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Register a new userThe corona algebra of stablized Jiang-Su algebra
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Let ${cal Z}$ be the Jiang-Su algebra and ${cal K}$ the C*-algebra of compact operators on an infinite dimensional separable Hilbert space. We prove that the corona algebra $M({cal Z}otimes {cal K})/{cal Z}otimes {cal K}$ has real rank zero. We actually prove a more general result.
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The parabolic algebra A_p is the weakly closed algebra on L^2(R) generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions e^{ilambda x}, lambda geq 0. This algebra is reflexive with an invariant subspace lattice, Lat A_p, which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). This identification is used here to classify strongly irreducible isometric representations of the partial Weyl commutation relations. The notion of a synthetic subspace lattice is extended from commutative to noncommutative lattices and it is shown that Lat A_p is nonsynthetic relative to the maximal abelian multiplication subalgebra of A_p. Also, operator algebras derived from isometric representations of A_p and from compact perturbations are defined and determined.
We define a groupoid from a labelled space and show that it is isomorphic to the tight groupoid arising from an inverse semigroup associated with the labelled space. We then define a local homeomorphism on the tight spectrum that is a generalization of the shift map for graphs, and show that the defined groupoid is isomorphic to the Renault-Deaconu groupoid for this local homeomorphism. Finally, we show that the C*-algebra of this groupoid is isomorphic to the C*-algebra of the labelled space as introduced by Bates and Pask.
We examine the common null spaces of families of Herz-Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in J. Funct. Anal. 266 (2014), 6473-6500 can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.
Given a connected and locally compact Hausdorff space X with a good base K we assign, in a functorial way, a C(X)-algebra to any precosheaf of C*-algebras A defined over K. Afterwards we consider the representation theory and the Kasparov K-homology of A, and interpret them in terms, respectively, of the representation theory and the K-homology of the associated C(X)-algebra. When A is an observable net over the spacetime X in the sense of algebraic quantum field theory, this yields a geometric description of the recently discovered representations affected by the topology of X.
In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompsons group $mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompsons semigroup $mathcal{S}$ and the free semigroup $mathcal{F}_n$ on $n$ generators ($geq2$). Let $mathfrak{B}(mathcal{S})$ and $mathfrak{B}(mathcal{F}_n)$ be the Banach algebras generated by the left regular representations of $mathcal{S}$ and $mathcal{F}_n$, respectively. It is proved that all derivations on $mathfrak{B}(mathcal{S})$ and $mathfrak{B}(mathcal{F}_n)$ are automatically continuous, and every derivation on $mathfrak{B}(mathcal{S})$ is induced by a bounded linear operator in $mathcal{L}(mathcal{S})$, the weak closed Banach algebra consisting of all bounded left convolution operators on $l^2(mathcal{S})$. Moreover, we show that the first continuous Hochschild cohomology group of $mathfrak{B}(mathcal{S})$ with coefficients in $mathcal{L}(mathcal{S})$ vanishes. These conclusions provide positive indications for the left amenability of Thompsons semigroup.
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