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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21 views

Definition p-multigraded Hilbert spaces

I'm having a hard time wrapping my head around the following definition taken form the book of Higson & Roe on k-homology. A Hilbert space is defined to have a $p$-multigrading if $H$ is $\mathbb{...
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30 views

Lipschitz Seminorms Associated to a Metric and Seminorms from Dirac Operators

I am learning to work with spectral triples and given a spectral triple $(A, H, D)$ with representation $\pi$, a seminorm can be defined on $A$ via $L_D := \|[ D, \pi(f) ]\|_{B(H)}$. View $A$ as $C(X)...
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76 views

$f(t) = \min\{1,t\}$ not operator monotone

I want to show that the function on $\mathbb{R}^+$ $f(t) = \min \{1,t\}$ is not operator monotone on the complex $2\times 2$ matrices. My plan is to find matrices $A$ and $B$, $B\geq A$, such that the ...
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61 views

smallest localising subcategory that contains a set of objects

Let $D$ a triangulated category closed under arbitrary coproducts and $\mathcal{R}$ a localising triangulated full subcategory that contains a set of objects $R$,which $R$ is closed under the ...
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32 views

Projection on C*-Algebra

I am trying to show that if A is an arbitrary C*-algebra, $p$ is a projection if and only if $p^{*}p = p$. Now since the C*-algebra is arbitrary, we only have a norm and not necessarily an inner ...
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54 views

tensor product in a k-linear category

Studying this paper i met a "tensor product" construction.At lemma 2.4.1 it is defined an object in D as a direct sum of $$E[n]\otimes_k H(E[n])$$ I searched on the internet and i found a comment ...
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29 views

Noncommutative geometry (Notation)

The representation π in (6.35) can be extended to a surjective map $$\mathbb{I}⊗π : CC(\mathcal{E}) → C(\mathcal{E})$$ , (8.13) namely, any compatible connection is the composition of π with a ...
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How to perform wedge product on a non-commutative algebra

I'm trying to calculate the covariant derivative of my connection $\mathbf{\mathcal{A}} $ (in this case $\mathbf{\mathcal{A}} $ is also a vector valued matrix but nevermind, I don't think is relevant ...
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71 views

The Operator $T= -i \frac{d}{dx}$

I read that the operator $T= -i \frac{d}{dx}$ on the Hilbert space $H_l = L^2([-l,l], (2l)^{-1}m)$, where $m$ is the normalized Lebesgue measure, can be defined as follows. Take the orthonormal basis ...
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Many definitions of Hochschild homology and cyclic homology

It appears that there are more definitions of cyclic homology than there are people working on cyclic homology. As a newcomer, this confuses me to no end. I've written a list of definitions that some ...
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120 views

What are some ideas on the fastest way to get from undergrad background in math to remotely reasonable understanding of noncommutative geometry.

What would be the fastest way to get from half of a class worth’s of (graduate) measure theory, same of functional analysis and generalized functions (or distributions), and only differential geometry ...
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275 views

Explicit 4-term commutator $[A,B,C,D]$

The commutator of two terms is: $$ [A,B]=AB-BA $$ The commutator of three terms is: $$ [A,B,C]=ABC+BCA+CAB-BAC-CBA-ACB $$ I was not able to google the four term commutator, not am I able to ...
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61 views

Why do we need noncommutative local rings? [closed]

Why are we need noncommutative local rings, is there any geometric meaning?
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Simplify the differential form $d [ A, B]$

How can we simplify the differential form in the expanded expression $$ d [ A, B]=? $$ Suppose $A$ is $p$-form, and $B$ is $q$-form, and $d$ is exterior derivative. My attempt: Generally I get $$...
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116 views

What is the definition of “quantum tori”?

I have learned that the non-commutative torus (2-dimensional) is $\mathcal{A}_\theta$. But now I'm going to read the paper written by Mania, which shows the equivalence of $\mathcal{QT}$ and $\mathcal{...
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Quasi-compactness of the Zariski topology on the spectrum of abelian category

I'm trying to understand a sufficient condition for the existence of a quasi-compact base of open sets of Zariski topology on the spectrum of an abelian category from https://sasharosenberg.com/?x-...
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60 views

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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120 views

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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56 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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49 views

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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121 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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105 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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159 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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55 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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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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94 views

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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141 views

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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68 views

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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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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112 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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83 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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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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125 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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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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148 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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262 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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161 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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47 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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163 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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92 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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65 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:\...