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Questions tagged [noncommutative-geometry]

Noncommutative geometry is a study of noncomutative algebras from geometrical point of view. The motivation of this approach is Gelfand representation theorem, which shows that every commutative C*-algebra is *-isomorphic to the space of continuous functions on some locally compact Hausdorff space.

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Pushforward of smooth section is smooth?

This is the main question: if $p:A \rightarrow B$ is a smooth vector bundle homomoprhism over base space $M$, then $pX$ is a smooth section of $B$, where $X \in \Gamma(A)$ is a smooth section of $...
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Modern introduction to $C^*$-algebras

Well, I am asking for the references on the subject for those who can't stand the Murphy's book. I have a background in functional analysis (including Banach algebras, functional calculus, Gelfand ...
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49 views

Piecewise linear structure as generators of $C(M)$

Let $M$ be a compact topological manifold and $C(M)$ the commutative unital $C^*$-algebra of complex valued continuous functions on $M$. Suppose that $M$ admits a piecewise linear structure. In the ...
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Analog of finite set in valued complete division ring

Let $K$ be a complete valued division ring and $S$ be a compact subset of $D$. It is easy to see that if $K$ is commuative, the $K$-algebra $\mathscr C(S,K)$ of continuous functions on $S$ in $K$ is $...
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66 views

The closed sphere $B_1(0)$ in $(C (X), d_\infty)$ is compact iff $X$ is finite

Let $C(X)$ be the set of all real-valued continuous functions on a set $X$. X is a compact topological space. Let $d_\infty(f,g):=\sup d(f(x),g(x))$, where $d$ is the standard distance on the real ...
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Regarding equality of norms in Hilbert bimodules

I was reading the book Elements of noncommutative geometry and in page 160 lemma 4.21 the authors state that in a Hilbert B-A bimodule $E$ the two norms induced by the two inner products coincide. ...
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93 views

Anzai flow in noncommutative geometry

Consider Anzai flows (cf. Anzai: Ergodίc 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 ...
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Toy example of deformed diffeomorphism group

Consider a toy example of a diffeomorphism group – the group of diffeomorphisms of a 1-dimensional manifold with a disconnected boundary (2 points). The group is a group of monotonically increasing ...
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Quantum principal bundles in physics

Recently I was reading in Stephen B. Sontz' "Principal bundles - The quantum case" and in contrast to "the classical case" he offered almost no connections with physical concepts. For quantum groups ...
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101 views

Lie algebra-valued differential forms, exactness, closedness, and the Moyal product

This is my attempt to prove something (I'm not even sure if it's true to begin with) using somewhat loose arguments. I present here all the steps and ideas and I would be extremely grateful if someone ...
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43 views

Does the following metric metrize the weak*topology on the state space?

Let A be a finite-dimensional C*-algebra. In a paper of Rieffel, it is shown that its state space, S(A), equipped with the energy metric, is isometrically embedded in a Hilbert space. Does this imply ...
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On the application and clarity of the statement of unitary implementation theorem

What is the essence of unitary implementation theorem that is given two von Neumann algebras are both $*$-isomorphic have cyclic and separating vectors in their represented Hilbert spaces are ...
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Does Pontryagin duality extend to class two nilpotent or maybe even metabelian locally compact groups?

A metabelian group $G$ is determined up to isomorphism by "abelian data" $(A,M,\alpha)$ -- an abelian group $A:=G/[G,G]$, an $A$-module $M:=[G,G]$, and a cocycle $\alpha:A\times A\to M$ giving the ...
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Doubt regarding Serre-Swan theorem and Hilbert modules

The Serre-Swan states that given a vector bundle $V$ with base $X$ (compact and Hausdorff), we have that $\Gamma(V)$ is a finitely generated projective $C(X)$-module. Moreover, all finitely generated ...
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Projections in the Cuntz algebra which have the same $K_0$ class

Assume that $e,f$ are two projections in the Cuntz algebra $\mathcal{O}_n$ , which have the same $K_0$ class. Are they necessarily Murray von Neumann equivalent ? The following post is the ...
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A separable $C^*$ algebra which contains all separable $C^*$ algebras.

Is there a unital separable $C^*$ algebra which unitaly contains all unital separable $C^*$ algebras? The motivation is that the answer is positive in the commutative case since every compact ...
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Fibres of $C_0(X)$-algebra. Clarification of Basics (Williams “Crossed Products of C*-Algebras”)

Definition. Let A be a $C^*$-algebra, $X$ - locally compact Hausdorff space. Then A is a $C_0(X)$-algebra if there is a homomorphism $Ф_A$ from $C_0(X)$ into the center $ZM(A)$ of the multiplier ...
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A “group” analogy of the Gelfand Naimark theorem

Are there two compact Hausdorff topological spaces $X,Y$ which are not homeomorphic spaces but the following two groups are isomorphic groups? $C(X, \mathbb{H}^2)$ and $C(Y,\mathbb{H}^2)$ where $\...
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On noncommutative dynamics

What is the connection between modular theory to study the W*-dynamics?? Takasaki theorem says the automorphism will commute with the modular automorphism group, further we know the classical action ...
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63 views

Nonconmutative linear vector space [closed]

I'm trying to understand what's a noncommutative space, I'm a undergraduate student so I'm not familiar with algebraic geometry. I think that the starting point of a noncommutative space would be a ...
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64 views

Diagonalizing a matrix of operators

Let $A,B,C\in\mathbb{C}[\partial_x,\partial_y]$ be differential operators. Diagonalize the matrix $$M=\begin{pmatrix}A+B&C\\-C&A-B\end{pmatrix}.$$ If $B,C$ were operators with purely complex ...
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66 views

Kähler differentials on $C^\infty(S^1)$

I'm trying to learn a bit about "de Rham homology" for algebras. Let us take $A=C^\infty(S^1)$. So if I understand well, the complex given by Kähler differentials and $d$ should compute de Rham ...
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182 views

What pre-requisites does non commutative geometry has?

I'm a masters student currently deciding in which area should I focus on. So far my primary interest has been C* algebras and operator algebras (already have some knowledge on K-theory for C* algebras ...
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106 views

Star products and Jacobi Identity

I'm having a "little" problem with one affirmation on Kontsevich's paper. He says that the second order terms $O(\hbar)$ implies, assuming that the associator $A(f,g,h)=0$, that the Jacobi Identity it'...
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87 views

Is the orthogonal complement a complementable submodule?

Let $H$ be a Hilbert $C^*$-module over some $C^*$-algebra $A$, and consider a closed submodule $M\subseteq H$. Then it is well-known that $M$ is not necessarily orthogonally complementable in $H$. ...
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115 views

AF C*-Algebras and Continuous Functions on Totally Disconnected Sets

Why do the continuous functions on a totally disconnected set, such as the Cantor set, form an AF C$^*$-algebra? Conversely, why do commutative AF C$^*$-algebras consist of continuous complex ...
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The reduced crossed product $A\rtimes_{\alpha,r}\Gamma$ doesn't depend on the choice of the faithful representation

The reduced crossed product $A\rtimes_{\alpha,r}\Gamma$ doesn't depend on the choice of the faithful representation $A \subset \mathbb{B}(\mathcal{H})$. [I am following the book by "Brown and Ozawa". ...
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204 views

Motivation for spectrum of an Abelian category

In his book Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Rosenberg defines (III.1.2 on page 111) the spectrum of an Abelian category $\mathbf{A}$ in the following way. ...
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Let $A$ be a $C^*$-algebra, and $S,T\in A$ be normal. If $ST=TS$, then must $S^*T=TS^*$?

I wish to show a kind of "simultaneous diagonalizability" statement for commuting normal operators $S,T\in\mathscr{L}(H)$ over a complex separable Hilbert space $H$. However, I've run into the hitch ...
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112 views

Diffeological isomorphisms between irrational tori.

This is exercise 4 from chapter 1 of the book "diffeology" by Patrick Iglesias-Zemmour. Let $\alpha \in \mathbb{R}$ and $\beta \in \mathbb{R}$ be irrational numbers. Define the irrational (non-...
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Completeness in the Noncommutative Geometry “Dictionary”

In numerous references on noncommutative geometry, one can find some sort of "dictionary" for translating concepts on the topological/geometric side into their corresponding algebraic counterparts (...
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45 views

Noncommutative (quantum, deformed, …) sphere interpolating between $S^n$ and $S^{n-2}$

Is there some example of such object? I have been looking some articles on quantum spheres, but a lot of families seem to be isospectral and then no questions are asked about limit cases. I am ...
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Representations of groupoid algebras

In reading through Khalkhali's Noncommutative Geometry text, I came across something I don't understand. Let $\mathfrak{G}$ be a discrete groupoid, and for each $x\in Obj(\frak{G})$, define the *-...
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144 views

Projection Lemma for Hilbert C-star modules

The projection lemma (for example, Rudin's functional analysis book theorem 12.4) says that if $M$ is a closed vector subspace of a $\mathbb{C}$-Hilbert space $\mathcal{H}$ then $$\mathcal{H}=M\oplus ...
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87 views

Characterisation of $C_{c}(X)$ as a dense subalgebra of $C_{0}(X)$.

Consider a non-unital commutative $C^{*}$-algebra $A$ and a dense subalgebra $B\subset A$. We know that $A\cong C_{0}(X)$, for some locally compact Hausdorff space $X$. I would like to know if we are ...
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56 views

Torsion property in noncommutative geometry

I'm reading this paper, and definition 2.2.1 (page 5) is the following: The torsion of a left $A$-covariant derivative $\nabla$ on $\Omega^1A$ is the left $A$-module map $T = \wedge \nabla - d:\...
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Is there a good notion of an étale topos associated to a noncommutative ring?

The title of my question basically sums up what I wish to know. I'm looking into trying to generalize étale cohomology to noncommutative rings (without having to go through some sort of De Rham or ...
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37 views

relationships of topological and C* concepts in noncommutative topology

According to wikipedia, noncommutative topology is " a term used for the relationship between topological and C*-algebraic concepts". Can somebody expand on this, give examples/theorems/results and ...
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Maps inducing identity in Hochschild and cyclic theories

Let $A$ be a unital algebra over $\mathbb{C}$, $M$ be an $A$ bimodule, $C^n(A,M)$ be a space off all $n$-linear maps $f:A^{n} \to M$ (to be called $n$ cochains) and define $b:C^n(A,M) \to C^{n+1}(A,M)$...
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Hochschild homology of a smooth manifold : sheaf?

It can be shown that the Hochschild cohomology of the algebra of smooth function on $X$ is naturally isomorphic to the sheaf of deRham current. But is it possible to see from the very definition of ...
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105 views

Most natural equivalence between $C^*$-algebras

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 ...
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1answer
794 views

Classification of vector bundles over the torus

In M. Rieffel's paper "The Cancellation Theorem for projective modules over irrational rotation $C^*$-algebras", he classifies finitely generated projective modules over the $C^*$-algebra $C(\mathbb{T}...
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38 views

grading of quantum plan

I can not understand a paragraph of this book:
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134 views

Learning noncommutative algebraic geometry

What are the prerequisites (I.e. Which textbooks/reference) and what's the recommended route at learning noncommutative algebraic geometry?
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143 views

A nilpotent element of an algebra which does not lie in the span of commutator elements.

What is an example of a $C^{*}$ algebra such that the span of nilpotent elements is not a sub vector space of the span of commutator elements. Obviously any such $C^{*}$ algebra would be a ...
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174 views

K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...
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67 views

Projective Resolution of $C^{\infty}(V)$ by Connes

In his article Noncommutative differential geometry (Inst. Hautes Études Sci. Publ. Math. No. 62 (1985), 257–360) A. Connes gives in Lemma 44 (p. 343f) a projective topological resolution of the ...
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1answer
91 views

Questions about the C* subalgebra $h(\Gamma)$ of $l^{\infty}(\Gamma)$

Given $(\Gamma,d)$ a metric space that is discrete. We know that $l^{\infty}(\Gamma)$ is a commutative $C^*$ algebra that is isometrically isomorphic to $C(\beta\Gamma)$. Let $h(\Gamma)$ be a subset ...
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Notation $A_q^{2|0}$ and $A_q^{0|2}$ in Manin's book.

In Manin's book: quantum groups and non-commutative geometry, there are two notations $A_q^{2|0}$ and $A_q^{0|2}$. Here $$ A_q^{2|0} = k<x,y>/(xy-q^{-1}yx), \\ A_q^{0|2} = k<\xi,\eta>/(\xi^...
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40 views

Existence of free operators, independent and with given distributions

I am trying to learn free probability from scratch, mostly by myself. I am trying to prove the following result. If $\mu$ and $\nu$ are compactly supported probability measures, then there ...