-ALGEBRAS ASSOCIATED WITH TWO-SIDED SUBSHIFTS

This paper is a continuation of the paper, Matsumoto [‘Subshifts, $\lambda $ -graph bisystems and $C^*$ -algebras’, J. Math. Anal. Appl. 485 (2020), 123843]. A $\lambda $ -graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift $\Lambda $ , there exists a $\lambda $ -graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ with shift automorphism $\rho _{\mathcal {L}}$ from a $\lambda $ -graph bisystem $({\mathcal {L}}^-,{\mathcal {L}}^+)$ , and define a $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ by the crossed product . It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If $\lambda $ -graph bisystems come from two-sided subshifts, these $C^*$ -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ and the K-theory formulas of the $C^*$ -algebras ${\mathcal {F}}_{\mathcal {L}}$ and ${\mathcal R}_{\mathcal {L}}$ . The K-group for the AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ is regarded as a two-sided extension of the dimension group of subshifts.

Collective Cohesion and Team Effi cacy in the Preparation Period and Team Performance

The aim of the study was to verify to what extent the eff ectiveness of sports teams throughout the season is conditioned by group processes, especially group cohesion and a sense of team effi cacy. Measurements of the analysed group processes were performed before the beginning of the main season, which allowed to obtain an answer to the question as to whether the level of group cohesion and the sense of team effi cacy developed before the start of league games is signifi cantly correlated with the team’s successes throughout the season proper. The study comprised 28 teams from 2 disciplines: basketball and volleyball. Both women and men participated in the study. Group cohesion was evaluated with the Polish version of the Group Environment Questionnaire (Polish adaptation according to Krawczyński, 1995), and the sense of team effi cacy was assessed with the Team Eff ectiveness Questionnaire (Polish version by Wałach-Biśta, 2015). The obtained results of simple regression analysis showed that the sense of team effi cacy is a signifi cant, strong and positive predictor of eff ectiveness on the pitch, both in women’s and men’s teams. Further analyses have indicated that the gender of athletes is a signifi cant moderator of the relationship between group cohesion in the GIS dimension (group social integration) and team performance. Hierarchical regression analysis demonstrated that gender, GIS, and gender interaction with GIS explain 20.2% of the variance regarding the dependent variable: effi ciency; and the overall model is statistically signifi cant (F(3, 24) = 3.28; p < 0.05). On the other hand, correlation analyses showed that in the men’s teams, along with the increase in social group integration, group eff ectiveness also signifi cantly increased (r = 0.436; p < 0.05). In the women’s teams, the correlation turned out to be signifi cant at the level of the statistical tendency, and the relationship between group eff ectiveness and the level of group social integration turned out to be negative and moderately strong (r = -0.432; p < 0.07).

Dimension groups for self-similar maps and matrix representations of the cores of the associated C*-algebras

We introduce a dimension group for a self-similar map as the $\mathrm {K}_0$ -group of the core of the C*-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit description of the core as the inductive limit using their matrix representations over the coefficient algebra, which can be described explicitly by the singularity structure of branched points. We compute that the dimension group for the tent map is isomorphic to the countably generated free abelian group ${\mathbb Z}^{\infty }\cong {\mathbb Z}[t]$ together with the unilateral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the $\mathrm {K}_0$ -groups are not automorphisms in general. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.

Automorphisms of the shift: Lyapunov exponents, entropy, and the dimension representation

Let $(X_{A},\unicode[STIX]{x1D70E}_{A})$ be a shift of finite type and $\text{Aut}(\unicode[STIX]{x1D70E}_{A})$ its corresponding automorphism group. Associated to $\unicode[STIX]{x1D719}\in \text{Aut}(\unicode[STIX]{x1D70E}_{A})$ are certain Lyapunov exponents $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$, which describe asymptotic behavior of the sequence of coding ranges of $\unicode[STIX]{x1D719}^{n}$. We give lower bounds on $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$ in terms of the spectral radius of the corresponding action of $\unicode[STIX]{x1D719}$ on the dimension group associated to $(X_{A},\unicode[STIX]{x1D70E}_{A})$. We also give lower bounds on the topological entropy $h_{\text{top}}(\unicode[STIX]{x1D719})$ in terms of a distinguished part of the spectrum of the action of $\unicode[STIX]{x1D719}$ on the dimension group, but show that, in general, $h_{\text{top}}(\unicode[STIX]{x1D719})$ is not bounded below by the logarithm of the spectral radius of the action of $\unicode[STIX]{x1D719}$ on the dimension group.

Nearly Approximate Transitivity (AT) for Circulant Matrices

By previous work of Giordano and the author, ergodic actions of $\mathbf{Z}$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in $\text{C}^{\ast }$-algebras and topological dynamics. Here we investigate how far from approximately transitive (AT) actions can be that derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, approximate transitivity arises. KIn addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided.

Al Said, Shakir Hassan (1925–2004)

Al Said was a prolific and influential artist. He was a founding member of the Baghdad Group for Modern Art (Jama’at Baghdad lil-fann al-hadith) in 1951, together with Jewad Selim and Jabra Ibrahim Jabra; and later, in 1971, of the One Dimension Group (al-Bu’d al-wahid). He wrote art manifestos for both groups in addition to his contemplative manifesto (al-bayan al-ta’ammuli), published in the cultural supplement of the Iraqi daily al-Jumhuriyya in 1966. The manifesto he wrote for the Baghdad Group for Modern Art was the first art manifesto of its kind in Iraq. It was read out at the group’s inaugural exhibition at the Museum of Ancient Costumes in Baghdad—an event that is considered by some the true birth of Iraqi modern art. The manifesto gives voice to the group’s commitment to both heritage and modernity. Its emphasis on local character drew inspiration from Islamic art, namely al-Wasiti’s thirteenth-century miniature paintings, but also from popular culture, like carpet production, and from the ancient civilizations of Mesopotamia. This meant distancing itself from the previous course of modern art in the Arab world, which was perceived as following European models, and setting out to ground modern art more firmly in a local context. It marked a reorientation in art that coincided with radical political change and the growth of Arab nationalism.

Homology for one-dimensional solenoids

Smale spaces are a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The definition was introduced to give an axiomatic description of the dynamical properties of Smale's Axiom A systems when restricted to a basic set. They include Anosov diffeomeorphisms, shifts of finite type and various solenoids constructed by R. F. Williams. The second author constructed a homology theory for Smale spaces which is based on (and extends) Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R. F. Williams.

Eigenvalues and strong orbit equivalence

We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues $E(X,T)$ of the minimal Cantor system $(X,T)$ is a subgroup of the intersection $I(X,T)$ of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated with $(X,T)$ is trivial, the quotient group $I(X,T)/E(X,T)$ is torsion free. We give examples with non-trivial infinitesimal subgroups where this property fails. We also provide some realization results.

Extensions of Cantor minimal systems and dimension groups

Given a factor map of Cantor minimal systems, we study the relations between the dimension groups of the two systems. First, we interpret the torsion subgroup of the quotient of the dimension groups in terms of intermediate extensions which are extensions of (Y,S) by a compact abelian group. Then we show that, by contrast, the existence of an intermediate non-abelian finite group extension can produce a situation where the dimension group of (Y,S) embeds into a proper subgroup of the dimension group of (X,T), yet the quotient of the dimension groups is nonetheless torsion free. Next we define higher order cohomology groups associated to an extension, and study them in various cases (proximal extensions, extensions by, not necessarily abelian, finite groups, etc.). Our main result here is that all the cohomology groups are torsion groups. As a consequence we can now identify as the torsion group of the quotient group .