Derived equivalent varieties with differing integral Mukai-Hodge structures?

For a smooth projective complex variety X of dimension n, let Hi(X) denote its integral Hodge structure of weight i. Define ~H0(X)=⨁H2i(X)⊗Z(i) and ~H1(X)=⨁H2i+1⊗Z(i), respectively. It is known that any derived equivalence Φ:Db(X)→Db(Y) induces isomorphisms of rational Hodge structures ~H0Q(X)≅~H0Q(Y) and ~H1Q(X)≅~H1Q(Y), however these isomorphisms are defined by characteristic classes whose coefficients aren’t necessarily integral. … Read more

Quantization of $S^2$ as $C^*$-algebra?

The general context for the question – is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695). The particular question is about the most simple example – sphere $S^2$ with the standard symplectic form. I think that corresponding $C^*$-algebra can be defined explicitly by generators and relations. … Read more

A continuous functional calculus on/positive elements in a Fréchet algebra?

I am trying to understand what (minimal) conditions one would need in order to obtain a functional calculus on a Fréchet algebra, which we demand to be equipped with an involution that leaves all semi-norms invariant. The motivation behind this is that I need to see if forcing a Fréchet algebra to have positive elements … Read more

Bott-type projections in C∗C^*-algebras

Let A be a unital C∗-algebra and a∈A. If aa∗+1 is invertible in A then the element β(a)=(aa∗+I)−1(aa∗aa∗I) is an idempotent. The formula is similar to the one defining the Bott projection in the classical case. I wanted to know what does the above β mean for the K-theory of A? Does it have any … Read more

Bounded self-adjoint perturbation of a p-summable spectral triple

I am new to the field of Noncommutative Geometry.I was reading the chapter on Spectral triple from the book ‘Elements of Noncommutative Geometry’ by Gracia-Bondía,Várilly and Figueroa.Now,after reading the basics of spectral triple,I have a felling that if I perturb the Dirac operator $\mathcal{D}$ of the Spectral triple $(\mathcal{A},\mathcal{H},\mathcal{D})$ by a bounded self-adjoint operator $\mathcal{S}$,then … Read more

Foliations, von Neumann algebras and measurability

In the excellent book Noncommutative Geometry by Alain Connes much of the first chapter is devoted to foliations. At the end of the first chapter the author discusses index theory on measured foliations (foliation equipped with the so called transverse measure). From the discussion there I didin’t get how this assumption is restrictive: I would … Read more

Definition of the GG-equivariant index map

My question concerns a statement on page 12 of the following paper of Baum, Connes, and Higson: http://www.mmas.univ-metz.fr/~gnc/bibliographie/BaumConnes/Baum-Connes-Higson.pdf about the definition of the G-index map from the G-equivariant K-homology group KG0(X) to K0(C∗r(G)), where X is a proper G-compact G-manifold. Suppose (H+,H−,F) is a G-equivariant abstract elliptic operator on X (defined on the previous page … Read more

Spectrum of the hypoelliptic transverse signature operator

Let D be the transverse signature operator constructed by Connes and Moscovici in the paper “Local index formula in Noncommutative Geometry”:this is first order hypoelliptic pseudodifferential operator D defined by the equality D|D|=Q where Q=(dVd∗V−d∗VdV)⊕(dH+d∗H) where dV,dH are vertical and horizontal exterior derivative. It acts on the sections of the bundle Λ(V∗)⊗Λ(p∗(T∗M)) over P:=GL+(M)/SO(n) (the … Read more

informative examples for understanding spectral triples

I am at the beginning of my thesis work and I am trying to understand spectral triples. I can recall the definition but I have no informative examples with which to make sense of it. What are some examples I can keep in mind to help me put this definition in context? Answer AttributionSource : … Read more

A relation between Hochschild cohomology of a $C^*$ algebra and its bidual

Let $A$ be a $C^*$ algebra and $A”$ be its bidual with the Arens product. Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A”$? Answer AttributionSource : Link , Question Author : Ali Taghavi , Answer Author : Community