# Tagged Questions

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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### relationships of topological and C* concepts in noncommutative topology

According to wikipedia, noncommutative topology is " a term used for the relationship between topological and C*-algebraic concepts". Can somebody expand on this, give examples/theorems/results and ...
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### Maps inducing identity in Hochschild and cyclic theories

Let $A$ be a unital algebra over $\mathbb{C}$, $M$ be an $A$ bimodule, $C^n(A,M)$ be a space off all $n$-linear maps $f:A^{n} \to M$ (to be called $n$ cochains) and define $b:C^n(A,M) \to C^{n+1}(A,M)$...
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### Hochschild homology of a smooth manifold : sheaf?

It can be shown that the Hochschild cohomology of the algebra of smooth function on $X$ is naturally isomorphic to the sheaf of deRham current. But is it possible to see from the very definition of ...
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### Most natural equivalence between $C^*$-algebras

I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism. Can someone explain this sentence or ...
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### Learning noncommutative algebraic geometry

What are the prerequisites (I.e. Which textbooks/reference) and what's the recommended route at learning noncommutative algebraic geometry?
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### A nilpotent element of an algebra which does not lie in the span of commutator elements.

What is an example of a $C^{*}$ algebra such that the span of nilpotent elements is not a sub vector space of the span of commutator elements. Obviously any such $C^{*}$ algebra would be a ...
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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} = k<\xi,\eta>/(\xi^...
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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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### 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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### 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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### 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 1-...
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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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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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### 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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### 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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### 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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### 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 ...