Mathematics

We describe a construction by Gábor Elek, associating C*-algebras with uniformly recurrent subgroups, in the language of groupoid C*-algebras. This allows us to simplify several proofs in the original paper and fully characterise their nuclearity. We furthermore relate our groupoids to the dynamics of the group acting on its uniformly recurrent subgroup.

We introduce the notion of Δ and σΔ− pairs for operator algebras and characterise Δ− pairs through their categories of left operator modules over these algebras. Furthermore, we introduce the notion of Δ -Morita equivalent operator spaces and prove a similar theorem about their algebraic extensions. We prove that σΔ -Morita equivalent operator spaces are stably isomorphic and vice versa. Finally, we study unital operator spaces, emphasising their left (resp. right) multiplier algebras, and prove theorems that refer to Δ -Morita equivalence of their algebraic extensions.

The Sierpinski gasket admits a locally isometric ramified self-covering. A semifinite spectral triple is constructed on the resulting solenoidal space, and its main geometrical features are discussed.

We present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation, inter-twining lifting, commuting and non-commuting dilations, BCL theorem etc. We point out some natural generalizations and variations.

We define a quantum poset to be a hereditarily atomic von Neumann algebra equipped with a quantum partial order in Weaver's sense. These quantum posets form a category that is complete, cocomplete and symmetric monoidal closed. This yields a quantum analogue of the inclusion order on a powerset. We show that every quantum poset can be embedded into its powerset via a quantum analogue of the mapping that takes each element of a poset to its down set.

We characterize completely bounded normal Jordan ∗ -homomorphisms acting on von Neumann algebras. We also characterize completely positive isometries acting on noncommutative L p -spaces.

Combing Elliott, Gong, Lin and Niu's result and Castillejos and Evington's result, we see that if A is a simple separable nuclear monotracial C ∗ -algebra, then A⊗W is isomorphic to W where W is the Razak-Jacelon algebra. In this paper, we give another proof of this. In particular, we show that if D is a simple separable nuclear monotracial M 2 ∞ -stable C ∗ -algebra which is KK -equivalent to {0} , then D is isomorphic to W without considering tracial approximations of C ∗ -algebras with finite nuclear dimension. Our proof is based on Matui and Sato's technique, Schafhauser's idea in his proof of the Tikuisis-White-Winter theorem and properties of Kirchberg's central sequence C ∗ -algebra F(D) of D . Note that some results for F(D) is based on Elliott-Gong-Lin-Niu's stable uniqueness theorem. Also, we characterize W by using properties of F(W) . Indeed, we show that a simple separable nuclear monotracial C ∗ -algebra D is isomorphic to W if and only if D satisfies the following properties:(i) for any θ∈[0,1] , there exists a projection p in F(D) such that τ D,ω (p)=θ ,(ii) if p and q are projections in F(D) such that 0< τ D,ω (p)= τ D,ω (q) , then p is Murray-von Neumann equivalent to q ,(iii) there exists a homomorphism from D to W .

We will show that separable unital AF-algebras whose Bratteli diagrams do not allow converging two nodes into one node, can be classified up to the tensor product with the universal UHF-algebra $\Q$ only by their trace spaces. That is, if $\A$ and $\B$ are such AF-algebras, then T(A)=T(B) if and only if $\A\otimes \Q \cong \B\otimes \Q$.

A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras. Moreover, it contains all unital simple separable amenable C*-algebras which satisfy the UCT and have finite rational tracial rank

We consider a set SPG(A) of pure split states on a quantum spin chain A which are invariant under the on-site action τ of a finite group G . For each element ω in SPG(A) we can associate a second cohomology class c ω,R of G . We consider a classification of SPG(A) whose criterion is given as follows: ω 0 and ω 1 in SPG(A) are equivalent if there are automorphisms Ξ R , Ξ L on A R , A L (right and left half infinite chains) preserving the symmetry τ , such that ω 1 and ω 0 ∘( Ξ L ⊗ Ξ R ) are quasi-equivalent. It means that we can move ω 0 close to ω 1 without changing the entanglement nor breaking the symmetry. We show that the second cohomology class c ω,R is the complete invariant of this classification.