User rsg - MathOverflow

Extensions of Carathéodory's theorem

2013-04-14T13:28:51Z

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex objects of any real manifold?

I believe the answers are negative. However I want to know whether there is a similar result like Carathéodory (i.e. an upper bound with respect to dimension of the space/manifold). I am not sure whether it should be asked here. I have already asked it in math.stackexchange without getting any reply. The only comment was regarding the Choquet Theory, but I am yet to get anything related to my question. I am new to this subject, and possibly overlooked the required section. Advanced thanks for any help/suggestion/reference which can be (relatively) easily understood. Feel free to ask (and also edit) if you want more clarification.

Arveson index and curvature

2013-03-26T05:24:11Z

Can someone explain me what is the intuitive idea behind Arveson Index and curvature of $E_0$ semigroups. I was reading the standard paper of Arveson, but is lost and yet to get intuition about it. An index is generally invariant under certain operations. Waht are the actions for which this index (and curvature) are invariant? Advanced thanks for any help suggestion.

Series of linear maps: on a paper by Evans and Hanche-Olsen

2012-11-19T15:12:40Z

I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.

Let $L$ be a bounded self-adjoint linear map on a unital $C^*$ algebra $\mathcal{A}$. The following conditions are equivalent

(iii) If $y\in\mathcal{A}_+$ (set of positive semi-definite objects), $a\in\mathcal{A}$ satisfy $ya=0$, then $a^*L(y)a\geq0$.

(iv) For some full invariant set of states $S$ on $\mathcal{A}$, that $y\in\mathcal{A}_+$, $f\in S$ with $f(y)=0$ imply $fL(y)\geq0$.

My question is on condition (iii). If $L$ is a positive map, the condition holds (by definition). However if it is not, what extra condition(s) a general linear bounded map $L$ must satisfy such that condition (iii) holds.

If we replace $\mathcal{A}$ by $\mathcal{B(H)}$ for some Hilbert space $\mathcal{H}$ (finite or infinite dimensions) can we find such conditions. If some works are already in literature, please refer. Advanced thanks for any help, suggestions etc.

Duality between extremal points and extremal maps

2012-09-10T06:36:24Z

Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set \begin{equation}\mathfrak{L}=\lbrace T:\mathbb{R}^n\rightarrow \mathbb{R}^n, \text{ $T$ is linear}, T(C)\subseteq C\rbrace\end{equation} Is there any relation between extremal points (as well as exposed points) of $C$ and extremal points (exposed points) of $\mathfrak{L}$?

I do not work in convex geometry, and so do not know whether the statement is making sense. I have asked this question here, but did not get any response (and so I decided to ask it here). Please give suggestions and feel free to correct (and edit as well), if I am wrong and/or there is some ambiguity. Advanced thanks for any help.

Fixed marginals of joint distribution: status

2012-07-28T17:58:05Z

One of the well-known problems of classical probability theory is the determination of the set of all extreme points in the convex set of all probability distributions in a product Borel space $\left (X \times Y, \:\:{\cal F} \times {\cal G} \right )$ with fixed marginal distributions $\mu$ and $\nu$ on $\left (X, {\cal F} \right )$ and $\left (Y, {\cal G} \right )$ respectively. Denote this convex set by $C(\mu, \nu).$

When $X=Y=${$1,2,\ldots,n$} and ${\cal F} = {\cal G}$ is the field of all subsets of $X$ and $\mu = \nu$ is the uniform distribution then the problem is answered by the Birkhoff and independent by von Neumann which deals with doubly stochastic matrices.

I do not work in probability theory. I have heard that the above result is the 'only' result in this direction. Can someone please tell me the present status of the above problem. It is nice, if anybody can point out survey article(s) on this matter also.

I have asked this question already math.stackexchange; but did not get any reply. Hence I decided to ask it here. I My motivation is to look at the analogues problem in a quantum set up. In particular see paper and its followup(s). Birkhoff's theorem and methods are used in some recent works regarding quantum channels as well. Hence I want to know the present status of the classical problem. Advanced thanks for all the helps.

Answer by RSG for Positive but not completely positive?

2012-01-28T05:51:10Z

Continuation of the answer of Benjamin Steinberg; this paper gives a list of such known maps.

On the structure of entanglement witnesses and new class of positive indecomposable maps: by Dariusz Chruscinski and Andrzej Kossakowski; http://xxx.imsc.res.in/abs/quant-ph/0606211v1 and the journal version (if you want) is here.

It is slightly outdated but gives a good starting point. Not many classes of such maps are known in literature.

Comment by RSG

2013-01-07T11:19:47Z

+1 for the last four, in particular for Kato and Arnold

Comment by RSG

2012-11-23T03:27:15Z

@Nik Weaver and @Ollie Margetts The conditions are all equivalent. If $L$ were a conditionally completely positive, from condition (i) $e^{tL}$ becomes completely positive. I wanted to relax the conditionally completely positive criteria, such that (iii) still holds, and I can get, perhaps not a completely positive but a general positive map.

Comment by RSG

2012-11-19T14:35:40Z

Not areal reply, however you may find some hints in the following book "Positive definite matrices" by Rajendra Bhatia (MR2284176). You may find some hints there (I do not work on that topic,and do not have a copy with me. However, I remember that there is a chapter devoted to matrix means).

Comment by RSG

2012-09-13T16:37:30Z

@Denis Serre Yes. Example $\lbrace (x,y,z)\in\mathbb{R}^3: ~x^2+y^2\leq z^2\rbrace$