Let H be a closed subgroup of a locally compact group G and let X=G/H be the quotient space of left cosets. Let C*X be the corresponding G-C*-algebra of continuous functions on X, vanishing at infinity. Suppose that L is a closed abelian subgroup of H and let f be a 2-cocycle on the dual group of L. Let G(f) be the Rieffel deformation of G. Using these data we may construct G(f)-C*-algebra C*X(f) - the Rieffel deformation of C*X. On the other hand we may perform the Rieffel deformation of the subgroup H obtaining the closed quantum subgroup H(f) of G(f) which in turn, by the results of Vaes, leads to the G(f)-C*-algebra G(f)/H(f). In this paper we show that G(f)/H(f) and C*X(f) are isomorphic G(f)-C*-algebras. We also consider the case where L is a subgroup of G but not of H, for which we cannot construct the subgroup H(f). Then C*X(f) cannot be identified with a quantum quotient. What may be shown is that it is a G(f)-simple object in the category of G(f)-C*-algebras.
With an action $α$ of $\mathbb{R}^n$ on a $C^*$-algebra $A$ and a skew-symmetric $n\times n$ matrix $Θ$ one can consider the Rieffel deformation $A_Θ$ of $A$, which is a $C^*$-algebra generated by the $α$-smooth elements of $A$ with a new multiplication. The purpose of this paper is to obtain explicit formulas for $K$-theoretical quantities defined by elements of $A_Θ$. We assume that there is a densely defined trace on $A$, invariant under the action. We give an explicit realization of Thom class in $KK$ in any dimension $n$, and use it in the index pairings. When $n$ is odd, for example, we …
We consider the $C^*$-algebra $\mathcal{E}_{n,m}^q$, which is a $q$-twist of two Cuntz-Toeplitz algebras. For the case $|q|<1$, we give an explicit formula which untwists the $q$-deformation showing that the isomorphism class of $\mathcal{E}_{n,m}^q$ does not depend on $q$. For the case $|q|=1$, we give an explicit description of all ideals in $\mathcal{E}_{n,m}^q$. In particular, we show that $\mathcal{E}_{n,m}^q$ contains a unique largest ideal $\mathcal{M}_q$. We identify $\mathcal{E}_{n,m}^q / \mathcal{M}_q$ with the Rieffel deformation of $\mathcal{O}_n \otimes \mathcal{O}_m$ and use a K-…
We give some sufficient conditions for the injectivity of actions of compact quantum groups on $C^{\ast}$-algebra. As an application, we prove that any faithful smooth action by a compact quantum group on a compact smooth (not necessarily connected) manifold is injective. A similar result is proved for actions on $C^{\ast}$- algebras obtained by Rieffel-deformation of compact, smooth manifolds.
Given a compact metric space X and a unital C*-algebra A, we introduce a family of seminorms on the C*-algebra of continuous functions from X to A, denoted C(X, A), induced by classical Lipschitz seminorms that produce compact quantum metrics in the sense of Rieffel if and only if A is finite-dimensional. As a consequence, we are able isometrically embed X into the state space of C(X,A). Furthermore, we are able to extend convergence of compact metric spaces in the Gromov-Hausdorff distance to convergence of spaces of matrices over continuous functions on the associated compact metric spaces i…
We consider $C^*$-algebra $\mathcal{E}_{n,m}^q$, which is a $q$-twist of two Cuntz-Toeplitz algebras. For the case $|q|<1$ we give an explicit formula, which untwists the $q$-deformation, thus showing that the isomorphism class of $\mathcal{E}_{n,m}^q$ does not depend of $q$. For the case $|q|=1$ we give an explicit description of all ideals in $\mathcal{E}_{n,m}^q$. In particular $\mathcal{E}_{n,m}^q$ contains unique largest ideal $\mathcal{M}_q$. Then we identify $\mathcal{E}_{n,m}^q / \mathcal{M}_q$ with the Rieffel deformation of $\mathcal{O}_n \otimes \mathcal{O}_m$ and use a K-theoretica…
Let $\mathbb B$ be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action $α$ of $\mathbb B$ on a Fréchet algebra $\mathcal A$. Denote by $\mathcal A^\infty$ the associated Fréchet algebra of smooth vectors for the action $α$. In the Abelian case $\mathbb B=\mathbb R^{2n}$ and $α$ isometric, Marc Rieffel proved that Weyl's operator symbol composition formula yields a deformation through Fréchet algebra structures ${\star_θ^α}_{θ\in\mathbb R}$ on $\mathcal A^\infty$. When $\mathcal A$ is a $C^\star$-algebra, every deformed algebra $(\…
In this paper, we review some recent developments of compact quantum groups that arise as $θ$-deformations of compact Lie groups of rank at least two. A $θ$-deformation is merely a 2-cocycle deformation using an action of a torus of dimension higher than 2. Using the formula (Lemma 5.3) developed in \cite{W2018}, we derive the noncommutative 7-sphere in the sense of Connes and Landi \cite{CL2001} as the fixed-point subalgebra.
