Spectrum of an algebra (Alain Connes paper)

[Heidi]

Hi Pfs,
If A is a c* algebra the members of A are operators on a Hilbert space and have eignevalues. We have a notion of spectrum on an operator. But here Alain Connes is talking about the spectrum
of an algebra , not of ifs members.
M and B are algebras what are spec(M), and spec(N)
He also talks about of the relative spectrum of M seen from N spec_N (M) ....
Have you other links about that?
thanks

https://arxiv.org/pdf/0810.2091.pdf

 

[fresh_42]

Hi Pfs,
If A is a c* algebra the members of A are operators on a Hilbert space and have eignevalues. We have a notion of spectrum on an operator. But here Alain Connes is talking about the spectrum
of an algebra , not of ifs members.
M and B are algebras what are spec(M), and spec(N)
He also talks about of the relative spectrum of M seen from N spec_N (M) ....
Have you other links about that?
thanks

https://arxiv.org/pdf/0810.2091.pdf

It is very likely the spectrum of the ring $M$. Every algebra is especially a ring and has therefore a spectrum:
https://en.wikipedia.org/wiki/Spectrum_of_a_ring

They play a central role in algebraic geometry. Do not forget to change to other language versions of the Wikipedia link, depending on which you speak. E.g. I found the German version easier to understand. Those language versions on Wikipedia differ sometimes a lot.

 

[martinbn]

https://en.wikipedia.org/wiki/Spectrum_of_a_C*-algebra

 

[Heidi]

We have all seen emission and absorption spectrum in physics.
Mathematically it is a map R -> {0,1}
I wonder why we have the same word with * algebras.
Is there a special case where we find a close analogy with the physical spectrum?

 

[fresh_42]

We have all seen emission and absorption spectrum in physics.
Mathematically it is a map R -> {0,1}
I wonder why we have the same word with * algebras.
Is there a special case where we find a close analogy with the physical spectrum?

Well, in a sense, both designations are related. The physical spectra are eigenvalues as are mathematical spectra. I could not find who coined the term, but Lagrange (1774) found certain simple roots of an equation while studying secondary inequalities of planets, and Laplace (1784), too.

In these problems, the imaginary parts of the exponents appear as the frequencies of the phenomena being studied.

You can find the papers in J.L. Lagrange, Oeuvres, 14, Gauthier-Villars, Paris, 1867-1892, and P.S. Laplace, Oeuvres, 14, Gauthier-Villars, Paris, 1878-1912. These are the citations in my (copy of Dieudonné's) book, History of Mathematics, 1700-1900.

 

[Heidi]

I found here in the decond answer
https://math.stackexchange.com/questions/2953190/how-to-understand-the-spectrum-of-a-c-algebra
what was called a nice result:
For any a in A and all phi in the dual of A then phi(a) is in the spectrum of a
How can it be derived?

 

[fresh_42]

I found here in the decond answer
https://math.stackexchange.com/questions/2953190/how-to-understand-the-spectrum-of-a-c-algebra
what was called a nice result:
For any a in A and all phi in the dual of A then phi(a) is in the spectrum of a
How can it be derived?

It makes no sense to discuss what has been said elsewhere. I even searched the SE page for "in the spectrum" to figure out what you were referring to. Without any hit! So not only that you ask about something someone else wrote somewhere else, but you also did not quote it correctly.

As mentioned on SE: "Wikipedia has a relatively good exposition of these topics." And @martinbn already gave you that link.

 

[martinbn]

For any a in A and all phi in the dual of A then phi(a) is in the spectrum of a
How can it be derived?

The spectrum of and element $a\in A$ is all $\lambda \in \mathbb C$ such that $a-\lambda$ is not invertible. So for any character $\varphi$ the value $\varphi(a)$ has to be in the spectrum, for otherwise the element $a-\varphi(a)$ will be invertable i.e. $(a-\varphi(a))b=1$ for some $b$. Then apply $\varphi$ to it and you get $(\varphi(a)-\varphi(a))\varphi(b)=1$, which is absurd. The other way around. Start with a $\lambda \in \mathbb C$ in the spectrum, and consider the ideal $\langle a-\lambda \rangle$, it is proper and $A/\langle a-\lambda \rangle \cong \mathbb C$. Then the homomorphism $A\rightarrow A/\langle a-\lambda \rangle \cong \mathbb C$ is a character, whos value at $a$ is exactly $\lambda$.

 

[martinbn]

ps The stackexchange Q/A is mostly about the commutative case, while what you asked in the beginning seems to be about non-commutative algebras.

 

[Heidi]

Thanks a lot for your answer #8.
you are right, the Connes paper is about non commutative geometry. I find it difficult. the origin of my interest was in a Kac question: can we hear the shape of a drum? there are different drums that are isospectral
It seems that Connes explains why the spectrum of the Dirac operator is not enough to see the shape of the drum.

 

[Heidi]

Could you help me about another thing:
In definition 2.5 i read
"a subset defined up to the gauge action of the unitary group U(N)"
Is N the algebra we were talking about and how does U(N) act?
thanks one more time.

 

[martinbn]

Could you help me about another thing:
In definition 2.5 i read
"a subset defined up to the gauge action of the unitary group U(N)"
Is N the algebra we were talking about and how does U(N) act?
thanks one more time.

I havn't read it, so i can only guess that this is the group of unitary operators of the algebra $N$.

If you are intersted in Kac question you don't need any of this. Just look for that instead. I am sure there are planty of lectures on the web. You can also look for spectral geometry.

 

[Heidi]

Milnor was the first who noticed that the frequencies of a drum are not enough to get its shape.
He gave two isospectral manifolds with different shapes.
So the proper values of the Laplacian does not give the complete information.
we have here two algebras M and N. N contains the spectral information,
and M contains the geometrical information. But M contains more than the missing information.
the C matrix in the article contains all the information and the aim is to eliminate
from it the information coming from N . So the missing information is in spec_N (M) the relative spectrum. Connes says that it is the information about the relative "positions" of M and N.

 

[Heidi]

There is still something that i do not understand in the wikipedia definition of the spectrum of
a C* algebra. the definition talks about sets of classes of representations, of ideals and so on.
I hoped that at the end we could get complex (or real) numbers. How to associate these numbers
to these representations, these ideals.....?

 

[martinbn]

There is still something that i do not understand in the wikipedia definition of the spectrum of
a C* algebra. the definition talks about sets of classes of representations, of ideals and so on.

I hoped that at the end we could get complex (or real) numbers.

Why?

How to associate these numbers
to these representations, these ideals.....?

Which numbers?

 

[Heidi]

Those we were talking about in #8

 

[martinbn]

Those we were talking about in #8

Which was about the spectrum of an element. Now, and at the begining, you were talking about the spectrum of a non-commutative algebra.

 

[Heidi]

You are right. the paper is about non commutative algebras but Alain Connes begins with
the commutative case. the wikipedia article is about the general case.
Is it possible, in the commutative case to associate homomorphisms from the algebra A to the set C of complex numbers instead of ideals, representations?

 

[martinbn]

You are right. the paper is about non commutative algebras but Alain Connes begins with
the commutative case. the wikipedia article is about the general case.

Is it possible, in the commutative case to associate homomorphisms from the algebra A to the set C of complex numbers instead of ideals, representations?

Yes.

 

[nicoo]

Here is a talk that addresses precisely your question.