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

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6
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2answers
254 views

Relationship between Hochschild cohomology and Drinfeld centers

Let $HH_*(A,N)$ (or $HH^*(A,N)$) be the Hochschild homology (or cohomology) of an associative algebra $A$ with coefficients in an $A$-bimodule $N$. I was reading nlab's entry on Hochschild cohomology ...
5
votes
1answer
299 views

The function algebra $C^{\infty}(M\#N)$ of the connected sum of two spaces

Operations such as taking union or Cartesian products of spaces have direct analogues in term of algebra of functions on them (direct sum and tensor product, respectively), my question is: Is there ...
8
votes
0answers
132 views

Commutative spectral triples not coming from manifolds

There is a very deep and remarkable theorem by Connes (the so called reconstruction theorem) which states that from a commutative spectral triple obeying certain axioms one can reconstruct a smooth ...
4
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1answer
118 views

Commutative spectral triples

The corresponence between compact Hausdorff topological spaces and commutative unital $C^*$-algebras is rather well known: Gelfand Najmark theorem gives perfect correspondence between these ...
2
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1answer
185 views

Hochschild cohomology of commutative quotients

Notation: Let $k$ be a commutative local ring and let $HH^{i}(A,N)$ denote the $i^{th}$ Hochschild cohomology $k$-module of a $k$-algebra A with coefficients in an $(A,A)$-bi-module $N$. If ...
3
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1answer
76 views

Local index formula for >ungraded< elliptic operators

Let $P\colon E \to F$ be an elliptic pseudodifferential operator over $M$. Assuming that $P$ defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic ...
4
votes
1answer
165 views

Nuclearity noncommutative torus

I read that the Noncommutative torus (rotation algebra) is nuclear when $\theta\in\mathbb{R}\setminus\mathbb{Q}$. Unfortunately, I haven't found a proof. Could someone give me a reference and/or an ...
4
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1answer
146 views

Noncommutative version of Littlewood's First Principle

There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle: Every Lebesgue ...
3
votes
1answer
97 views

Strong and weak equivalence of C$^∗$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
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0answers
113 views

Noncommutativization of fixed point theory

What papers or references have been devoted for a noncommutativization of "Fixed point theory". Here the terminology Noncommutativiztion, as usual, indicates to that famous table with 2 columns: ...
2
votes
1answer
218 views

Non commutative topological manifolds

Assume that $A$ is a Banach algebra with two closed two sided ideals $I$ and $J$ such that $I$ and $J$ are commutative and $A=I+J$. Does this implies that $A$ is commutative? For the ...
12
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2answers
2k views

Mysterious quotes (at least for me)

I heard two quotes, one from Alain Connes and an other one from Orlov. Alain Connes was talking about noncommutative geometry and he said the following: " a noncommutative algebra creates its own ...
3
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0answers
49 views

Boundedness Spectral Triple Axioms for de Rham Complex

In Connes' axioms for a spectral triple $(A,H,D)$, they have a representation of an algebra $A$ in bounded operators on a Hilbert space $H$, and (unbounded) operator $D$, such that $[D,a]$ is bounded. ...
3
votes
1answer
93 views

Is the square diagram of index and exponential maps in $K$-theory of $C^*$-algebras anti-commutative?

Assume we have a $3\times 3$ grid with rows and columns being short exact sequences of $C^*$-algebras. This gives a grid of 6-term exact sequences: 3 "horizontal" sequences and 3 "vertical" ...
1
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2answers
126 views

Deformation quantization of a closed Riemann surface with genus >1

Quantization of of an elliptic curve can be done in different ways. In C^*-algebraic version, one can start with the C^*-algebra ...
3
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0answers
66 views

The 4-th generator of $K_1$ group for 3-dimensional NC tori algebra

An $n$-dimensional NC torus algebra $A_\theta^{(n)}$ is defined for any antisymmetric $n\times n$ matrix $\theta$ of real numbers as the universal $C^*$-algebra, generated by unitaries ...
2
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0answers
102 views

A topological space extracting from a group action

Let $G$ be a compact abelian topological group with invariant measure $\mu$ which acts on a compact Hausdorff space $X$. A $G$-odd function is a continuous function $f:X\to ...
6
votes
0answers
185 views

Noncommutative geometry and line length

I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...
0
votes
0answers
43 views

One dimensional foliation of surfaces with prescribed graph of foliation

According to the definition of the graph of a foliation by Winkelnkemper we ask the following questions: Let $G$ be one of the following non hausdorff 3 dim manifold 1) $G$ is a ...
1
vote
1answer
325 views

Are these two “FUNCTORS” adjoint?

I am considering the following correspondence: Let $X$ be quasi compact quasi separated schemes.Consider a pseudo functor \begin{equation}Sch\rightarrow CAT :U\mapsto Qcoh(U),f:U\rightarrow V\mapsto ...
10
votes
2answers
643 views

If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?

The other direction is well known. I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove. I am also wondering ...
1
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1answer
124 views

An unconventional definition of the $ C^{*} $-algebraic reduced crossed product

Let $ (A,G,\alpha) $ be a $ C^{*} $-dynamical system, i.e., $ A $ is a $ C^{*} $-algebra, $ G $ is a locally compact Hausdorff group and $ \alpha $ is a strongly continuous action of $ G $ on $ A $ by ...
3
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0answers
133 views

$S^{3}$-valued harmonic analysis

Edit: Note that $S^{3}$ with the quaternion operation is a group. For a locally compact Abelian group $\Gamma$ we consider $$\tilde{\tilde{\Gamma}}=\{\phi:\Gamma \to S^{3} \mid ...
5
votes
1answer
207 views

C*-bimodules: the mess with definitions

I used to participate in a seminar that taught students about foundations of non-commutative geometry. It isn’t very complicated to define a C*-module $\mathcal E$ (also known as C* Hilbert module) ...
5
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1answer
125 views

“The” kronecker foliation or “a” kronecker foliation?

Consider the following two foliations of torus: 1)The Kronecker foliation with slope $\sqrt{2}$ 2)The Kronecker foliation with slope $\pi$ As I learn from the literature, these two foliations are ...
5
votes
1answer
135 views

Generators of the $ K_{0} $-group of the non-commutative torus $ A_{\theta} $ with $ \theta \in \mathbb{Q} $ (i.e. rational rotation algebra)

I am studying the non-commutative torus $ A_{\theta} $. When $ \theta $ is irrational, $ {K_{0}}(A_{\theta}) $ is generated by $ [1] $ and $ [p_{\theta}] $. (Note: $ p_{\theta} $ is a projection in ...
3
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0answers
163 views

Are two $W^*$-algebras $A$ and $B$ that are Morita equivalent (in the sense of Rieffel) also Morita equivalent in the algebraic sense (as rings)?

Let $A$ and $B$ be $C^*$-algebras, I had a question about Morita equivalence for $C^*$-algebras and $W^*$-algebras, I've come across 3 concepts: Morita equivalence for $C^*$-algebras: Equivalence ...
2
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1answer
264 views

Reference request for instantons

I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...
6
votes
2answers
283 views

Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background: There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...
5
votes
1answer
264 views

Noncommutative HKR theorem

What is the analog of HKR theorem in the noncommutative world? Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a smooth commutative algebra $A$ of finite type ...
0
votes
3answers
229 views

Specific Reference? Noncommutative topology and C^* algebras [closed]

I stumbled across this "dictionary for noncommutative topology" http://planetmath.org/noncommutativetopology and I would be very interested in learning more on the subject, particularly I'd like to ...
0
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2answers
141 views

Smooth Affine algebras are Calabi-Yau

Are all smooth affine algebras over a field Calabi-Yau? I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/
5
votes
0answers
214 views

Is the crossed product $\mathcal{K} \rtimes G$ a groupoid algebra?

Suppose G, a discrete group acting on the compact operators $\mathcal{K}$ by automorphism of C*-algebra $\mathcal{K}$. Can we view the crossed product as a groupoid C*-algebra of some groupoid? This ...
4
votes
1answer
189 views

A question on $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which satisfies each of the following equivalent conditions: All elements of $A$ are left zero divisor. All elements are right zero divisor. All elements ...
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1answer
74 views

Unitization via “End points compactification”

We know that every compactification of locally compact Hausdorff spaces correspond to a unitization of $C^{*}$ algebras. For example the one point compactification corresponds to the minimal ...
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0answers
102 views

Local Index Formula vs Atiyah Singer Index Theorem

I have a question concerning the so called Local Index Formula by Connes in noncommutative geometry. First issue: why it is called Index Formula? I spoke to one person about this and he gave me the ...
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0answers
119 views

Number of connected components of a $C^{*}$ algebra

Motivating by the concept in the following post What are these compact sets called? We introduce the following concept: Let $A$ be a unital $C^{*}$ algebra. We consider the unitary equivalent ...
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0answers
244 views

A noncommutative vector bundle

We know that a noncommutative vector bundle is a finitely generated projective $A$-module where $A$ is a non commutative $C^{*}$ algebra. In this question we introduce a particular non commutative ...
5
votes
1answer
249 views

NonCommutative Baire theorem

The classical Baire theorem says that the intersection of a sequence of open dense subsets of $X$, is dense, if the space is compact Hausdorff. In the language of $C^{*}$ algebras this is ...
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0answers
172 views

Soft Question: What does periodic cyclic theory measure?

Ex1) The cyclic homology of $\mathbb{C}[X,Y]$ and that of the algebra of functions on the sphere $S^2$ have the same periodic cyclic homology, clearly however these objects are topologically very ...
0
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0answers
165 views

A noncommutative analogy of the tube lemma

Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to ...
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0answers
49 views

Quick question about conjugate equivalence bimodules and inner products

let $A$ and $B$ be $W^{*}$-algebras, let $X$ be an $A-B$-equivalence bimodule (according to the definition given in "Morita equivalence for $C^*$-algebras and $W^*$-algebras" by Rieffel, ...
1
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1answer
128 views

Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?
7
votes
2answers
340 views

Projective modules over noncommutative tori?

It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class ...
2
votes
0answers
64 views

Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false) If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality? All i need to find is a protective ...
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0answers
78 views

Explicit calculation of module of derivations on noncommutative polynomial ring

Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$. Explicitly how would one go about computing ...
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1answer
114 views

Jacobi-Zariski exact sequence question

Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ ...
3
votes
1answer
219 views

Why should noncommutative CYs be dgas?

For a (commutative) CY manifold of dimension $n$, Serre duality implies that there are ifunctorial isomorphisms $$\operatorname{Hom}_{D^b(X)}(E,F)\cong\operatorname{Hom}_{D^b(X)}(F,E[n])^*$$ in the ...
3
votes
0answers
94 views

Noncommutative Poincare inequalities

This is a question on how (or if) people in the community think about the Poincare inequality in noncommutative geometry. In geometry, the Poincare inequality (when it exists) gives a bound on a ...
13
votes
2answers
644 views

Why is “naive” definition of non-commutative spectrum bad?

It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative ...