**6**

votes

**0**answers

54 views

### Compute the index of the Dirac operator on $C_0(R^2)$ to obtain Bott element in $K_0$

I am studying the paper of Baum-Connes-Higson to understand the Connes-Kasparov conjecture. In example 4.23, they discuss the case $G=\mathbb{R}^2$. I have constructed the Dirac operator, but I’m ...

**3**

votes

**1**answer

262 views

### K theory long exact sequence

(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset ...

**17**

votes

**0**answers

246 views

### Topological loops vs. algebro-geometric suspension in Hochschild homology

Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...

**7**

votes

**2**answers

453 views

### $H^{*}$ algebras as a generalization of $C^{*}$ algebras

Let $H$ be the quaternions algebra. An $H^{*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $*$ with the following properties:
$\forall \lambda ...

**2**

votes

**1**answer

81 views

### Existence of free operators, independent and with given distributions

Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free ...

**7**

votes

**1**answer

143 views

### Hopf Galois extensions and conditional expectations for C* algebras

Suppose that $H$ is a Hopf algebra with normalised invariant integral (appropriate side) $\int:H\to \mathbb{C}$. The $H$ right comodule algebra $P$ is a Hopf Galois extension, so the canonical map ...

**4**

votes

**1**answer

162 views

### Current status of the linearity of mapping class group

In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...

**5**

votes

**1**answer

175 views

### Are all transversely oriented, transversely measured foliations given by closed forms?

This paper, "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the ...

**8**

votes

**1**answer

397 views

### Analytical formula for topological degree

At the first page of the following article http://arxiv.org/pdf/1004.1018v1.pdf [edit: the formula on the arXiv differs from the formula in the published paper, and the formula displayed below is the ...

**2**

votes

**1**answer

107 views

### Do representations of the universal enveloping algebra $\mathrm{U}\mathfrak{su}_2$ retain the Hopf algebra structure?

A Lie algebra $\mathfrak{g}$ generates its universal enveloping algebra $\mathrm{U}\mathfrak{g}$, which has the structure of a Hopf algebra. Modules of $\mathrm{U}\mathfrak{g}$ are exactly the of ...

**0**

votes

**2**answers

135 views

### Example of a covariant derivative on a non-projective bundle

I was looking for a simple example of a covariant derivative on a bundle, where the bundle is not projective. If necessary, the example could be from complex or noncommutative geometry, but I would ...

**0**

votes

**1**answer

240 views

### Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes [closed]

Question
I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...

**12**

votes

**1**answer

275 views

### Is there something like “Noncommutative geometry internal to a category”?

I have heard that one can do algebraic geometry internal to symmetric monoidal categories. Topological quantum field theories also exist internal to symmetric monoidal categories, and the usual ...

**5**

votes

**1**answer

290 views

### A survey for various $K$-homology theories and their relationship

The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology ...

**10**

votes

**1**answer

206 views

### NCG with all noncommutativity in a nilpotent ideal

While in general non-commutative geometry behaves rather differently from commutative geometry when it comes to local-to-global properties (descent), there are versions of "mild" noncommutative ...

**0**

votes

**1**answer

160 views

### A $C^{*}$ algebra associated to a group [closed]

Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar ...

**6**

votes

**1**answer

348 views

### Formula for the distance in noncommutative geometry

Probably the most famous formula in noncommutative geometry is the following formula allowing one to compute distance of two points using the operator theoretic data:
$$(1) \ \ ...

**1**

vote

**0**answers

92 views

### A noncommutative vector bundle associated with a codimension one foliation

Assume that we have a codimension one foliation of a manifold $M$ which is generated by a one form $\alpha$. So the following $\phi$ satisfies $\phi \circ \phi =0$:$$\phi:\Omega^{i}(M)\to ...

**7**

votes

**1**answer

284 views

### Can a finite von Neumann algebra be strongly morita equivalent to a properly infinite von neumann algebra?

Can a finite (by finite I mean when the projection $1$ is finite) von Neumann algebra be strongly morita equivalent to a properly infinite von Neumann algebra?
(Strong morita equivalence is the same ...

**4**

votes

**1**answer

485 views

### What is the current state of generalizations Noether's theorem?

The well-known Noether's theorem is a vital tool in classical physics. But it assumes some hypothesis, many of which could be removed by a detailed look.
So my question is: In what directions has ...

**0**

votes

**0**answers

110 views

### Product of two foliations

1.What is an example of a manifold $M$ with two foliations $F$ and $F'$ which are not topological equivalent but the product foliations $F\times F$ and $F'\times F'$, as foliations on $M\times M$, ...

**4**

votes

**0**answers

265 views

### Flat Connections on Ring Spectra

So first I'll try to give a really quick reminder of the classical description of these things when one is doing non-commutative descent theory. In the setting of discrete algebra, if we have a ...

**3**

votes

**1**answer

204 views

### The Jordan Plane and Enveloping Algebras

Let $k$ denote a field of characteristic $0$ (assume algebraically closed for convenience). Define $J=k\langle x,y|[x,y]=y^{2}\rangle$. This noncommutative algebra (which can be viewed as a derivation ...

**6**

votes

**1**answer

194 views

### Connes' correspondences of two $L^\infty$-algebras

In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu_X)$ and ...

**11**

votes

**0**answers

418 views

### Lifting DG-categories to characteristic zero

The question of lifting (smooth projective) varieties from an algebraically closed field $k$ of characteristic $p$ to characteristic zero (i.e., to the Witt vectors $W(k)$) is a classical one. It's ...

**4**

votes

**0**answers

199 views

### Can we obtain a derived category from an additive category? Like a category of Banach modules?

Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...

**6**

votes

**2**answers

451 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

**1**answer

327 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

**0**answers

166 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 ...

**5**

votes

**1**answer

223 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**

votes

**1**answer

194 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**

votes

**1**answer

94 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

**1**answer

185 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**

votes

**1**answer

166 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

**1**answer

113 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 ...

**1**

vote

**0**answers

121 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

**1**answer

242 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**

votes

**2**answers

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**

votes

**1**answer

131 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**

vote

**2**answers

168 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**

votes

**0**answers

81 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 ...

**3**

votes

**0**answers

111 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 ...

**7**

votes

**0**answers

252 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 ...

**1**

vote

**1**answer

344 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 ...

**12**

votes

**2**answers

713 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**

vote

**1**answer

162 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**

votes

**0**answers

145 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

**1**answer

257 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) ...

**7**

votes

**1**answer

212 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

**1**answer

196 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 ...