Questions tagged [noncommutative-geometry]
Noncommutative geometry in the sense of Connes and beyond: noncommutative algebras viewed as functions on a noncommutative space.
459
questions
5
votes
1
answer
202
views
Noncommutative symmetric spaces
I am recently studying ergodic actions of Lie groups acting on Riemannian symmetric spaces. Since I am also interested in operator algebras, it makes me wonder if there are some very natural ...
- 1,765
3
votes
0
answers
210
views
Reference on noncommutative PDE
I would like to ask if there is reference on semi-linear parabolic PDE (or more generally any kinds of PDE) with non-commutative unknown variable. For example, assume $u$ is a matrix-valued function (...
- 31
6
votes
0
answers
234
views
For what kind of $C^*$ algebra $A$ every normal element $y\in A$ has a normal lift for every given surjective $C^*$ morphism $\phi:B\to A$
Is there a terminology for the following property of $C^*$ algebra $A$:
For every $C^*$ algebra $B$ and surjective $C^*$ morphism $\phi: B\to A$, every normal element $y\in A$ admits a normal ...
- 312
2
votes
1
answer
257
views
A question on quantum tori
Let $\mathbb T_\theta^2$ be quantum tori generated by two unitary operators $u,v$. can $u,v$ be finite dimensional?
- 1,765
5
votes
1
answer
482
views
Embedding of Cuntz algebras $O_2\subseteq O_3$?
The Cuntz algebra $O_n$ is the (universal) C*-algebra generated by n-isometries $s_1,...,s_n$ such that
$$\sum_{i=1}^n s_is_i^\ast =\mathbf{1}, \ \hbox{and}\ s_i^\ast s_j=\delta_{ij} \mathbf{1}\ (\...
- 155
7
votes
1
answer
475
views
Localization of symmetric monoidal categories and geometry
I have a series of vague questions, related to localization of symmetric monoidal categories.
Here is the context. Say we are working over a field of characteristic zero. Then the "one category ...
- 7,902
9
votes
1
answer
228
views
A non nuclear $C^*$ algebra $A$ for which the algebraic tensor product $A\otimes A$ admits a unique $C^*$ norm
Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide?
A somewhat similar question is discussed here.
- 312
6
votes
0
answers
200
views
What is a quantum analogue of the fact that the second fundamental group of every Lie group is trivial?
What is an appropriate version of the following fact in terms of Hopf algebras and quantum groups:
"For every connected Lie group $G$ the second fundamental group $\pi_2(G)$ is trivial?"
Is there ...
- 312
4
votes
0
answers
241
views
Non-commutative analogue of a certain fact in differential geometry
In the literature, is there a non-commutative analogue of the fact that every Riemannian manifold whose isometry group has sharp dimension must be a constant curvature manifold?
- 312
2
votes
0
answers
121
views
Relative de Rham Cohomology groups of k-algebra
Let $A$ be a commutative unital $k$-algebra. Then we have de Rham complex given as:
$C_{\ast}(A)$ : $ 0 \rightarrow A \rightarrow \Omega_{A \lvert k}^{1} \rightarrow \Omega_{A \lvert k}^{2} \...
- 609
3
votes
1
answer
151
views
Reduced compact quantum group and left and right multiplication
Let $(A,\Delta)$ be a compact quantum group in the sense of Woronowicz, and let $A_0$ be its dense Hopf subalgebra. We can construct from the Haar state $h:A \to \mathbb{C}$ an inner product
$$
\...
- 1,144
4
votes
2
answers
245
views
$K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $p:A \to B$ be a surjective norm-decreasing $*$-homomorphism which is injective on a dense $*$-sub-algebra of $A$. Can such a map have non-trivial kernel,...
- 949
8
votes
1
answer
498
views
Vanishing of Hochschild homology of a category
Let $A$ be a dg- or $A_{\infty}$-category (with $\mathbb{Z}$-graded Hom sets, over a field of characteristic $0$). Let $HH_*(A)$ be the Hochschild homology of $A$.
Suppose that $HH_n(A)=0$ for all $n ...
- 927
2
votes
1
answer
151
views
Automorphism of algebras with certain initial conditions on given idempotents
The First question
Let $A$ be a Banach or a $C^*$ algebra. Assume that $e,f$ are two idempotents or prjections in $A$ which satisfy $ef=fe=0$. Assume that there are two automorphisms $\phi, \psi: A \...
- 312
4
votes
0
answers
326
views
Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)
The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...
- 312
0
votes
0
answers
96
views
Smooth sections of finite dimensional bundle and covering space
Let $G$ be a discrete finitely generated group which acts properly and freely on a smooth manifold $M$ with compact quotient $M/G$. Is it right to consider any function $f \in C^{\infty}_c(M)$ (with ...
4
votes
1
answer
338
views
On differential equation $Z'=Z^2-Z$ on a $C^*$ algebra
Let $A$ be a Banach or a $C^*$ algebra. We consider the differential equation $$(*)\;\;\;\;Z'=Z^2-Z$$ on $A$.
Obviously the singularities of this systems are just the idempotents of the ...
- 312
3
votes
0
answers
139
views
a closed projection on a C*-algebra is compact iff it is closed on the multiplier algebra
I'm trying to understand the proof for the equivalence of (i) and (v) in the following picture. I don't quite understand what the highlighted sentence means. I want to know why there is a surjection ...
- 135
2
votes
0
answers
60
views
Integrals in noncommutative graded algebras which are not necessarily Hopf
Let $\mathbf{k}$ be a field. Let $A$ be a finite dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbf{k}$-algebra such that $A^0=\mathbf{k}1$. Let $m$ be the maximal non-negative integer such that $A^m\...
- 1,064
8
votes
1
answer
717
views
A question regarding Kadison-Kaplansky idempotent conjecture (A nearest group element $g$ to a nontrivial self adjoint unitary element u )
Edit: According to answer and comments by Prof. Valette we edite the question.
The Kadison Kaplansky conjecture says:
Kadison-Kaplansky conjecture: If $G$ is a torsion-free discrete group then $C^*_{\...
- 312
2
votes
1
answer
221
views
$C^*$-algebras appearance in study of Lie groupoids and differentiable stacks
I am reading Differentiable stacks, gerbes, and twisted K-Theory by Ping Xu.
To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All ...
- 5,931
3
votes
0
answers
254
views
Homotopicity of $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ as morphisms from $A$ to $A\otimes A$
let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.
Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\...
- 312
1
vote
0
answers
109
views
Commutative subalgebras of $B(H)$ whose all automorphisms are in the form of unitary conjugation
Let $H$ be a complex Hilbert space.
Is there a compact Hausdorff space $X$ such that $C(X)$ is embeded in $B(H)$ and for every homeomorphism $\alpha$ of $X$ there exist a unitary operator $u\in B(H)$ ...
- 312
4
votes
0
answers
113
views
Can every dynamical system be interpreted in terms of (unitary) conjugation in an operator algebra
Let $H$ be a Hilbert space and $X$ be a compact Hausdorff space with a homeomorphism $\alpha: X \to X$. Assume that $C(X)$ is a commutative sub algebra of $B(H)$, namely $C(X)$ is embedded in $B(H)$...
- 312
3
votes
0
answers
53
views
The number of minimal components of a dynamical system via certain invariants of corresponding cross product $C^*$ algebra, some precise examples
Let $X$ be a compact Hausdorff space and $\alpha$ be a homeomorphism of $X$.
So we have a natural action of $\mathbb{Z}$ on $C(X)$ which generates the cross product algebra $C^*(X,\alpha)$...
- 312
0
votes
1
answer
210
views
Quantum (group) version of ${\mathbb Z}^n$?
As we know there are quantum analogue of tori called quantum tori generated by noncommuting operators $(A_1,\dots,A_n)$ with $A
_iA_j=A_jA_ie^{2\pi i\alpha}$ where $\alpha$ is a irrational number as a ...
- 1,765
11
votes
0
answers
650
views
Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'
It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version ...
- 419
2
votes
1
answer
117
views
Twisted canonical commutation relations
I am dealing with universal C*-algebra generated by $x,y$ with the following relations: $xy = qyx$, $x^{*}y = qyx^{*}$, $y^{*}x = qxy^{*}$, $x^{*}x = q^2xx^{*} - (1-q^2)yy^{*}$, $y^{*}y = q^2yy^{*} - (...
- 161
6
votes
2
answers
267
views
Finite-dimensional Hilbert $C^*$-modules
Does there exist a classification, or characterization, of finite-dimensional Hilbert $C^*$-modules? More generally, does there exist a characterization of countable direct sums of finite-dimensional ...
- 973
5
votes
1
answer
288
views
Abstract transverse measure theory
After reading Noncommutative Geometry book (see here) I came across the notion of the so called abstract transverse measure theory which is a generalization of standard measure theory well adapted to ...
- 9,150
1
vote
0
answers
144
views
Equivalence of two approaches to transverse measures for a foliation
Suppose that $(V,F)$ is a foliated manifold. There are three equivalent approaches to the notion of transverse measure as described in this book (see pages 65-69). I would like to understand the last ...
- 9,150
9
votes
0
answers
343
views
Geometric motivation behind the Fredholm module definition
If $A$ is an involutive algebra over the complex numbers $\mathbb{C}$, then a Fredholm module over $A$ consists of an involutive representation of $A$ on a Hilbert space $H$, together with a self-...
- 973
3
votes
1
answer
172
views
Noncommutative Leray - Hirsch theorem in the context of noncommutative principal bundles
In the literature, are there some researchs on non commutative analogy of Leray-Hirsch theorem in the context of non commutative Principal bundles?
- 312
14
votes
1
answer
636
views
Is every finite quantum group a quantum symmetry group?
This post is basically a quantum extension of Is every finite group a group of “symmetries”?
Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra.
Frucht's theorem ...
17
votes
2
answers
2k
views
Quantum corrections to geometry
In this video Alain Connes made a comment about the ,,quantum corrections'' of the geometry. I would like to understand this notion in some details since I haven't found anything about this in the ...
- 9,150
5
votes
1
answer
212
views
Zero divisors in compact quantum groups
Let $\mathcal{G}$ be compact quantum group in the sense of S. L. Woronowicz. As is well-known, every compact quantum group contains a dense Hopf algebra, called the polynomial Hopf algebra Pol$(\...
- 949
5
votes
1
answer
272
views
Dirac operator on a Morita equivalent algebra
Let $(A,H,D)$ be a spectral triple and let $B$ be an algebra which is Morita equivalent to $A$. Then there exists a finitely generated, projective $A$ module $E$ such that $B=End_A(E)$. Endow $E$ with ...
- 9,150
2
votes
1
answer
140
views
Is the algebra of Schwarz functions on a noncommutative torus the maximal algebra of smooth functions?
Let $\theta$ be a real number. We define $A_{\theta}$, the algebra of continuous functions on a noncommutative $2$-torus, to be the universal $C^*$-algebra generated by two generators $U$ and $V$ ...
- 8,707
5
votes
1
answer
171
views
(Noncommutative) Tietze $C^*$ algebras
A unital $C^*$ algebra $A$ is said a Tietze algebra if it satisfies the following:
For every ideal $I$ of $A$ and every unital morphism $\phi: C[0,1] \to A/I$ there is a unital morphism $\tilde{\phi}:...
- 312
8
votes
1
answer
532
views
Comparing the definitions of $K$-theory and $K$-homology for $C^*$-algebras
In Higson and Roe's Analytic K-homology, for a unital $C*$-algebra $A$, the definitions of K-theory and K-homology have quite a similar flavor.
Roughly, the group $K_0(A)$ is given by the ...
- 949
7
votes
0
answers
164
views
How does the $C^\ast$ algebra of an orbifold grupoid relate to the corresponding orbifold?
My question is in nature a bit vague but let me try to make it concrete. Given a Lie grupoid $G$ that is étale and proper (called an orbifold grupoid) we have an associated orbifold $X$; this is ...
- 233
8
votes
1
answer
270
views
Noncommutative Fredholm operators
Let $A$ be a unital $C^*$-algebra and $F:H_A\rightarrow H_A$ a Fredholm operator on the standard Hilbert $A$-module $H_A:=l^2(A)$. Is it true that $\mbox{ker}(F)$ and $\mbox{coker}(F)$ are finitely ...
- 99
6
votes
1
answer
361
views
Real rank 0 implies stable rank 1 on $C^\ast$-algebras?
A $C^\ast$ algebra has defined stable rank (https://www.univie.ac.at/nuhag-php/bibtex/open_files/2079_Rieffel-StableRank.pdf) and real rank (https://core.ac.uk/download/pdf/82123484.pdf), which are ...
- 233
1
vote
1
answer
492
views
$P$ is projective if and only if $P\otimes N\cong Hom(Hom(P,R),N)$
Let $R$ be a commutative Noetherian ring and $P$ be finitely generated $R$-module.
How to prove the following.
$P$ is projective if and only if $P\otimes N\cong Hom(Hom(P,R),N)$ for all finitely ...
- 323
3
votes
0
answers
64
views
Understanding order structure and trace on $K_0(T_\theta)$, a non-commutative torus
I've been working with Rieffel's "Projective Modules over Higher-dimensional Non-commutative Tori" and I'm struggling with a few basic questions.
I know that when the dimension $d=2$ we have that $...
- 111
4
votes
1
answer
230
views
Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma
I would like to understand the proof of Lemma 1, page 339 in this book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator ...
- 9,150
4
votes
0
answers
145
views
A relation between Hochschild cohomology of a $C^*$ algebra and its bidual
Let $A$ be a $C^*$ algebra and $A''$ be its bidual with the Arens product.
Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$...
- 312
7
votes
1
answer
207
views
$*$-algebras, completions, and $K$-theory
What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $...
- 973
3
votes
0
answers
201
views
Anzai flow in noncommutative geometry
Consider Anzai flows (cf. Anzai: Ergodic Skew Product Transformations on the Torus, Osaka Math. J. 3 (1951), 83-99) on the two dimensional torus $T^2$. I would like to know if there exists some ...
1
vote
0
answers
61
views
A 2- cocycle $\tau$ which is not cyclic but it still satisfies the stability of $\tau(e,e,e)$ for idempotent $e$
I learned the following statement from page $20$ of the book Noncommutative Geometry by Alain Connes:
Let $\tau$ be a $2$-cyclic cocyle on a $C^*$ algebra. Then for every smooth curve $e(t)$ of ...
- 312