arXiv subjects
Publications and source records attributed to Joachim Zacharias.
Noncommutative topological entropy estimates are obtained for polynomial gauge invariant endomorphisms of Cuntz algebras, generalising known results for the canonical shift endomorphisms. Exact values for the entropy are computed for a class of permutative endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura.
Higher-rank graph generalisations of the Popescu-Poisson transform are constructed, allowing us to develop a dilation theory for higher rank operator tuples. These dilations are joint dilations of the families of operators satisfying relations encoded by the graph structure which we call $Λ$-contractions or $Λ$-coisometries. Besides commutant lifting results and characterisations of pure states on higher rank graph algebras several applications to the structure theory of non-selfadjoint graph operator algebras are presented generalising recent results in special cases.
Let X be a Hilbert bimodule over a C*-algebra A and $O_X= A \rtimes_X \Z$. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on $s_ξ s_η^*$ as Schur multipliers converging to the identity. This shows immediately that for a finitely generated X the algebra $O_X$ inherits any standard approximation property such as nuclearity, exactness, CBAP or OAP from A. We generalise this to certain general Pimsner algebras by proving semi-splitness of the Toeplitz extension under certain conditions and discuss some examples.
Let O_{Lambda} be a higher rank graph C*-algebra of rank r. For every tuple p of non-negative integers there is a canonical completely positive map Phi^p on O_{Lambda} and a subshift T^p on the path space X of the graph. We show that ht(Phi^p)=h(T^p), where ht is Voiculescu's approximation entropy and h the classical topological entropy. For a higher rank Cuntz-Krieger algebra O_M we obtain ht(Phi^p)= log r(M_1^{p_1}M_2^{p_2} ... M_r^{p_r}), r being the spectral radius. This generalises Boca and Goldstein's result for Cuntz-Krieger algebras.
Higher-rank versions of Wold decomposition are shown to hold for doubly commuting isometric representations of product systems of C*-correspondences over N^k, generalising the classical result for a doubly commuting pair of isometries due to M.Slocinski. Certain decompositions are also obtained for the general, not necessarily doubly commuting, case and several corollaries and examples are provided. Possibilities of extending isometric representations to fully coisometric ones are discussed.
Given a contractive tuple of Hilbert space operators satisfying certain $A$-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximal $A$-relation piece. We define a maximal piece more generally for a finite set of polynomials in $n$ noncommuting variables. We classify all representations of Cuntz-Krieger algebras ${\mathcal O}_A$ obtained from dilations of commuting tuples satisfying $A$-relations. The universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.