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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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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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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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} = ...
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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 ...
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What, and how can, topological invariants can be computed from a space's algebra of functions?

The Gelfrand duality says that the category of locally compact Hausdorff spaces (with proper continuous functions) is equivalent to the category of commutative $C^*$ algebras (with proper ...
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51 views

3d parametric spiral to 3D goldean mean spiral

I know I can create a 3d parametric spiral with the formula below but How can I do the same thing with goldean spiral? I looked at https://en.wikipedia.org/wiki/Golden_spiral but I don't see how to ...
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95 views

Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
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What is the most general notion of “Fourier transform?”

I know the definition of a classical Fourier transform that maps a function f(x) on the real line X to a function F(p) on a dual space (here another real line and borrowing some physics notation) P. ...
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59 views

Noncommutative torus with $\theta = 0$.

According to Wikipedia, one can construct noncommutative tori as follows: Letting $\theta \in \mathbb{R}$ be a parameter, consider the hilbert space $H = L^2(\mathbb{T})$, where $\mathbb{T}$ is the ...
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Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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Noncommutative manifold: Spectral triples on noncommutative quotients

I'm interested in taking the noncommutative quotient of a manifold, and endowing it with a kind of noncommutative smooth structure. More formally I'm interested in the question: is there a canonical ...
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31 views

Question about Noncommutative quotients

I want to understand noncommutative quotients. Now the book Basic Noncommutative Geometry by M. Khalkhali gives two different constructions of the noncommutative quotient and claims they are ...
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25 views

How are $C(S^1)$ and the crossed product algebra $C(\mathbb{R})\ltimes \mathbb{Z}$ Morita equivalent?

In Connes' Noncommutative geometry one construct "noncommutative quotients" by taking certain crossproduct algebra's. Given a group $G$ acting on a set $X$ through an action $\alpha$ we can form the ...
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187 views

Good starting point for learning noncommutative geometry?

Currently, I am attempting to learn noncommutative geometry. My interests mostly lie on the boundaries of pure mathematics and theoretical physics, so I am not only interested in the mathematical ...
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46 views

Algebra of matrix valued functions on the sphere

I was watching a video of a lecture by Alain Connes, and at around 8:00 he very briefly mentions a way to think about the algebra of 2x2 matrix functions over the 2-sphere, i.e. the maps from $S^2 ...
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Reference for proof of Hochschild-Kostant-Rosenberg for Hochschild cohomology

Is there a place where there is a full proof of the Hochschild-Kostant-Rosenberg Theorem for Hochschild cohomology? I am aware of many places where the result is proven for Hochschild homology i.e. ...
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NCG with all noncommutativity in a nilpotent ideal

While in general non-commutative geometry behaves rather differently from commutative geometry when it comes to local-to-global properties (descent), there are versions of "mild" noncommutative ...
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Can the Berezinian be used instead of the determinant of a block matrix?

Suppose we have a $2N \times 2N$ matrix $(N\ge 2)$. For example we can consider a block matrix: $$X= \left[\begin{matrix} A & B \\ C & D \end{matrix}\right]$$ with $A,B,C,D=$ $2\times 2$ ...
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pullback of global sections with respect to an automorphism of schemes

Let $X$ be a projective scheme and $\sigma:X\to X$ an automorphism of $X$. Is there a natural pullback of global sections map $H^0(X,\mathcal{F}) \to H^0(X,\sigma^*\mathcal{F})$ for ...
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There is only one interesting measure space

On page 52 of Noncommutative Geometry (available here: http://www.alainconnes.org/docs/book94bigpdf.pdf), Alain Connes states, "This wealth of transformations of a measure space $X$ is bound up in ...
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Noncommutative version of Littlewood's First Principle

There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle: Every Lebesgue ...
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108 views

Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
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Boundedness Spectral Triple Axioms for de Rham Complex

In Connes' axioms for a spectral triple $(A,H,D)$, they have a representation of an algebra $A$ in bounded operators on a Hilbert space $H$, and (unbounded) operator $D$, such that $[D,a]$ is bounded. ...
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Construction of noncommutative torus

In short, how do we get the formula for the NC torus? I find the equations in many places (including here) but I still have no idea for how this comes from the torus. If my understanding is correct, ...
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Example of a regular element with a commutative quotient

Whats an example of ring $A$ which is not commutative and contains an element $x\in Z(A)$ such that the left multiplication by $x$ is an injection and $x$ is not a unit and $A/(x)$ is commutative?
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Example of a regular element in noncommutative rings

Whats an example of ring $A$ which is not commutative and contains an element $x\in Z(A)$ such that the left multiplication by $x$ is an injection and $x$ is not a unit?
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Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
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Global dimension of translation algebra

What is the Hochschild cohomological dimension of the "translation algebra": $\mathbb{C}\langle x,y\rangle/(xy-yx-x)$? I expect it to be $2$, but I haven;t found a serious argument as to why this ...
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Isn't the center of a von Neumann algebra on a separable Hilbert space a hyperfinite von Neumann subalgebra?

this is a very quick, probably dumb, question, I was reading this chapter from "Hochschild cohomology of von Neumann algebras" by Allan Sinclair and Roger M. Smith and I came across this theorem on ...
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Roe algebra of a countably infinite set of points

First let me state some definitions. Let $X$ be a second countable, proper metric space. Let $H$ be a separable Hilbert space equipped with a nondegenerate $*$-representation $C_{0}(X)\rightarrow ...
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The first Weyl algebra is Calabi-Yau

Why is the Weyl algebra $A_1(k)$ over a field $k$ Calabi-Yau? (My definition of Calabi-Yau is Ginzburg's)
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Example of excision in Hochschild homology

The excision theorem for Hochschild homology introduced by Wodzicki seems like a very powerful tool (as scision was hyper-useful in topology). However, I cannot actually seem to think of a result ...
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47 views

Tor of submodule

Let $R$ be a $CRing$. If $i:A \rightarrow B$ is the inclusion of a $R$-subalgebra A into an $R$-algebra $B$, then what is ther relationship between: $Tor_{A^e}$ and $Tor_{B^e}$?
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Coproducts and Hochschild

I $\{X_i\}$ is a small family of associative $\mathbb{C}$-algebras and $X$ is their free product. Then I have two questions: 1) Why is $X$ their coproduct? 2) Is the Hochschild homology of X ...
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421 views

Obtaining rotation matrix from Euler angles if all three rotations happen at once. Does order of multiplication matter?

I'm having a problem getting my head around Euler Angles. Specifically if I wish to obtain a rotation matrix for a system where pitch, roll and yaw have all changed at once by various values... how ...
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Strong and weak equivalence of $C^*$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
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Monoidal categories and tensor products

Does the multiplication $-\square-$ biendofunctor in a Monoidal category, $\mathfrak{C}$ necessarily commute with coproducts? This is true in some familiar categories, such as $_RMod$, $Grp$ $CRings$ ...
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Smooth algebras and quasi-freeness

Let A be a unital associative algebra over a field k. Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...
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Is there a discrete version of non-commutative geometry (yet)?

I wonder if mathematicians have developed a discrete version of non-commutative geometry, a bit like graphs, simplicial complexes etc may be seen as a discrete version of (Riemannian) geometry (of ...
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Reference for proof of homotopy invarance of Cyclic cohomolgy

I'm looking for a good reference for a proof of the homotopy invariance of cyclic (co)homology. I'm following a refernce book by Joachim Cuntz, the proofs are ommited therein, or only shown in the ...
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Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
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Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
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Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...
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A lattice-theoretic question related to noncommutative tori

[NCG] So I'm trying to pin down a fairly well-known bit of noncommutative-geometric folklore that says that for $\Theta \in M_N(\mathbb{Q})$ skew-symmetric, the corresponding noncommutative $N$-torus ...
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Physical interpretation of $q$-deformation

I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a ...
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Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
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Specific (algebraic) directions in NCG

Since my initial question (How much algebra is there in Noncommutative Geometry?), I got to study the basics of NCG and I now consider starting a PhD. program in NCG. I read Khalkhali's Basic NCG and ...
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Exactness of $\Gamma^\infty$ Functor

Does anybody know a reference for fact that the Functor $\Gamma^\infty$, assigning to every smooth vector bundle $\mathcal{E}\to M$ the $C^\infty(M)$-module $\Gamma^\infty(\mathcal{E})$ of smooth ...
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Left-ratio and right-ratio in (not necessarily commutative) field

I am reading Birkhoff–Neumann (1936) about (not necessarily commutative) field $F$. The authors use terms right-ratio and left-ratio in section 13. Right-ratio is denoted as $[x_1, x_2, ... ...