CONDITION (K) FOR BOOLEAN DYNAMICAL SYSTEMS

We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

Correction: Gušić, D. Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One. Mathematics 2020, 8, 1762

The author wishes to make the following correction to the paper [...]

On Some Classical andWeighted Estimates for SL4

The purpose of this paper is two-sided. First, we obtain the correct estimate of the error term in the classical prime geodesic theorem for compact symmetric space SL4. As it turns out, the corrected error term depends on the degree of a certain polynomial appearing in the functional equation of the attached zeta function. This is in line with the known result in the case of compact Riemann surface, or more generally, with the corresponding result in the case of compact locally symmetric spaces of real rank one. Second, we derive a weighted form of the theorem. In particular, we prove that the aforementioned error term can be significantly improved when the classical approach is replaced by its higher level analogue.

C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

We study homeomorphisms of a Cantor set with $k$ ( $k<+\infty$ ) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where $k\geq 2$ , the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition $(X,\unicode[STIX]{x1D70E})$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist of infinitesimals.

The classification of certain ASH C*-algebras of real rank zero

In this paper, using ordered total K-theory, we give a K-theoretic classification for the real rank zero inductive limits of direct sums of generalized dimension drop interval algebras.

Rokhlin dimension: duality, tracial properties, and crossed products

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of $C^{\ast }$-algebras, including: $D$-absorbing $C^{\ast }$-algebras, where $D$ is a strongly self-absorbing $C^{\ast }$-algebra; stable $C^{\ast }$-algebras; $C^{\ast }$-algebras with finite nuclear dimension (or decomposition rank); $C^{\ast }$-algebras with finite stable rank (or real rank); and $C^{\ast }$-algebras whose $K$-theory is either trivial, rational, or $n$-divisible for $n\in \mathbb{N}$. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_{2}$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

THE INFLUENCE OF LEADERSHIP AND STRATEGIC PLANNING TOWARD LOCAL REVENUE (AT WEST BANDUNG REGENCY)

The goal of this research is to describe the influence of leaderships and strategic planning toward local revenue at West Bandung Regency. Research method is Quantative research and collect data by quessioner, interview , observation and documentation.The variables were leaderships and strategic planning. as the independent variables, the dependent variable was local revenue. Data have been analyzed through(Multiple Regresion).The result of this research showed that leaderships and strategic planning. influence the local revenue The influence of leaderships and strategic planning toward local revenue is positive it means that the more higher leaderships and the strategic planning the more higher local revenue and it has mean in real rank p < 0.01.By this research, the hypothesis, which has been proposed, leaderships and strategic planning will influence local revenue. The result has showed that the mean of local revenue is 0. 4570 or the grade variation of local revenue is 45,70 percent depending on leaderships and strategic planning

THE INFLUENCE OF LEADERSHIP AND STRATEGIC PLANNING TOWARD LOCAL REVENUE (AT WEST BANDUNG REGENCY)

The goal of this research is to describe the influence of leaderships and strategic planning toward local revenue at West Bandung Regency. Research method is Quantative research and collect data by quessioner, interview , observation and documentation.The variables were leaderships and strategic planning. as the independent variables, the dependent variable was local revenue. Data have been analyzed through(Multiple Regresion).The result of this research showed that leaderships and strategic planning. influence the local revenue The influence of leaderships and strategic planning toward local revenue is positive it means that the more higher leaderships and the strategic planning the more higher local revenue and it has mean in real rank p < 0.01.By this research, the hypothesis, which has been proposed, leaderships and strategic planning will influence local revenue. The result has showed that the mean of local revenue is 0. 4570 or the grade variation of local revenue is 45,70 percent depending on leaderships and strategic planning