4.3. (i) Since HomA(A (I),M) ∼= M (I) for a set I and ⊕ is exact, it is clear that HomA(A (I),−) is exact, i.e. A(I) is projective. (ii) “Only if” part: Let A(P ) = ⊕ p∈P Ap be the free A-module indexed by P and ψ : A (P ) → p be the A-homomorphism such that ψ(1p) = p. Then ψ is surjective. If P is projective, then ψ has a section, i.e. an A-homomorphism φ : P → A(P ) such that ψ ◦ φ = idP . Th...

4.3. (i) Since HomA(A (I),M) ∼= M (I) for a set I and ⊕ is exact, it is clear that HomA(A (I),−) is exact, i.e. A(I) is projective. (ii) “Only if” part: Let A(P ) = ⊕ p∈P Ap be the free A-module indexed by P and ψ : A (P ) → p be the A-homomorphism such that ψ(1p) = p. Then ψ is surjective. If P is projective, then ψ has a section, i.e. an A-homomorphism φ : P → A(P ) such that ψ ◦ φ = idP . Th...

In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to exploit the deeper properties of Gabor frames. Furthermore, we a...

In this note we introduce the notion of Grassmann convexity analogous to the wellknown notion of convexity for curves in real projective spaces. We show that the curve in G2,4 osculating to a convex closed curve in P is Grassmann-convex. This proves that the tangent developable (i.e. the hypersurface formed by all tangents) of any convex curve in P has the ‘degree’ equal to 4. Here by ‘degree’ ...

I will build some standard resolutions for Mackey functors which are projective relative to p-subgroups. Those resolutions are closely related to the poset of p-subgroups. They lead to generalizations of known results on cohomology. They give a way to compute the Cartan matrix for Mackey functors, in terms of p-permutation modules, and to precise the structure of projective Mackey functors. The...

Abstract Let $H$ be a Hilbert space and $P(H)$ the projective of all quantum pure states. Wigner’s theorem states that every bijection $\phi \colon P(H)\to P(H)$ preserves angle between is automatically induced by either unitary or an antiunitary operator $U\colon H\to H$. Uhlhorn’s generalizes this result for bijective maps $ are only assumed to preserve $\frac{\pi }{2}$ (orthogonality) in bot...