# Questions tagged [noncommutative-geometry]

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

360
questions

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### 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 (...

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### 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 ...

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**1**answer

139 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?

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230 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}\ (\...

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192 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 ...

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**1**answer

121 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.

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189 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 ...

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156 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?

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59 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} \...

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**1**answer

83 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
$$
\...

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146 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,...

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204 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 ...

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**1**answer

137 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 \...

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242 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 ...

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### 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 ...

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301 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 ...

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119 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 ...

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54 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\...

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454 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^*...

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47 views

### Daniell integral of “generalized (of some sort)” functions?

Let $E$ be a (Dedekind $\sigma$-complete) Riesz space and $H\subseteq E$ a subspace. A Daniell integral $I\colon H\to\mathbb R$ is defined to be a positive linear functional which is continuous with ...

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145 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 ...

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250 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 $\...

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75 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 $u\in B(H)$ ...

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106 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 $B(H)$...

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### 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 $C^*(X,\alpha)$...

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### 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 ...

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313 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 two is that Rosenberg's version of noncommutative algebraic geometry mainly concerns as ...

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### 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^{*} - (...

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151 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 ...

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### 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 ...

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### 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 ...

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### On differentiable dynamical system for group actions on von Neumann algebras

For actions of groups on von Neumann algebras is there any existing notion of the differentiable dynamical system? How one can say automorphism coming from the action are differentiable at appropriate ...

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### 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-...

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### 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?

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247 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 ...

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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 ...

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### 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$(\...

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### 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 ...

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### 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 $V$ ...

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### (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}:...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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308 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 ...

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### 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 $...

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198 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 ...

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135 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 $A''$...

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### $*$-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 $...

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### 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 ...