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.
Questions tagged [noncommutative-geometry]
199 questions
51
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A fleshed-out version of the Noncommutative Geometry proof of the Gauss-Bonnet Theorem?
In Connes's book on noncommutative geometry, he outlines a rather short "algebraic" proof of the Gauss-Bonnet theorem that uses multilinear forms. (Start reading on page 19 of the book) This is given as motivation for cyclic cohomology.
Where can I…
Jon Bannon
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4 answers
Why study Hopf Algebras?
I'm looking for reasons that motivate the study of Hopf Algebra, like its applications in other branches of mathematics or maybe with physics. The first I've got is that they're interesting by themselves. But I'll want to motivate the undergaduate…
Math.mx
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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 geometrical/topological context and vice versa. I…
user829347
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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 other hand, the school of Alain Connes developped a…
Bruno Joyal
- 55,975
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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 the existence, up to isomorphism, of only one…
Ben
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15
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3 answers
Applications or uses of the Serre-Swan theorem
The Serre-Swan theorem states (at least in one form) that the category of real vector bundles over a compact Hausdorff space $M$ is equivalent to the category of finitely generated projective modules over the ring $C(M)$ of continuous functions on…
Zach Conn
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Noncommutative algebraic geometry in the case of skew fields
Noncommutative algebraic geometry is a developing field. Things have not yet got the final form as in commutative geometry.
But one might wonder whether things are any better in the case of skew-fields, ie division rings, ie possibly noncommutative…
user977
13
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1 answer
Does this notion of morphism of noncommutative rings appear in the ring theory literature?
Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on commutative subrings of $R$.
This definition is…
Qiaochu Yuan
- 468,795
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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 Google searches have given me, although I am…
nagger
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What prerequisites does noncommutative geometry have?
I'm a Masters student currently deciding which area to focus on. So far, my primary interest has been $C^*$-algebras and operator algebras (I already have some knowledge of $K$-theory for $C^*$-algebras and Hilbert modules), but I always had some…
Julio Cáceres
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What is the commutative analogue of a $C^*$-subalgebra?
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring…
Rasmus
- 18,946
10
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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.
$\to$ It's noncommutative geometry "à la Alain…
Sebastien Palcoux
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10
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Minimal spectrum of a commutative ring
Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I remember that this is true but can't seem to prove…
Justin Campbell
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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 counter example to the question $2$ of the following …
Ali Taghavi
- 925
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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, then you take the torus and look at its algebra of…
user46348
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