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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Question Regarding the Eigenvalues of Matrix Commutators
Let us consider a commutator of two diagonizable $n\times n$, i.e. square matrices, $A$ and $B$. The eigenvalues of $A$ are $\lambda^1,\lambda^2,...,\lambda^{n}$ The eigen values of $B$ are $\mu^1,\...
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Smooth vectors for the torus action on the irrational rotation algebra
There exists a canonical action of the group $S^1\times S^1$ on the irrational rotation algebra $A_\theta$, which is the universal C*algebra generated by two unitaries $u$ and $v$ satisfying $uv=e^{i2\...
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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 ...
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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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Spectral triple on $\mathcal C(M)$ where $M$ is a compact Riemannian manifold, not necessarily spin
I have been reading Alain Connes' Compact metric spaces, Fredholm modules and hyperfiniteness.
In proposition 1, it is mentioned that an unbounded Fredholm module (nowadays: spectral triple) over $C(M)...
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Hochschild homology of stable categories as topological chiral homology
Let $\mathscr{C}_0$ be a small idempotent complete stable category tensored over some symmetric monoidal category $\mathcal{E}$.
Its Ind-completion $\mathscr{C} := \operatorname{Ind}(\mathscr{C}_0)$ ...
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Spectral triple for a (real) full matrix algebra
Let $\mathcal A = \mathbb R^{N \times N}$ be the real full matrix algebra, $N \in \mathbb N_{> 1}$, which is represented by the Hilbert space $H := \mathbb R^N$ (that is, $\mathcal A \to B(H)$, $A \...
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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 ...
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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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What is the definition of $f(a)$ where $a$ is in a unital $C^*$-algebra?
Suppose that $A$ is a unital C$^*$-algebra. An element $a \in A$ is said to be normal if $a^*a=a^*a$. Further, let $\sigma(a)$ be the spectrum of $a$. I keep seeing these references to $f(a)$ where $f ...
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``Quantum''/ non-commutating extension of polytopes
Are there non-commutative (ie. quantum) extensions of polytopes?
More specifically I was wondering if there are some deformation, say $\hbar$, to polytopes which when is taken to be zero, one gets ...
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Simple modules of quantum planes
Let $k$ be an algebraically closed field.
Let $R := k\langle x,y \rangle/(yx-qxy) (q \in k^*)$.
We often call $R$ a quantum plane.
If $q$ is a primitive $n$-th root, then for any $(\zeta, \xi) \in k^* ...
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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 $\sigma$...
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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 ...
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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
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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 ...
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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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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $C^{*}$ ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 $C(X)...
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$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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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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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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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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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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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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