# Questions tagged [noncommutative-geometry]

Noncommutative geometry in the sense of Connes and beyond: noncommutative algebras viewed as functions on a noncommutative space.

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### equivalent definition of k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...

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### Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma

I would like to understand the proof of Lemma 1, page 339 in this book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator ...

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### A relation between Hochschild cohomology of a $C^*$ algebra and its bidual

Let $A$ be a $C^*$ algebra and $A''$ be its bidual with the Arens product.
Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$...

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### $*$-algebras, completions, and $K$-theory

What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $...

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### Anzai flow in noncommutative geometry

Consider Anzai flows (cf. Anzai: Ergodic 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 ...

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### A 2- cocycle $\tau$ which is not cyclic but it still satisfies the stability of $\tau(e,e,e)$ for idempotent $e$

I learned the following statement from page $20$ of the book Noncommutative Geometry by Alain Connes:
Let $\tau$ be a $2$-cyclic cocyle on a $C^*$ algebra. Then for every smooth curve $e(t)$ of ...

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### Is this a 2-cyclic cocycle ? Does it have a nontrivial geometric interpretation?

Let $S$ be a surface in $\mathbb{R}^3$.
Inspired by the $2$-cyclic cocycle defined in page 20 of the book "Non-commutative geometry" by Alain Connes, we consider the following $3$-linear map on the ...

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### Kernel of multiplication in noncommutative 2-torus

What is $\Omega^1 (A_{\theta})$, that is, what is the kernel of the multiplication map $m:A_{\theta} \otimes A_{\theta} \to A_{\theta}$ where $A_{\theta}$ is the noncommutative 2-torus with parameter $...

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### informative examples for understanding spectral triples

I am at the beginning of my thesis work and I am trying to understand spectral triples. I can recall the definition but I have no informative examples with which to make sense of it. What are some ...

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### Constructing a noncommutative algebra from a commutative algebra

I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...

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### Separability of compact quantum groups

In the theory of compact quantum groups due Woronowicz, we assume usually that the C*-algebra of the compact quantum group is separable. Is the assumption essential in the theory? Will it eventually ...

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### Hilbert space separability for spectral triples

A spectral triple $({\cal A},{\cal H},D)$ consists of a unital $*$-algebra ${\cal A}$ represented as bounded operators on a Hilbert space ${\cal H}$, together with an unbounded operator $D$ having ...

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### What quantum groups admit quantum topography space structure?

Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian ...

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### Categorical features of Hilbert spaces

Does the category of Hilbert spaces and bounded maps have any particular categorical feature which can be studied systematically?
I mean, I know that it's a $*$-category, but it seems to have much ...

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### Path algebras are formally smooth

In Ginzburg's notes Lectures on Noncommutative Geometry, he claim that the path algebra of a quiver is formally smooth (See Section 19.2).
I have two questions. First, how to show this claim and ...

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### Some generalizations of the graph of a foliation

The graph of a foliation $\mathcal{F}$ of a manifold $M$ is the space of all triple $(x,y,[\gamma])$ where $x,y$ lies on the same leaf $F$ and $[\gamma]$ is the equivalent class of a ...

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### number of indecomposable summands of an extension of two modules

I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of ...

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### C*-algebra of a singular surface foliation

Noncommutative geometry associates a $C^*$-algebra $C^*(S,{\cal F})$ with a foliation $\cal F$ on a manifold $S.$
Did somebody study this construction for noncompact surfaces $S$?
What I am really ...

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### Simple $C^*$ algebras with invariant subspace property

Edit: According to the valuable comment of Yemon Choi I revise the question by replacing "faithful" with "irreducible".
We say that a $C^*$ algebra $A$ satisfies the invariant subspace ...

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### KMS-states of Bost-Connes type system

I have some struggles with understanding theorem 25 in the paper "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory".
More precisely, there is ...

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### Example of a prime action on a compact Hausdorff Space

Suppose that a discrete group $\Gamma$ acts on a compact Hausdorff space $X$ via homeomorphisms. This action induces an action on $C(X)$, the space of all continuous functions from $X$ to $\mathbb{C}$,...

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### Is the algebra $\mathcal{C}^{\infty}(M,\mathbb{R})$ a smooth algebra in the sense of algebraic geometry?

First of all, let me fix some terminology: I will follow the definitions that can be found in the book "Cyclic Homology" of J.L. Loday (second edition) page 102 in the special case of $K$ a field.
...

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### Kazhdan constant and finite index subgroups

I am wondering if there is some general relation between Kazhdan constants of a group and it finite index subgroups?
Let $G$ be a finitely generated group with a generating set $\Sigma$ that ...

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### Connes-Chern pairing, compatibility with periodicity operator in the odd case

Let $A$ be an algebra (say unital). For an odd (say $2n-1$) cyclic cocycle $\varphi$ and a class in $K_1(A)$ represented by invertible $u$ we define
$$\langle [\varphi],[u] \rangle:=\frac{2^{-(2n+1)}}...

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### Causal fermion systems fromm fractal geometry

Okay, first off- I apologise if this is a stupid question. I'm mainly a very young physics guy, but this has primarily math basis. I'm trying to build a theory that is, long story short, some ...

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### Second order signature operator in diffeomorphism invariant geometry as an image under right regular representation

I would like to understand the following statement taken from this paper, dealing with the so called Transverse Index Theory or in other words with the index theory for diffeomorphism invariant ...

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### Pairing between cyclic cohomology and $K$-theory: the odd case

I would like to understand the proof of Proposition 15 (see page 70 in this link ). More precisely: I would like to understand a particular step in the proof namely:
Why $\frac{d}{dt}(\varphi \# ...

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### Two approaches to periodic cyclic cohomology

Cyclic cohomology may be defined in several ways: the easiest way to define it is via a subcomplex $C^*_{\lambda}$of Hochschild complex consisting from cyclic cochains. There are also other ...

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### preliminary reading recommendation before embarking on Connes non commutative geometry book?

I want to try to understand non commutative geometry by reading Connes's book
..and I am discovering it is a hard book to read :-) as I miss a lot of background specially in operator algebra and ...

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### Is Hochschild homology invariant under A-infinity quasi isomorphism?

If A and B are two A-infinity algebra, A is A-infinity quai-isomorphic to B. Do we have HH(A)=HH(B)?

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### Question on a paper by U. Krähmer (“Dirac operators on quantum flag manifolds”)

I don't know if this is an adequate question for MO. But I cannot understand many aspects of the said paper
https://link.springer.com/content/pdf/10.1023%2FB%3AMATH.0000027748.64886.23.pdf
by ...

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### Relative version of Hopf cyclic cohomology

In this paper Connes and Moscovici introduced the Hopf algebra of transverse differential operators in order to compute the index formula for the diffeomorphism invariant geometry. They developed the ...

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### range of trace on projections: beyond rotation algebras

In a rotation algebra, $A_\theta=C(S^1)\rtimes \mathbb{Z}$, there is a tracial state $\tau$ coming from the invariant measure $\mu$ on the circle.
There is a projection $p\in M_n( A_\theta)$ (we can ...

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### A quantity associated with a smooth groupoid

Assume that $(G,G^0,r,s)$ is a smooth groupoid such that $G$ is a compact connected manifold.
The graph of "source" and "range" maps $s, r: G \to G^0$ are compact submanifolds $S$ and $R$ of $G\times ...

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### Pseudodifferential calculus for the Diffeomorphism Invariant Geometry

In the paper "Local Index Formula in Noncommutative Geometry" Connes and Moscovici build the spectral triple $(A,H,D)$ where $A=C^{\infty}_c(P) \rtimes \Gamma$ where $\Gamma$ is an arbitrary subgroup ...

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### Spectrum of the hypoelliptic transverse signature operator

Let $D$ be the transverse signature operator constructed by Connes and Moscovici in the paper "Local index formula in Noncommutative Geometry":this is first order hypoelliptic pseudodifferential ...

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### Baum Connes conjecture and abstract isomorphism

Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...

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### Regularilty of Commutative Spectral Triples

In Connes' approach to non-commutative geometry, the notion of a spectral triple is said to generalize compact Riemannian manifolds to the non-commutative setting. Motivating classical examples ...

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### Equivalence of rational Voevodsky motives: partial Converse to Conjecture of Orlov

There is a conjecture of Orlov stating that if $X$ and $Y$ are smooth projective complex varieties that are derived equivalent (equivalent bounded derived categories of coherent sheaves), then their ...

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### Dense subalgebra of continuous functions with same K -theory

Suppose $X$ is a compact metric space. Is there a good candidate for a dense subalgebra $A\subseteq C(X)$, such that the inclusion induces an isomorphism in $K$-theory?
For example, if $X$ was a ...

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### Normalization of cyclic cocycles

This question is a continuation of the discussion
Normalization of Hochschild cocycles
but this time in the cyclic context. I would like to ask whether the following is true:
The inclusion of ...

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### Hodge theoretic mirror symmetry and DG-BV algebras

Consider two Calabi-Yau manifold $X$ and $\check{X}$ which are meant to be mirror partners. Motivated by "classical MS", In DGBV Algebras and Mirror Symmetry, the following enhancement is proposed:
...

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### Geometric or conceptual way to understand supersymmetry algebra

Is there any geometric or more direct conceptual way to understand a supersymmetry algebra, rather than starting from a Lagrangian including boson and fermion fields, deriving all the expressions ...

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### Definition of the $G$-equivariant index map

My question concerns a statement on page 12 of the following paper of Baum, Connes, and Higson:
http://www.mmas.univ-metz.fr/~gnc/bibliographie/BaumConnes/Baum-Connes-Higson.pdf
about the definition ...

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### direct images of states in $C*$ algebras

Take a unital cp map $f:B\to A$ between unital $C^*$ algebras. Given a state $\psi:B\to \mathbb{C}$ what conditions are necessary for there to exist a state $\phi:A\to \mathbb{C}$ so that $\phi\circ f=...

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### Foliations, von Neumann algebras and measurability

In the excellent book Noncommutative Geometry by Alain Connes much of the first chapter is devoted to foliations. At the end of the first chapter the author discusses index theory on measured ...

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### Could we extend the star product on a Poisson manifold from its ring of smooth functions to its de Rham complex?

Let $M$ be a smooth manifold with a Poisson bracket $\{-,-\}$. Kontsevich proved that there exists a deformation quantization of $M$, i.e. let $C^{\infty}(M)[[\hbar]]=C^{\infty}(M)\otimes_{\mathbb{R}}\...

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### Do almost commutative flat degenerations induce equality in K-theory? (Or: Is the characteristic variety actually a support of a class in $K$-theory?)

I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not ...

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### Hochschild coboundary on the space of alternative forms

Assume that $A$ is a complex algebra. By $C^{n}(A)$ we mean the space of all $n-$linear map $\phi:A^n \to \mathbb{C}$. An alternative $k-$ form is
an element $\phi \in C^{k}(A)$ ...

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### Perfect Complexes on Tangent Bundle

Suppose $X$ is a $k$-variety of dimension $d$, and suppose $TX$ is its tangent bundle. Consider the (triangulated, stable $\infty$-,...) categories of perfect complexes $\text{Perf}(X)$ and $\text{...