arXiv subjects
Publications and source records attributed to Beatriz Abadie.
We give a definition of Toeplitz matrix acting on the $\ell^2$-space of an imprimitivity bimodule $X$ over a $C^*$-algebra $A$. We characterize the set of Toeplitz matrices as the closure in a certain topology of the image of the left regular representation of the crossed product $A\rtimes X$.
We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and $D^c_{μ' ν'}$ are isomorphic if and only if the parameters $(μ,ν)$ and $(μ',ν')$ belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.
With each Fell bundle over a discrete group G we associate a partial action of G on the spectrum of the unit fiber. We discuss the ideal structure of the corresponding full and reduced cross-sectional C*-algebras in terms of the dynamics of this partial action.
We associate an algebra $\Gami(\fA)$ to each bornological algebra $\fA$. The algebra $\Gami(\fA)$ contains a two-sided ideal $I_{S(\fA)}$ for each symmetric ideal $S\triqui\elli$ of bounded sequences of complex numbers. In the case of $\Gami=\Gami(\C)$, these are all the two-sided ideals, and $I_S\mapsto J_S=\cB I_S\cB$ gives a bijection between the two-sided ideals of $\Gami$ and those of $\cB=\cB(\ell^2)$. We prove that Weibel's $K$-theory groups $KH_*(I_{S(\fA)})$ are homotopy invariant for certain ideals $S$ including $c_0$ and $\ell^p$. Moreover, if either $S=c_0$ and $\fA$ is a local $C^*$-algebra or $S=\ell^p,\ell^{p\pm}$ and $\fA$ is a local Banach algebra, then $KH_*(I_{S(\fA)})$ contains $K_*^{\top}(\fA)$ as a direct summand. Furthermore, we prove that for $S\in\{c_0,\ell^p,\ell^{p\pm}\}$ the map $K_*(Γ^\infty(\fA):I_{S(\fA)})\to KH_*(I_{S(\fA)})$ fits into a long exact sequence with the relative cyclic homology groups $HC_*(Γ^\infty(\fA):I_{S(\fA)})$. Thus the latter groups measure the failure of the former map to be an isomorphism.
We discuss the crossed product by the dual action of the circle on the crossed product of a C*-algebra A by a Hilbert C*-bimodule X. When X is an A-A Morita equivalence bimodule, the double crossed product is shown to be Morita equivalent to the C*-algebra A.
A notion of unique ergodicity relative to the fixed-point subalgebra is defined for automorphisms of unital C*-algebras. It is proved that the free shift on any reduced amalgamated free product C*-algebra is uniquely ergodic relative to its fixed-point subalgebra, as are autormorphisms of reduced group C*-algebras arising from certain automorphisms of groups. A generalization of Haagerup's inequality, yielding bounds on the norms of certain elements in reduced amalgamated free product C*-algebras, is proved.
Given a correspondence X over a C*-algebra A, we construct a C*-algebra and a Hilbert C*-bimodule over it whose crossed product is isomorphic to the augmented Cuntz-Pimsner C*-algebra of X. This construction enables us to establish a condition for two augmented Cuntz-Pimsner C*-algebras to be Morita equivalent.
We discuss Morita equivalence within the family of quantum Heisenberg manifolds. The main tool employed is the generalization of a result of P. Green and M. Rieffel about Morita equivalence of transformation groups to crossed products by Hilbert bimodules.
A method for deforming C*-algebras is introduced, which applies to C*-algebras that can be described as the cross-sectional C*-algebra of a Fell bundle. Several well known examples of non-commutative algebras, usually obtained by deforming commutative ones by various methods, are shown to fit our unified perspective of deformation via Fell bundles. Examples are the non-commutative spheres of Matsumoto, the non-commutative lens spaces of Matsumoto and Tomiyama, and the quantum Heisenberg manifolds of Rieffel. In a special case, in which the deformation arises as a result of an action of R^{2d}, assumed to be periodic in the first d variables, we show that we get a strict deformation quantization.
A study of Hilbert $C^*$-bimodules over commutative $C^*$-algebras is carried out and used to establish a sufficient condition for two quantum Heisenberg manifolds to be isomorphic.
By means of techniques from the Morita equivalence theory, we get finitely generated and projective modules over the quantum Heisenberg manifolds. This enables us to get some information about the range of the trace of these algebras, at the level of the K_{0}-group, in terms of the Poisson bracket in whose direction the manifolds are deformed.
Let G and H be two locally compact groups acting on a C*-algebra A by commuting actions. We construct an action on the crossed product AXG out of a unitary 2-cocycle u and the action of H on A. For A commutative, and free and proper actions of G and H, we show that if the roles of these two actions are reversed, and u is replaced by u*, then the corresponding generalized fixed-point algebras, in the sense of Rieffel, are strong-Morita equivalent. We apply this result to the computation of the K-theory of quantum Heisenberg manifolds.