The non-commutative torus $C^{*}({\mathbb{Z}}^n,\omega )$ is realized as the $C^{*}$-algebra of sections of a locally trivial $C^{*}$-algebra bundle over $\widehat{S_{\omega }}$ with fibres isomorphic to $C^{*}({\mathbb{Z}}^n/S_{\omega },{\omega }_1)$ for a totally skew multiplier ${\omega }_1$ on ${\mathbb{Z}}^n/S_{\omega }$. D. Poguntke [9] proved that $A_{\omega }$ is stably isomorphic to $C\left(\widehat{S_{\omega }}\right)\otimes C^{*}({\mathbb{Z}}^n/S_{\omega },{\omega }_1)\cong C\left(\widehat{S_{\omega }}\right)\otimes A_{\varphi }\otimes M_{kl}\left(ℂ\right)$ for a simple non-commutative torus $A_{\varphi }$ and an integer $kl$. It is well-known that a stable isomorphism of two separable $C^{*}$-algebras is equivalent to the existence of equivalence bimodule between them. We construct an $A_{\omega }$-$C\left(\widehat{S_{\omega }}\right)\otimes A_{\varphi }$-equivalence bimodule.

Let $𝒜={\left\{A_t\right\}}_{t\in G}$ and $ℬ={\left\{B_t\right\}}_{t\in G}$ be $C^{*}$-algebraic bundles over a finite group $G$. Let $C={⨁}_{t\in G}{A}_t$ and $D={⨁}_{t\in G}{B}_t$. Also, let $A=A_e$ and $B=B_e$, where $e$ is the unit element in $G$. We suppose that $C$ and $D$ are unital and $A$ and $B$ have the unit elements in $C$ and $D$, respectively. In this paper, we show that if there is an equivalence $𝒜-ℬ$-bundle over $G$ with some properties, then the unital inclusions of unital $C^{*}$-algebras $A\subset C$ and $B\subset D$ induced by $𝒜$ and $ℬ$ are strongly Morita equivalent. Also, we suppose that $𝒜$ and $ℬ$ are saturated and that $A^{\text{'}}\cap C=𝐂1$. We show that if $A\subset C$ and $B\subset D$...

We show that the family of Podleś spheres is complete under equivariant Morita equivalence (with respect to the action of quantum SU(2)), and determine the associated orbits. We also give explicit formulas for the actions which are equivariantly Morita equivalent with the quantum projective plane. In both cases, the computations are made by examining the localized spectral decomposition of a generalized Casimir element.

We present the review of noncommutative symmetries applied to Connes' formulation of spectral triples. We introduce the notion of equivariant spectral triples with Hopf algebras as isometries of noncommutative manifolds, relate it to other elements of theory (equivariant K-theory, homology, equivariant differential algebras) and provide several examples of spectral triples with their isometries: isospectral (twisted) deformations (including noncommutative torus) and finite spectral triples.

Using the Nagy dilation of linear contractions on Hilbert space and the Stinespring’s theorem for completely positive maps, we prove that any quantum dynamical system admits a dilation in the sense of Muhly and Solel which satisfies the same ergodic properties of the original quantum dynamical system.