User sergio a. yuhjtman - MathOverflow

Answer by Sergio A. Yuhjtman for Nonseparable disintegration theory: references

2013-01-04T17:00:16Z

In case this recent article arxiv.org/abs/1212.6192 is correct, it is surely relevant.

Proper morphisms of C*-algebras / Nondegenerate representations

2012-11-20T18:54:17Z

Let $A \to B$ be a proper morphism of $C^*$-algebras. A nondegenerate representation of $B$ induces a nondegenerate representation of $A$. Does the converse hold?

I.e.: let $A \to B$ be a morphism of $C^*$-algebras such that every nondegenerate representation of $B$ induces a nondegenerate representation of $A$. Does the morphism result proper? I guess not.

Answer by Sergio A. Yuhjtman for Sheafification via hypercovers

2012-04-30T19:40:36Z

One way of constructing the associated sheaf in one step is written here: http://cms.dm.uba.ar/academico/carreras/licenciatura/tesis/yuhjtman.pdf (in spanish) page 19, (3.2). The key idea (due to Eduardo Dubuc) is to consider "locally compatible families" instead of just "compatible families".

Weak topology restricted to the unitary group of a von Neumann algebra

2012-04-25T19:56:47Z

Consider the weak topology in a von Neumann algebra (weak in the sense of Banach spaces).

Does this topology coincide with the rest of the "weak" topologies when restricted to the unitary group?

Mathematical foundations of Quantum Field Theory

2012-04-14T04:39:42Z

Is there any reasonable approach, essentially different from Wightman's axioms and Algebraic Quantum Field Theory, aimed at obtaining rigorous models for realistic Quantum Field Theories? (such as Quantum Electrodynamics).

EDIT: the reason for asking "essentially different" is the following. It is possible to intuitively think "states" as solutions of the equations of motion (in some sense, in a "multiparticle space"). In realistic interacting QFT, the equations of motion are nonlinear. So, according to my chosen intuitive concept, a reasonable state space should be nonlinear (something like a Hilbert manifold). Meanwhile, in Wightman or AQFT frameworks, state spaces are Hilbert spaces. This seems to correspond with the fact that it is very very difficult to construct interacting QFT's in these frameworks. So, as a desire... there should be a different, more interaction-friendly framework where realistic models arise in a more natural way.

Does something in this direction already exist?

Where is the error in this argument?

2012-04-12T15:36:01Z

Let $G$ be a locally compact Hausdorff group. It is known that $G$ can be topologically embedded in $W^{\ast}(G)$ , its universal $W^{\ast}$-algebra (with the $\sigma$-weak topology). An element $T \in W^{\ast}(G)$ is a function assigning to each representation $\pi$ a bounded operator $T(\pi) \in B(H_{\pi})$. This $T$ must be compatible with interwiners and $T(\pi)$ must be uniformly bounded.

This was done (in a slightly different language) by J. Ernest here: http://www.jstor.org/stable/2373020

Now define $G_{\otimes}= \{ T \in W^*(G)_{\neq 0} / T(\pi_1 \otimes \pi_2) = T(\pi_1) \otimes T(\pi_2) \}$

It's not hard to see that elements in $G_{\otimes}$ are unitaries. This is briefly proven here: http://cms.dm.uba.ar/Members/sergioyuhjtman/WG2.pdf/download (proposition 4.2).

Now Tatsuuma's duality theorem applies (Tatsuuma, proposition 2) so $G=G_\otimes$. But $G_\otimes$ is closed and inside the unit ball, so it is compact (always $\sigma$-weak topolgy). Therefore $G$ is compact.

Are the categories of representations of G and C*(G) isomorphic?

2012-04-10T16:39:39Z

Let G be a locally compact Hausdorff group, and C*(G) the full group C* algebra.

I found in some books that "representation theory of both is the same". Can this be expressed as "the categories are equal"?

(Here "representation of G" means a unitary weakly continuous representation on a Hilbert space, and representations of C*(G) are nondegenerate. Morphisms in those categories are the interwiner bounded operators).

Comment by Sergio A. Yuhjtman

2012-11-29T18:06:09Z

Thanks a lot! .

Comment by Sergio A. Yuhjtman

2012-11-21T18:16:28Z

It is a morphism that maps an approximate unit to an approximate unit (equivalently: it maps every approximate unit to an approximate unit).

Comment by Sergio A. Yuhjtman

2012-11-05T23:41:54Z

Does anyone have a reference for this equivalence? Thanks...

Comment by Sergio A. Yuhjtman

2012-04-30T21:10:03Z

This proof has no surprising ideas other than the definition of "locally compatible families". From there it is like the usual ++ construction. I'm not familiar with the hypercovers approach.

Comment by Sergio A. Yuhjtman

2012-04-30T19:29:49Z

It is possible to construct the associated sheaf functor in only one step. This is due to Eduardo Dubuc. The key idea is to consider "locally compatible families" instead of "compatible families". You can read the details here: cms.dm.uba.ar/academico/carreras/licenciatura/… (spanish, sorry) Page 19, (3.2).

Comment by Sergio A. Yuhjtman

2012-04-18T16:54:36Z

Interesting. Where can I find a good rigorous treatment of the anharmonic oscillator? (as simple as possible, of course)

Comment by Sergio A. Yuhjtman

2012-04-17T05:41:48Z

For the hydrogen atom, double-well, etc, (i.e: one non-relativistic quantum particle) the interaction seem to be of a different nature than in QFT, because it is external. The object interacting with the particle is outside the model! Could this be the origin of difficulties in order to obtain rigorous models? Why the mathematical apparatus should be necessarily essentially the same?

Comment by Sergio A. Yuhjtman

2012-04-17T05:29:56Z

OK. So the equations of motion in QFT apply to the fields and not the states.

Comment by Sergio A. Yuhjtman

2012-04-15T16:23:29Z

According to what I know, Schroedinger and Heisenberg picture are trivially equivalent. However, you say that the hydrogen atom and double-well have "non-linear (Heisenberg) equations of motion that live perfectly...". I would like to know about those nonlinear equations. Any reference?

Comment by Sergio A. Yuhjtman

2012-04-15T16:16:35Z

Schroedinger equation for the Hydrogen atom and double well is a linear PDE. So I still don't see a valid objection. "In short, the standard approaches to constructive QFT already incorporate non-linear interactions in a natural way." I disagree.

Comment by Sergio A. Yuhjtman

2012-04-14T04:57:21Z

I didn't find it between the 52 tagged as qft.

Comment by Sergio A. Yuhjtman

2012-04-14T01:28:58Z

OK, I'll accept the answer to be polite and grateful (actually I want to thank everyone who got involved!) but I still think as before.

Comment by Sergio A. Yuhjtman

2012-04-13T15:33:02Z

This is a good reference, thank you! But the real answer is what Jesse Peterson said.

Comment by Sergio A. Yuhjtman

2012-04-13T03:28:00Z

Thank you! That sounds right. I will check it.

Comment by Sergio A. Yuhjtman

2012-04-12T19:44:54Z

I think this is wrong: "it suffices to describe an element $T$ of $G_\otimes$ on irreducibles" because $T$ might not commute with direct integrals. However, in case Tatsuuma is wrong, it is still possible to reach a contradiction. Thit is explained here: cms.dm.uba.ar/Members/sergioyuhjtman/WG2.pdf/…