We define two such pairs (X, i), (Y, j) to be equivalent if there is an isogeny α : X → Y such that if α̃ : End(X) ' End(Y ) is the induced isomorphism, we have α̃ ◦ i = j. It is easy to check that this is an equivalence relation. We write X ∼ Y if (X, i) is equivalent to (Y, j). Let AF be the collection of equivalence classes. We will classify such equivalence classes. Let D be any R-algebra. A ...

Let R be a Dedekind domain, G an affine flat R-group scheme, and B a flat R-algebra on which G acts. Let A → BG be an R-algebra map. Assume that A is Noetherian. We show that if the induced map K ⊗ A → (K ⊗ B)K⊗G is an isomorphism for any algebraically closed field K which is an R-algebra, then S ⊗A→ (S ⊗B)S⊗G is an isomorphism for any R-algebra S.

We introduce and study bimeasurings from pairs of bialge-bras to algebras. It is shown that the universal bimeasuring bialgebra construction, which arises from Sweedler's universal measuring coalgebra construction and generalizes the finite dual, gives rise to a contravariant functor on the category of bialgebras adjoint to itself. An interpretation of bimeasurings as algebras in the category o...

In this note, we generalize a result of [4] (see also [9]) and set the isomorphism between the iterated cross product algebra H∨#(H#A) and braided analog of an A-valued matrix algebra H∨⊗A⊗H for a Hopf algebra H in the braided category C and for an H-module algebra A. As a preliminary step, we prove the equivalence between categories of modules over both algebras and category whose objects are ...

For every hyperbolic group Γ with Gromov boundary ∂Γ, one can form the cross product C∗-algebra C(∂Γ)⋊Γ. For each such algebra we construct a canonical K-homology class, which induces a Poincaré duality map K∗(C(∂Γ)⋊Γ) → K (C(∂Γ)⋊Γ). We show that this map is an isomorphism in the case of Γ = F2 the free group on two generators. We point out a direct connection between our constructions and the ...

The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups. We show that in many cases, isomorphisms of Cuntz semigroups that respect this additional structure can be ...

For certain families of compact subsets the plane, isomorphism class algebra absolutely continuous functions on a set is completely determined by homeomorphism set. This analogous to Gelfand–Kolmogorov theorem for C(K) spaces. In this paper, we define family sets comprising finite unions convex curves and show that has ‘Gelfand–Kolmogorov’ property.

In this paper we determine the number of endomorphism rings superspecial abelian surfaces over a field $\mathbb{F}_q$ odd degree $\mathbb{F}_p$ in isogeny class corresponding to Weil $q$-number $\pm\sqrt{q}$. This extends earlier works T.-C. Yang and present authors on isomorphism classes these surfaces, also generalizes classical formula Deuring for supersingular elliptic curves. Our method is...

In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras andΦ : A→ B is a linear map onto B that preserves the spectrum of elements, thenΦ is a Jordan isomorphism if either A or B is a C∗-al...

It is well known that a measured groupoid G defines a von Neumann algebra W ∗(G), and that a Lie groupoid G canonically defines both a C∗-algebra C∗(G) and a Poisson manifold A∗(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C∗-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of obj...