# 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.

141 questions
Filter by
Sorted by
Tagged with
21 views

76 views

### $f(t) = \min\{1,t\}$ 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 ...
61 views

### 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 ...
32 views

### 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 ...
54 views

### 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 ...
29 views

### 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 ...
25 views

### 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 ...
71 views

### 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 ...
70 views

### 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 ...
120 views

### What are some ideas on the fastest way to get from undergrad background in math to remotely reasonable understanding of noncommutative geometry.

What would be the fastest way to get from half of a class worth’s of (graduate) measure theory, same of functional analysis and generalized functions (or distributions), and only differential geometry ...
275 views

### Explicit 4-term commutator $[A,B,C,D]$

The commutator of two terms is: $$[A,B]=AB-BA$$ The commutator of three terms is: $$[A,B,C]=ABC+BCA+CAB-BAC-CBA-ACB$$ I was not able to google the four term commutator, not am I able to ...
61 views

### Why do we need noncommutative local rings? [closed]

Why are we need noncommutative local rings, is there any geometric meaning?
51 views

### Characterisation of $C_{c}(X)$ as a dense subalgebra of $C_{0}(X)$.
Consider a non-unital commutative $C^{*}$-algebra $A$ and a dense subalgebra $B\subset A$. We know that $A\cong C_{0}(X)$, for some locally compact Hausdorff space $X$. I would like to know if we are ...
I'm reading this paper, and definition 2.2.1 (page 5) is the following: The torsion of a left $A$-covariant derivative $\nabla$ on $\Omega^1A$ is the left $A$-module map \$T = \wedge \nabla - d:\...