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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
29 views

Metric connection as map from $\Omega^1 \, \to \, \Omega^1 \otimes \, \Omega^1$

I was reading Shahn Majid's and Edwin Begg's book on Quantum Reimannian Geometry. In this they define the metric to be a map $$g:\Omega^1 \otimes \Omega^1 \to A.$$ I do not understand this definition. ...
user avatar
19 votes
1 answer
1k views

How to understand what a 'noncommutative space' is

Whilst acquainting myself with the fundamentals of $C^*$-algebras and their $K$-theory, I read about how Gelfand Duality allows various $C^*$-algebraic concepts to be directly translated into a ...
user avatar
  • 2,530
3 votes
0 answers
93 views

Simplifications in definition of KK groups ("Fredholm picture")

I want to understand the relation between the $KK$ groups (in the sense of Kasparov) and ordinary operator $K$-theory, i. e. I would like to prove something like this: For $B$ trivially graded $\sigma$...
user avatar
  • 305
1 vote
0 answers
50 views

Principal bundles in the sense of Cartan and Steenrod.

i am starting to learn Noncommutative Geometry, but I did not attend course of Algebraic Topology during studies. For next week I need to know what is principal bundle, in both senses of Cartan and ...
user avatar
3 votes
0 answers
120 views

1-dimensional foliation of surfaces with prescribed graph of foliation

Definition of the graph of a foliation Let we have a $k$ dimensional foliation of an $n$ dimensional manifol $M$. One associates to this foliated manifold a (not necessarily Hausdorff) ...
user avatar
  • 1,730
0 votes
1 answer
46 views

About the definition of $A^{e}-$mudule structure of an $A$-bimudule $M$

Let $R$ be a conmutative ring (with unit $1$) and $A$ an associative $R$-algebra (with unit $1_{A}$). Let $A^{e}=A\otimes_{R}A^{opp}$ the eveloping algebra of $A$. Let $M$ be an $A$-bimodule. I want ...
user avatar
1 vote
0 answers
74 views

Trouble verifying covariance of differential calculus of the two dimensional quantum plane

I'm looking at page 4 and 5 of https://arxiv.org/pdf/math-ph/0105002.pdf. I am wanting to use (14) and (17) to verify the covariance of (14). But I'm having trouble, in particular, I'm having trouble ...
user avatar
  • 11
1 vote
1 answer
48 views

Does the category of $C^*$-algebras embed into the category of frames?

The category of commutative $C^*$-algebras is dual to the category of LCH topological spaces. My understanding is that authors in operator algebras often understand a 'noncommutative LCH space' to be ...
user avatar
  • 872
1 vote
1 answer
60 views

Compactly generated stable categories are dualizable.

Let $1-\operatorname{Cat}^{\operatorname{St,cocmpl}} _{\operatorname{cont}}$ denote the ($\infty$-)category of stable cocomplete ($\infty$-)categories. The Ind-completion $\operatorname{Ind}(C_0)$ of ...
user avatar
  • 1,341
1 vote
0 answers
27 views

Correspondence between projective algebra and (not necessarily commutative) fields

So currently I'm reading the Birkhoff–Neumann (1936) quantum logic paper 1, and I'm basically stuck in the interpretation of section (13) (p. 833-834); for what I understand, they are trying to ...
user avatar
1 vote
1 answer
107 views

Representations of a semigroup over a Hilbert space

A representations of a discrete semigroup $S$ with an involution $\star$ over a Hilbert space $H$ is a semigroup homomorphism $\varphi : S \to B(H)$ that preserve the involution. That is, for any $a, ...
user avatar
  • 16.3k
1 vote
1 answer
41 views

Showing the existence of a right-inverse in a von Neumann algebra

In this paper, specifically Theorem 4.1 on page 10, one of the last steps in the proof involves saying that a particular right-inverse exists. I'll try to restate what I think are the relevant pieces ...
user avatar
  • 7,909
2 votes
0 answers
67 views

Geometry vs Topology

I am a beginner in functional analysis and operator algebras. Recently I stumbled upon the two subjects, non-commutative topology and non-commutative geometry. What are the similarities and ...
user avatar
  • 170
1 vote
0 answers
36 views

symmetry group for noncommutative manifold

(This post is cross-post in MathOverflow https://mathoverflow.net/questions/381731/symmetry-group-for-noncommutative-manifold-from-spectral-triple) Is there any notion of symmetry group for ...
user avatar
  • 899
1 vote
1 answer
39 views

Is the KO dimension of commutative real spectral triple agree with it's dimension of the manifold?

Given a spectral triple with some conditions(such as the algebra is commutative), Connes's reconstruction theorem states a we can recover a Riemannian manifold with spin structure.(see here) Now I ...
user avatar
  • 899
2 votes
0 answers
131 views

what is noncommutative geometry studying if "noncommutative space" does not exist?

(If this is not a suitable question in here, I will delete it.) I am new to noncommutative geometry. I know the motivation is to try to generalize the Gelfrand duality to noncommutative $C^{*}$ ...
user avatar
  • 899
1 vote
0 answers
54 views

supersymmetry and quantum groups

I have some background in non-commutative geometry (in particular, I am doing research in quantum groups) and I have of course heard several times about the concept of supergeometry and supergroups. ...
user avatar
  • 358
0 votes
0 answers
152 views

Grassman variable and Grassmannian?

Grassman variables are anticommuting number or supernumber, is an element of the exterior algebra over the complex numbers. Grassmannian $Gr(k, V)$ is a space that parameterizes all $k$-dimensional ...
user avatar
2 votes
1 answer
77 views

Hilbert bundle and set of sections.

Let $\Lambda$ be a manifold and $p:H\to\Lambda$ a continuous Hilbert bundle with $H(\lambda):=p^{-1}(\lambda)$. Suppose $\Gamma_0^0(\Lambda)$ is the space of continuous sections vanishing at infinity ...
user avatar
  • 371
3 votes
1 answer
83 views

Is there any reference treating explicitly Lie-Rinehart pairs over non-commutative base algebras?

Wondering around in the literature, any reference to Lie-Rinehart algebras define them as pairs $(A,L)$ where $A$ is a commutative algebra over some field $\Bbbk$ (or even commutative ring) and $L$ is ...
user avatar
2 votes
0 answers
158 views

How to read Alain Connes's Noncommutative Geometry [closed]

I'm a undergraduate student interested in noncommutative geometry and quantum field theory. And I have learned basic functional analysis,theoretical mechanics,basic quantum mechanics, now, I'm ...
user avatar
  • 21
1 vote
1 answer
36 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{...
user avatar
  • 518
1 vote
1 answer
97 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)...
user avatar
  • 105
3 votes
1 answer
93 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 ...
user avatar
  • 1,133
1 vote
1 answer
98 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 ...
user avatar
2 votes
1 answer
149 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 ...
user avatar
  • 1,133
2 votes
1 answer
174 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 ...
user avatar
1 vote
1 answer
40 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 ...
user avatar
  • 127
0 votes
1 answer
51 views

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 ...
user avatar
1 vote
0 answers
83 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 ...
user avatar
  • 105
9 votes
0 answers
134 views

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 ...
user avatar
  • 111
2 votes
2 answers
155 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 ...
user avatar
  • 665
2 votes
2 answers
324 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 ...
user avatar
  • 1,983
0 votes
1 answer
63 views

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 $$...
user avatar
1 vote
0 answers
273 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{...
user avatar
2 votes
0 answers
94 views

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-...
user avatar
  • 545
1 vote
0 answers
128 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 $...
user avatar
  • 8,954
1 vote
1 answer
306 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 ...
user avatar
  • 2,267
1 vote
0 answers
57 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 ...
user avatar
  • 2,190
1 vote
0 answers
54 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 $...
user avatar
  • 937
1 vote
1 answer
171 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 ...
user avatar
  • 85
3 votes
0 answers
48 views

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. ...
user avatar
2 votes
0 answers
111 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 ...
user avatar
4 votes
0 answers
79 views

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 ...
user avatar
1 vote
0 answers
80 views

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 ...
user avatar
  • 1,431
4 votes
0 answers
225 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 ...
user avatar
  • 152
0 votes
0 answers
60 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 ...
user avatar
  • 105
3 votes
0 answers
57 views

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 ...
user avatar
1 vote
0 answers
146 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 ...
user avatar
2 votes
2 answers
166 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 ...
user avatar
  • 1,730