Commutativity and continuity conditions for the Moore-Penrose inverse and the "conorm" are established in a C*-algebra; moreover, spectral permanence and B*-properties for the conorm are proved.

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...

We consider circulant graphs having $p$ vertices, with $p$ prime. To any such graph we associate a certain number $k$, that we call type of the graph. We prove that for $p\gg k$ the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.

We tackle R. V. Kadison’s similarity problem (i.e. any bounded representation of any unital C*-algebra is similar to a *-representation), paying attention to the class of C*-unitarisable groups (those groups G for which the full group C*-algebra C*(G) satisfies Kadison’s problem) and thereby we establish some stability results for Kadison’s problem. Namely, we prove that $A{\otimes }_{min}B$ inherits the similarity problem from those of the C*-algebras A and B, provided B is also nuclear. Then we prove that G/Γ is...