Equivariant -absorption theorem for exact groups

We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal {O}_{2}$ -absorbing, pointwise outer action on the Cuntz algebra $\mathcal {O}_{2}$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal {O}_{2}$ -absorption theorem for these groups.

ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

Rohlin Actions of Finite Groups on the Razak–Jacelon Algebra

Let $A$ be a simple separable nuclear C$^*$-algebra with a unique tracial state and no unbounded traces, and let $\alpha $ be a strongly outer action of a finite group $G$ on $A$. In this paper, we show that $\alpha \otimes \textrm{id}$ on $A\otimes \mathcal{W}$ has the Rohlin property where $\mathcal{W}$ is the Razak–Jacelon algebra. Combing this result with the recent classification results and our previous result, we see that such actions are unique up to conjugacy.

On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras

We show that an outer action of a finite abelian group on a simple Cuntz-Krieger algebra is strongly approximately inner in the sense of Izumi if the action is given by diagonal quasi-free automorphisms and the associated matrix is aperiodic. This is achieved by an approximate cohomology vanishing-type argument for the canonical shift restricted to the relative commutant of the set of domain projections of the canonical generating isometries in the fixed point algebra.

On pathological properties of fixed point algebras in Kirchberg algebras

We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

Spectral gap characterization of full type III factors

We give a spectral gap characterization of fullness for type {\mathrm{III}} factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if M is a full factor and {\sigma:G\rightarrow\mathrm{Aut}(M)} is an outer action of a discrete group G whose image in {\mathrm{Out}(M)} is discrete, then the crossed product von Neumann algebra {M\rtimes_{\sigma}G} is also a full factor. We apply this result to prove the following conjecture of Tomatsu–Ueda: the continuous core of a type {\mathrm{III}_{1}} factor M is full if and only if M is full and its τ invariant is the usual topology on {\mathbb{R}}.

Group actions on Smale space -algebras

Group actions on a Smale space and the actions induced on the $\text{C}^{\ast }$-algebras associated to such a dynamical system are studied. We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra. We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algebra to the induced actions on the stable and unstable $\text{C}^{\ast }$-algebras. In each of these cases, we discuss the preservation of properties (such as finite nuclear dimension, ${\mathcal{Z}}$-stability, and classification by Elliott invariants) in the resulting crossed products.

Outer Partial Actions and Partial Skew Group Rings

We extend the classical notion of an outer action α of a group G on a unital ring A to the case when α is a partial action on ideals, all of which have local units. We show that if α is an outer partial action of an abelian group G, then its associated partial skew group ring A *α G is simple if and only if A is G-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

Studying Web 2.0 Interactivity

With more than one third of the world's population being online, the Internet has increasingly become part of modern living, giving rise to popular literature that often takes a teleological and celebratory perspective, heralding the Internet and Web 2.0 specifically, as an enabler of participation, democracy, and interactivity. However, one should not take these technological affordances of Web 2.0 for granted. This article applies an interaction framework to the analysis of two Web 2.0 websites viewed as spaces where interaction goes beyond the mere consultation and selection of content, i.e., as spaces supporting the (co)creation of content and value. The authors' approach to interactivity seeks to describe websites in objective, structural terms as spaces of user, document, and website affordances. The framework also makes it possible to talk about the websites in subjective, functional terms, considering them as spaces of perceived inter-action, intra-action and outer-action affordances. Analysis finds that both websites provide numerous user, document, and website affordances that can serve as inter-action or social affordances.