The generalized non-commutative torus $T_{\varrho }^k$ of rank n is defined by the crossed product $A_{m/k}{\times }_{\alpha ₃}\mathbb{Z}{\times }_{\alpha ₄}...{\times }_{\alpha ₙ}\mathbb{Z}$, where the actions ${\alpha }_i$ of ℤ on the fibre $M_k\left(ℂ\right)$ of a rational rotation algebra $A_{m/k}$ are trivial, and $C*\left(k\mathbb{Z}\times k\mathbb{Z}\right){\times }_{\alpha ₃}\mathbb{Z}{\times }_{\alpha ₄}...{\times }_{\alpha ₙ}\mathbb{Z}$ is a non-commutative torus $A_{\varrho }$. It is shown that $T_{\varrho }^k$ is strongly Morita equivalent to $A_{\varrho }$, and that $T_{\varrho }^k\otimes M_{p^{\infty }}$ is isomorphic to $A_{\varrho }\otimes M_k\left(ℂ\right)\otimes M_{p^{\infty }}$ if and only if the set of prime factors of k is a subset of the set of prime factors of p.

Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections $P_a$ determined by the different involutions ${}_a$ induced by positive invertible elements a ∈ A. The maps $\phi :P\to P_a$ sending p to the unique $q\in P_a$ with the same range as p and ${\Omega }_a:P_a\to P_a$ sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| < 1 such that...