algebra.
Completeness of the isomorphism problem for separable C*algebras
, 2013
"... We prove that the isomorphism problem for separable nuclear C*algebras is complete in the class of orbit equivalence relations. In fact, already the isomorphism of simple, separable AI C*algebras is a complete orbit equivalence relation. This means that any isomorphism problem arsing from a cont ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
We prove that the isomorphism problem for separable nuclear C*algebras is complete in the class of orbit equivalence relations. In fact, already the isomorphism of simple, separable AI C*algebras is a complete orbit equivalence relation. This means that any isomorphism problem arsing from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple, separable AI C*algebras. This sheds new light on the classification problem for separable C*algebras, which dates back to the 1960’s. In particular, this answers questions posed by Elliott, Farah, Hjorth, Paulsen, Rosendal, Toms and Törnquist.
ORDERS OF ACCUMULATION OF ENTROPY
"... Abstract. For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show that every countable ordinal is realized as the order of ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if M is a metrizable Choquet simplex, we bound the ordinals that appear as the order of accumulation of entropy of a dynamical system whose simplex of invariant measures is affinely homeomorphic to M. These bounds are given in terms of the CantorBendixson rank of ex(M), the closure of the extreme points of M, and the relative CantorBendixson rank of ex(M) with respect to ex(M). We also address the optimality of these bounds. 1.
ON KREINSMULIAN THEOREM FOR WEAKER TOPOLOGIES
"... Abstract. We investigate possible extensions of the classical KreinSmulian theorem to various weak topologies. In particular, if X is a WCG Banach space and is any locally convex topology weaker than the normtopology, then for every compact normbounded set H, conv H is compact. In arbitrary B ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We investigate possible extensions of the classical KreinSmulian theorem to various weak topologies. In particular, if X is a WCG Banach space and is any locally convex topology weaker than the normtopology, then for every compact normbounded set H, conv H is compact. In arbitrary Banach spaces, the normfragmentability assumption on H is shown to be sucient for the last property to hold. A new proof to the following result is given: if a Banach space does not contain a copy of `1[0; 1], then the KreinSmulian theorem holds for every topology induced by a norming set of functionals. We conclude that in such spaces a normbounded set is weakly compact if it is merely compact in topology induced by a boundary. On the other hand, the same statement is obtained for all C(K) and `1() spaces. 1.
A Framework for Optimization under Ambiguity
, 2008
"... In this paper, single stage stochastic programs with ambiguous distributions for the involved random variables are considered. Though the true distribution is unknown, existence of a reference measure ˆ P enables the construction of nonparametric ambiguity sets as Kantorovich balls around ˆ P. The ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
In this paper, single stage stochastic programs with ambiguous distributions for the involved random variables are considered. Though the true distribution is unknown, existence of a reference measure ˆ P enables the construction of nonparametric ambiguity sets as Kantorovich balls around ˆ P. The resulting robustified problems are infinite optimization problems and can therefore not be solved computationally. To solve these problems numerically, equivalent formulations as finite dimensional nonconvex, semi definite saddle point problems are proposed. Finally an application from portfolio selection is studied for which methods to solve the robust counterpart problems explicitly are proposed and numerical results for sample problems are computed.