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Noncommutative geometry in the sense of Connes and beyond: noncommutative algebras viewed as functions on a noncommutative space.

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Constructing a noncommutative algebra from a commutative algebra

I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
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1answer
114 views

Separability of compact quantum groups

In the theory of compact quantum groups due Woronowicz, we assume usually that the C*-algebra of the compact quantum group is separable. Is the assumption essential in the theory? Will it eventually ...
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59 views

Hilbert space separability for spectral triples

A spectral triple $({\cal A},{\cal H},D)$ consists of a unital $*$-algebra ${\cal A}$ represented as bounded operators on a Hilbert space ${\cal H}$, together with an unbounded operator $D$ having ...
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68 views

What quantum groups admit quantum topography space structure?

Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian ...
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0answers
117 views

Categorical features of Hilbert spaces

Does the category of Hilbert spaces and bounded maps have any particular categorical feature which can be studied systematically? I mean, I know that it's a $*$-category, but it seems to have much ...
5
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2answers
179 views

Path algebras are formally smooth

In Ginzburg's notes Lectures on Noncommutative Geometry, he claim that the path algebra of a quiver is formally smooth (See Section 19.2). I have two questions. First, how to show this claim and ...
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36 views

Some generalizations of the graph of a foliation

The graph of a foliation $\mathcal{F}$ of a manifold $M$ is the space of all triple $(x,y,[\gamma])$ where $x,y$ lies on the same leaf $F$ and $[\gamma]$ is the equivalent class of a ...
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1answer
101 views

number of indecomposable summands of an extension of two modules

I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of ...
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0answers
51 views

C*-algebra of a singular surface foliation

Noncommutative geometry associates a $C^*$-algebra $C^*(S,{\cal F})$ with a foliation $\cal F$ on a manifold $S.$ Did somebody study this construction for noncompact surfaces $S$? What I am really ...
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1answer
183 views

Simple $C^*$ algebras with invariant subspace property

Edit: According to the valuable comment of Yemon Choi I revise the question by replacing "faithful" with "irreducible". We say that a $C^*$ algebra $A$ satisfies the invariant subspace ...
4
votes
1answer
107 views

KMS-states of Bost-Connes type system

I have some struggles with understanding theorem 25 in the paper "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory". More precisely, there is ...
5
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1answer
124 views

Example of a prime action on a compact Hausdorff Space

Suppose that a discrete group $\Gamma$ acts on a compact Hausdorff space $X$ via homeomorphisms. This action induces an action on $C(X)$, the space of all continuous functions from $X$ to $\mathbb{C}$,...
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152 views

Is the algebra $\mathcal{C}^{\infty}(M,\mathbb{R})$ a smooth algebra in the sense of algebraic geometry?

First of all, let me fix some terminology: I will follow the definitions that can be found in the book "Cyclic Homology" of J.L. Loday (second edition) page 102 in the special case of $K$ a field. ...
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2answers
293 views

Kazhdan constant and finite index subgroups

I am wondering if there is some general relation between Kazhdan constants of a group and it finite index subgroups? Let $G$ be a finitely generated group with a generating set $\Sigma$ that ...
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0answers
89 views

Connes-Chern pairing, compatibility with periodicity operator in the odd case

Let $A$ be an algebra (say unital). For an odd (say $2n-1$) cyclic cocycle $\varphi$ and a class in $K_1(A)$ represented by invertible $u$ we define $$\langle [\varphi],[u] \rangle:=\frac{2^{-(2n+1)}}...
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73 views

Causal fermion systems fromm fractal geometry

Okay, first off- I apologise if this is a stupid question. I'm mainly a very young physics guy, but this has primarily math basis. I'm trying to build a theory that is, long story short, some ...
3
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0answers
30 views

Second order signature operator in diffeomorphism invariant geometry as an image under right regular representation

I would like to understand the following statement taken from this paper, dealing with the so called Transverse Index Theory or in other words with the index theory for diffeomorphism invariant ...
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0answers
144 views

Pairing between cyclic cohomology and $K$-theory: the odd case

I would like to understand the proof of Proposition 15 (see page 70 in this link ). More precisely: I would like to understand a particular step in the proof namely: Why $\frac{d}{dt}(\varphi \# ...
5
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1answer
171 views

Two approaches to periodic cyclic cohomology

Cyclic cohomology may be defined in several ways: the easiest way to define it is via a subcomplex $C^*_{\lambda}$of Hochschild complex consisting from cyclic cochains. There are also other ...
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1answer
297 views

preliminary reading recommendation before embarking on Connes non commutative geometry book?

I want to try to understand non commutative geometry by reading Connes's book ..and I am discovering it is a hard book to read :-) as I miss a lot of background specially in operator algebra and ...
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0answers
149 views

Is Hochschild homology invariant under A-infinity quasi isomorphism?

If A and B are two A-infinity algebra, A is A-infinity quai-isomorphic to B. Do we have HH(A)=HH(B)?
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1answer
94 views

Question on a paper by U. Krähmer (“Dirac operators on quantum flag manifolds”)

I don't know if this is an adequate question for MO. But I cannot understand many aspects of the said paper https://link.springer.com/content/pdf/10.1023%2FB%3AMATH.0000027748.64886.23.pdf by ...
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Relative version of Hopf cyclic cohomology

In this paper Connes and Moscovici introduced the Hopf algebra of transverse differential operators in order to compute the index formula for the diffeomorphism invariant geometry. They developed the ...
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range of trace on projections: beyond rotation algebras

In a rotation algebra, $A_\theta=C(S^1)\rtimes \mathbb{Z}$, there is a tracial state $\tau$ coming from the invariant measure $\mu$ on the circle. There is a projection $p\in M_n( A)$ (we can have $n=...
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2answers
312 views

A quantity associated with a smooth groupoid

Assume that $(G,G^0,r,s)$ is a smooth groupoid such that $G$ is a compact connected manifold. The graph of "source" and "range" maps $s, r: G \to G^0$ are compact submanifolds $S$ and $R$ of $G\times ...
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51 views

Pseudodifferential calculus for the Diffeomorphism Invariant Geometry

In the paper "Local Index Formula in Noncommutative Geometry" Connes and Moscovici build the spectral triple $(A,H,D)$ where $A=C^{\infty}_c(P) \rtimes \Gamma$ where $\Gamma$ is an arbitrary subgroup ...
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32 views

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 ...
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135 views

Baum Connes conjecture and abstract isomorphism

Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...
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190 views

Regularilty of Commutative Spectral Triples

In Connes' approach to non-commutative geometry, the notion of a spectral triple is said to generalize compact Riemannian manifolds to the non-commutative setting. Motivating classical examples ...
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1answer
378 views

Equivalence of rational Voevodsky motives: partial Converse to Conjecture of Orlov

There is a conjecture of Orlov stating that if $X$ and $Y$ are smooth projective complex varieties that are derived equivalent (equivalent bounded derived categories of coherent sheaves), then their ...
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0answers
151 views

Dense subalgebra of continuous functions with same K -theory

Suppose $X$ is a compact metric space. Is there a good candidate for a dense subalgebra $A\subseteq C(X)$, such that the inclusion induces an isomorphism in $K$-theory? For example, if $X$ was a ...
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1answer
169 views

Normalization of cyclic cocycles

This question is a continuation of the discussion Normalization of Hochschild cocycles but this time in the cyclic context. I would like to ask whether the following is true: The inclusion of ...
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0answers
99 views

Hodge theoretic mirror symmetry and DG-BV algebras

Consider two Calabi-Yau manifold $X$ and $\check{X}$ which are meant to be mirror partners. Motivated by "classical MS", In DGBV Algebras and Mirror Symmetry, the following enhancement is proposed: ...
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4answers
699 views

Geometric or conceptual way to understand supersymmetry algebra

Is there any geometric or more direct conceptual way to understand a supersymmetry algebra, rather than starting from a Lagrangian including boson and fermion fields, deriving all the expressions ...
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0answers
77 views

Definition of the $G$-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 ...
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1answer
133 views

direct images of states in $C*$ algebras

Take a unital cp map $f:B\to A$ between unital $C^*$ algebras. Given a state $\psi:B\to \mathbb{C}$ what conditions are necessary for there to exist a state $\phi:A\to \mathbb{C}$ so that $\phi\circ f=...
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93 views

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 ...
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112 views

Could we extend the star product on a Poisson manifold from its ring of smooth functions to its de Rham complex?

Let $M$ be a smooth manifold with a Poisson bracket $\{-,-\}$. Kontsevich proved that there exists a deformation quantization of $M$, i.e. let $C^{\infty}(M)[[\hbar]]=C^{\infty}(M)\otimes_{\mathbb{R}}\...
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220 views

Do almost commutative flat degenerations induce equality in K-theory? (Or: Is the characteristic variety actually a support of a class in $K$-theory?)

I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not ...
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0answers
90 views

Hochschild coboundary on the space of alternative forms

Assume that $A$ is a complex algebra. By $C^{n}(A)$ we mean the space of all $n-$linear map $\phi:A^n \to \mathbb{C}$. An alternative $k-$ form is an element $\phi \in C^{k}(A)$ ...
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136 views

Perfect Complexes on Tangent Bundle

Suppose $X$ is a $k$-variety of dimension $d$, and suppose $TX$ is its tangent bundle. Consider the (triangulated, stable $\infty$-,...) categories of perfect complexes $\text{Perf}(X)$ and $\text{...
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106 views

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 ...
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2answers
351 views

Reference for de Rham cohomology for physicists

Do you know a basic reference to introduce an undergraduate student with more physical rather than mathematical background to De Rham cohomology? The Student (from a Bachelors ...
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1answer
456 views

Derived noncommutative geometry includes derived, or spectral algebraic geometry?

Let $k$ be a commutative ring. In derived noncommutative (algebraic) geometry a "noncommutative space over $k$" is a $k$-linear $\mathrm{DG}$-category. This is motivated by the fact that homological ...
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1answer
219 views

Homotopy groups of noncommutative spaces

In the approach to noncommutative geometry of Alain Connes any Hausdorff compact space $X$ is replaced by its algebra of complex valued continuous functions $C^0(X)$, and one regard general (that is, ...
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1answer
114 views

Différences between KKO and KKR in Kasparov theory

In Kasparov article : The operator K functor and extensions of $C^*$algebras there is the definition of the two bifunctors $KKO : ralg^{op} \times ralg \to Ab$ and $KKR : Ralg^{op}_r \times Ralg_r \to ...
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1answer
97 views

Non-commutative Ito Formula

Does there exist a formula of the Ito lemma for matrix valued processes under matrix multiplication? That is where $$ \Delta X_{t+\Delta t} \neq X_{t+\Delta t} - X_t $$ but instead $$ \Delta X_t = ...
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1answer
674 views

How to understand the explicit formula for zeta function?

The explicit formula for the zeta function, e.g. $$\psi_0(x) = \dfrac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma-i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}ds=x-\sum_\rho\frac{x^\...
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3answers
160 views

Smallest norms on crossed product $C^*$-algebras

Let $A$ be a commutative $C^*$-algebra with a discrete group $G$ acting on it. The reduced crossed product is the completion of the algebraic crossed product $C_c(G,A)$ in the reduced norm $\Vert \...
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0answers
61 views

Renormalization on noncommutative torus

I am reading a paper of renormalization of field theory on noncommutative torus. At the end of chapter 6 there is the following statement "Although our analysis is far from being exhaustive, we ...