**2**

votes

**0**answers

114 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

159 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**

votes

**1**answer

83 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

104 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 ...

**2**

votes

**0**answers

145 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**

votes

**1**answer

210 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

**2**answers

251 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

**1**answer

236 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

**3**answers

198 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**

votes

**2**answers

129 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

**0**answers

195 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 ...

**3**

votes

**1**answer

151 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 ...

**1**

vote

**1**answer

65 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 ...

**1**

vote

**0**answers

75 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 ...

**1**

vote

**0**answers

112 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 ...

**0**

votes

**0**answers

234 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

**1**answer

242 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 ...

**1**

vote

**0**answers

158 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**

votes

**0**answers

163 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 ...

**0**

votes

**0**answers

46 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**

vote

**1**answer

111 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?

**5**

votes

**1**answer

240 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

**0**answers

60 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 ...

**0**

votes

**0**answers

73 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 ...

**0**

votes

**1**answer

99 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

**1**answer

208 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

**0**answers

89 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

**2**answers

551 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 ...

**7**

votes

**0**answers

196 views

### Non Commutative Hyperspaces

Let $X$ be a compact metric space. Recall that the hyperspace $2^{X}$ is the set of all non empty compact subsets of $X$ with the Hausdorff metric. Assume that $\mathcal{C}$ is the category of all ...

**6**

votes

**1**answer

240 views

### Morita equivalence for operator algebras and tensor products, question about proof

This is a bit of a dumb question I know, but I was reading "Morita equivalence for C*-algebras and W*-algebras" by Rieffel, in this section about Morita equivalences and how they relate to forming ...

**1**

vote

**0**answers

75 views

### Non Commutative analogues of a commutative fact

What is a relevant non commutative analogues for the following fact, in term of spectral triples and cyclic cohomology?:
"If $M$ is a compact oriantable manifold without boundary and $X\subset M$ is ...

**3**

votes

**0**answers

227 views

### 2 - Calabi Yau algebras and bimodule coherence

Let $\Pi:=\Pi(Q)$ be the preprojective algebra of a connected non-Dynkin quiver over an algebraically closed field of characteristic zero.
In H. Minamoto "Ampleness of two-sided tilting complexes", ...

**1**

vote

**1**answer

345 views

### Computing noncommutative geometries

I find myself needing to construct some noncommutative geometries. I want to take various (algeba-) geometric objects and look at their noncommutative analogs. Is there a constructive way to do this? ...

**5**

votes

**1**answer

233 views

### A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...

**1**

vote

**0**answers

110 views

### Non commutative analogy of compact-open topology

Let $R$ be a ring, define a topology on $AUT(R)$(Or End(R)) with the following subbase:
For every two 2-sided Ideal $I$ and $J$, a subbase element is $B(I,J)=\{f\in AUT(R) \mid f(I)+J=R\}$.
We can ...

**1**

vote

**1**answer

150 views

### On non-unital ring and algebraic geometry

When I learned abstract algebra many years ago,I noticed the author deals with commutative ring say,$A$ has the proposition:$A^2=A$(without assuming it has identity).It seems that many proposition of ...

**1**

vote

**0**answers

50 views

### When is the dual module isomorphic to conjugate module of a *-algebra

Let $A$ be a $\ast$-algebra, and let ${\cal M}$ be a module over $A$. As usual, we denote the dual module of $M$ by $M^*$. Moreover, we denote the conjugate module of $M$ by $\overline{M}$, where as ...

**6**

votes

**0**answers

237 views

### Reconstructing the Chow ring from the derived category

Let $X$ be a smooth projective variety and write $\mathbf{D}(X)$ for its triangulated category of perfect complexes of quasi-coherent sheaves.
Recall that $\mathbf{D}(X)$ determines the Grothendieck ...

**1**

vote

**1**answer

173 views

### algebra-geometry duality

For topological spaces $S$ and $T$, denote by $C(S)$ and $C(T)$ the corresponding algebras of continuous real-valued functions. What are the necessary conditions that we need to impose on $S$ and $T$ ...

**4**

votes

**0**answers

141 views

### “Definitive” Noncommutative Space

Let $Y$ be a (locally compact) non-Hausdorff topological space. I want to know if there is a necessary and/or sufficient condition for $Y=X/G$, that is, $Y$ is the orbit space of a locally compact ...

**6**

votes

**2**answers

239 views

### Expression of a non-orthogonal projection in a $C^*$ algebra via an orthogonal one

A paper I'm currently reading uses the following fact. If $A$ is a unital $C^*$-algebra, $P=P^2\in A$, then there are $T, F\in A$ s.t. $F$ is an orthogonal projection ($F=F^*=F^2$) and
...

**3**

votes

**1**answer

152 views

### example of a compact quantum group at a root of unity?

In Woronowicz's theory of compact quantum groups, the most well-known example is $SU_q(2)$, for $q$ a real number. Moreover, all the other examples of compact quantum groups, based some ...

**14**

votes

**3**answers

1k views

### What's a noncommutative set?

This issue is for logicians and operator algebraists (but also for anyone who is interested).
Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]):
Let $H$ ...

**4**

votes

**1**answer

200 views

### Noncommutative Erlangen Program

Has Klein's "Erlangen Program" been generalized/extended to the noncommutative setting (say, à la Connes)? Is there a classification of "noncommutative klein geometries" at least in very low ...

**19**

votes

**1**answer

1k views

### What is about nonassociative geometry ?

At the end of a conference given by Alain Connes in 2000 (here is a video in French), a member of the audience asked a question. I transcribed and translated it for you below:
Audience: You showed ...

**4**

votes

**1**answer

202 views

### Do we have a “topological assembly map” in the Baum-Connes conjecture?

In the equivariant Atiyah-Singer index theorem, when $G$ is a compact group acting on a manifold $M$ and $R(G)$ is the representation ring of $G$. We have the analytic index
$$
\text{a-ind}: ...

**2**

votes

**1**answer

385 views

### C* Algebras, Foliations and Dynamical Systems

I am a Ph.D student involved in topics like integrability of foliations arising from center stable bundles of partially hyperbolic dynamical systems. These are generally only continuous bundles so one ...

**9**

votes

**3**answers

457 views

### The “right” $C^*$ algebraic proof of Bott Periodicity

In learning about the K-theory of $C^*$-algebras, I have encountered the following 3 proofs of Bott periodicity:
$\bullet$ An argument based on Moyal quantization found in "Elements of Noncommutative ...

**4**

votes

**0**answers

187 views

### Non-commutative Formal Group Laws

Does anyone know of a good, complete reference for non-commutative formal group laws (i.e. construction of a "Lazard ring," discussion of non-commutative formal groups, perhaps some discussion of ...

**2**

votes

**2**answers

434 views

### Possible directions in noncommutative geometry

I recently came across noncommutative geometry and found it rather interesting. I should mention that I'm a graduate student considering options for my research and if I were to name an area which I'm ...