# 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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### Is the preimage of a Hopf subalgebra a Hopf subalgebra?

The following must be simple, but I have no intuition here, so excuse me. Let $F$ and $G$ be Hopf algebras over a field $k$ (in the usual sence, i.e. Hopf algebras in the category of vector spaces ...
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### Prove that the subalgebra of functions that identify two points and $2\times 2$ matrices are the same.

In this introductory lecture on non-commutative geometry, Connes gives an elementary example of how to identify two points. We take a two-point space $S:=\{a,b\}$ and we wish to identify (i.e. ...
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### Concerning the q-deformations of semisimple Lie groups

Recently, I came across a question on the q-analougs of finite groups of Lie type. Some people say that there is no q-deformations of finite groups in the category of quantum groups which I am not ...
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### Are there observables $X_1,\ldots,X_m$ and a state $\rho$ in a Hilbert space $H$ of dimension $n$. Prove that rank$(\text{tr}\ \rho X_iX_j))\le n^2$

This is from the exercise 5.7 of the book An Introduction to Quantum Stochastic Calculus by K.R. Parthasarathy. By observable we mean Hermitian Operators on a Hilbert space $H$ and by state we mean ...
1 vote
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### Non commutative Atiyah-Singer theorem

It is well known that for any unital C*-algebra $A$, $K_0(A)\cong \pi_0(\text{Fred}(\mathcal{H}_A))$ (the non-commutative Atiyah-Jänich theorem). In the monograph "Lectures on Operator Theory&...
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### Multiplier algebra as idealizer in fourth dual

The multiplier algebra $M(A)$ of a $C^*$-algebra $A$ can be identified with the idealizer of $A$ in $A^{**}$. I am wondering if the same identification extends to the fourth dual (seen as the ...
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### Are compactly supported states preserved by adjoints of proper *-homomorphisms?

Given a (non-unital) $C^*$-algebra $A$, I'll call a state $\phi$ on $A$ compactly supported if $\phi$ attains its norm on $A$. For example, every pure state is compactly supported (which follows from ...
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### Resolvent of an unbounded self-adjoint operator affiliated to a semi-finite algebra could be unbounded? [closed]

Let $\mathcal{M}$ be a semi-finite von Neumann algebra and $H$ is an unbounded self-adjoint operator affiliated to $\mathcal{M}$. Then is it the true fact that $(H-iI)^{-1}\in\mathcal{M}$ ?
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### How to Learn Noncommutative Geometry

This is a more direct, and precise question of a previous one that I have deleted. Obligatory Information/Background I am currently a first-year undergraduate studying at a frequented Mathematical ...
1 vote
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### Seeing that a product of quasi-free algebras is quasi-free in terms of lifting homomorphisms

A $k$-algebra over a field $k$ is quasi-free if for any bimodule $M$ the second Hochschild cohomology $H^2(A,M)$ vanishes or, equivalently, if for any square-zero extension $R/M=A$ there is a lifting ...
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### Unitary isomorphism between balanced tensor product spaces on $L^2(X)$

Some definitions Recall that given two ($i=1,2$) Hilbert $B_i$-modules $E_i$ and a $*$-homomorphism $$\phi:B_1\to \mathcal{L}(E_2),$$ where $\mathcal{L}(E)$ denotes the space of adjointable operators ...
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### 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. ...
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### 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 ...
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### 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 $\sigma$...
1 vote
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### 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 ...
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### 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) ...
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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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