Questions tagged [noncommutative-geometry]
Noncommutative geometry in the sense of Connes and beyond: noncommutative algebras viewed as functions on a noncommutative space.
475 questions
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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 ...
- 1,903
4
votes
1
answer
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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=...
- 1,143
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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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7
votes
0
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139
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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}}\...
- 9,009
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0
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264
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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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1
vote
0
answers
104
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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 $\phi \in C^{k}(A)$ ...
- 356
3
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0
answers
148
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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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4
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0
answers
126
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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 ...
- 41
3
votes
2
answers
825
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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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8
votes
1
answer
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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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2
votes
1
answer
386
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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, ...
user95283
2
votes
1
answer
181
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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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1
vote
1
answer
241
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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
$$
\Delta X_t = ...
- 5,407
6
votes
1
answer
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How to understand the explicit formula for zeta function?
The explicit formula for the zeta function, e.g.
$$\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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3
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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 \...
3
votes
0
answers
87
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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 ...
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6
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0
answers
147
views
What is the meaning of complex values/multiplicities in dimension spectrum?
If we have a manifold $M$ (say smooth, closed) it can be equipped with the Laplace operator $\Delta$. One can consider the function $\textrm{trace}(\Delta^{-s})$ where $s$ is complex parameter and $\...
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votes
0
answers
227
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What is the motivation behind the definition for a smooth differential graded category?
Let $\mathcal{A}$ be an $\mathbb{F}$-linear differential graded category. It is said to be smooth if it is a perfect complex over the differential graded category $\mathcal{A}^\circ\otimes_\mathbb{F}\...
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0
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194
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New homotopy groups
In the book "Noncommutative Geometry" by Alain Connes (link to the book) on page 162 the author defines new homotopy groups $\pi_{n,k}(X,*)$ for a locally compact pointed space $(X,*)$ as the group of ...
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votes
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241
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A problem on automatic groups and geodesic paths on the Cayley graph
Let $\Gamma = \langle S \mid R \rangle$ be a finitely generated group, with the neutral element $e \not \in S= S^{-1}$.
Let $\ell : \Gamma \to \mathbb{N}$ be the world length related to $S$.
For ...
1
vote
0
answers
117
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The holonomy groupoid of certain one dimensional foliations of 2 dimensional Euclidean regions
What Is the first fundamental group of each of the following $3$ dimensional Hausdorff manifolds? What about homology groups of these 3-manifolds? Is the first one a contractible manifold?
The ...
- 356
10
votes
1
answer
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Baum Connes conjecture and Atiyah-Singer index theorem
Baum Connes conjecture is considered as a far generalisation of the Atiyah Singer index theorem (in K-theoretical formulation). I would like to understand how the latter follows from this conjecture. ...
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13
votes
1
answer
669
views
Is a "smooth" finite-dimensional algebra separable modulo its radical?
Let $k$ be a field, and let us write the "unadorned" tensor $\otimes$ in place of $\otimes_k$. For a unital finite-dimensional $k$-algebra $A$, let $A^e = A \otimes A^{op}$ denote the enveloping ...
- 5,407
3
votes
0
answers
98
views
Quasi isomorphism for bicomplexes both defining cyclic cohomology
Let $A$ be a complex unital algebra. Consider the cyclic (cohomological) bicomplex $\mathcal{C}(A)$. This is a bicomplex
where in the $p$-th row one has $C^p(A)$ (the space of all $p+1$-linear forms) ...
- 9,330
4
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0
answers
398
views
Bott-type projections in $C^*$-algebras
Let $A$ be a unital $C^*$-algebra and $a\in A$. If $aa^*+1$ is invertible in $A$ then the element
$$\beta(a)=(aa^*+I)^{-1}\left(\begin{array}{cc}aa^* & a \\a^* & I\end{array}\right)$$
is an ...
- 41
7
votes
0
answers
300
views
Injectivity of the Chern character in $K$-homology
Let $(\pi,H,F)$ be a Fredholm module: here $\pi:A \to B(H)$ is a representation of an algebra on the Hilbert space $H$ and $F$ is a self adjoint operator with square one such that for each $a \in A$ ...
- 9,330
9
votes
2
answers
293
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Spectral triples which are not $\theta$-summable
I would like to see an example of a spectral triple $(A,H,D)$ such that the underlying algebra $A$ is commutative but this spectral triple is not $\theta$-summable in the sense that $e^{-tD^2}$ is ...
- 9,330
4
votes
0
answers
211
views
Inner automorphisms acts as identity on Hochschild homology
Let $A$ be a unital algebra and $u \in A$ be an invertible element. Let us consider $u_n(a_0 \otimes a_1 \otimes ... \otimes a_n):=ua_0u^{-1} \otimes ua_1u^{-1} \otimes ... \otimes ua_nu^{-1}$. Then $(...
- 9,330
8
votes
1
answer
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Isomorphism in cyclic cohomology vs isomorphism in Hochschild cohomology
Let $A$ be a unital algebra over a field $K$, $C^n(A)$ a space of all $n+1$ linear maps into scalar field $k$ (I'm interested in case $k=\mathbb{C}$) and
$$(bf)(a_0,...,a_{n+1})=\sum_{i=0}^n(-1)^if(...
- 9,330
4
votes
1
answer
193
views
isomorphism of noncommutative tori
I have a kind of vague question.
Two non-degenerate symplectic vector space of same dimension (say $\mathbb{R}^{2n}$) are isomorphic. Then why all noncommutative tori of same dimension aren't ...
- 103
2
votes
1
answer
569
views
Why $k[x,y]$ is not a formally smooth algebra?
We could talk about the formal smoothness of an algebra. See for example Ginzburg's lecture notes For an associative algebra $A$ over a field $k$ we define
$$
D(A)=T(A+\bar{A})/(\bar{ab}=a\bar{b}+\bar{...
- 9,009
1
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0
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147
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Is there a non-integer in the dimension spectrum for the Heisenberg group?
Let $\Gamma = \langle a,b,c \ | \ c=aba^{-1}b^{-1}, \ ac=ca, \ bc = cb \rangle$ be the discrete Heisenberg group.
Let $\ell: \Gamma \to \mathbb{N} $ be the word length on $\Gamma$. This group has a ...
5
votes
0
answers
149
views
Algebras involved to define spectral triple
A spectral triple consist from $(A,H,D)$ where $A$ is some unital $*$-subalgebra of $B(H)$ and $D$ is unbounded operator with compact resolvent such that for all $a\in A$ the commutator $[D,a]$ is ...
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8
votes
1
answer
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views
K theory for pre $C^*$-algebras
In noncommutative geometry when one want to go to the differentiable level, one is forced to work with algebras which are no longer $C^*$. It is nice if we don't loose much information by the ...
- 9,330
7
votes
0
answers
435
views
K theory as the fundamental group
There are several ways in which one can define $K$-theory for $C^*$-algebras: for $K_0(A)$ group two aproaches: algebraic (using idempotents) and topological (using projections, i.e. self-adjoint ...
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5
votes
1
answer
506
views
reference for KK theory
I wanted to ask you, if you have any good references (book or pdf) to learn about the KK theroy of Kasparov. I think the presentation of Blackadar is too close from the commutative theory.
I was ...
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6
votes
1
answer
157
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The Quantum Group ${\cal O}_q(SL(n))$, for $q>1$
For the quantum group ${\cal O}_q(SL(n))$, $q\in \mathbb{R}$, I have read, without a proof, that for $p>1$, there exists a $q\in (0,1)$ such that
$$
{\cal O}_p(SL(n)) \simeq {\cal O}_q(SL(n)).
$$
...
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4
votes
0
answers
206
views
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-...
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7
votes
1
answer
263
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Most natural equivalence between $C^*$-algebras in NCG
I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism.
Can someone explain this sentence or ...
- 561
9
votes
1
answer
563
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Further directions of index theory
The Atiyah-Singer theorem is a major achievement of twentieth century mathematics. It has inspired a lot of work and people started to develop generalizations of this theorem. I would like to know the ...
- 9,330
8
votes
2
answers
707
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Quantum Grassmannians?
In noncommutative algebraic geometry a commonly studied family of objects are quantum projective spaces. Theses are certain deformations of the homogeneous coordinate ring of $\mathbb{CP}^n$. For ...
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3
votes
2
answers
1k
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Reference for Connes Bourbaki membership or otherwise
Alain Connes being a leading French mathematician today one could ask whether he is a member of the Bourbaki group. Is there a published reference that would either refute or confirm this?
- 16.6k
13
votes
1
answer
765
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Atiyah-Singer index theorem, pairing between K-homology and K-theory and Chern character
There is a general (abstract) index theorem in noncommutative geometry: you take a
K-theory class and K-homology class (which is represented by a triple $(A,H,F)$) and
you pair them together. This ...
- 9,330
5
votes
5
answers
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Elementary linear algebra over a (possibly skew) field $K$
I have a number of questions which seem linked to me, about basic (?) linear algebra:
Given a field (possibly skew) $K$, and an superfield $L$, one can do linear matrix algebra with coefficients in $...
- 1,555
2
votes
0
answers
254
views
isomorphism of Chern character in kk-theory
Suppose we work with Fréchet algebras. Cuntz defined kk-theory for those algebras and hence we have the notions of K-theory and K-homology for those algebras. Now suppose Chern character is ...
- 493
4
votes
0
answers
337
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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 ...
- 24.8k
9
votes
0
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Existence/characterization/properties of $C^*$-algebras which "are" quantization of compact symplectic manifolds?
Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
- 24.8k
69
votes
4
answers
13k
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What is a foliation and why should I care?
The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...
- 9,330
6
votes
2
answers
690
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Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively?
Dictionary between algebra and geometry is somewhat one of the main concepts in modern mathematics. So commutative $C^*$ algebras are one-to-one with locally compact Hausdorff spaces.
So it is ...
- 24.8k
4
votes
1
answer
362
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$K$-Theory of finite dimensional Banach algebras
Is there a finite dimensional Banach algebra $A$ for which $K_{0}(A)$ is a finite group?
I asked this question in MSE but I received no answer
https://math.stackexchange.com/questions/1624250/...
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