User michael - MathOverflow

Non-Abelian Duistermaat-Heckman Measure (not just a reference request)

2011-09-02T21:23:45Z

Let $V$ be a representation of some torus $T$. It is then well-known that the Duistermaat-Heckman measure for $P(V)$ is the weak limit of the properly rescaled distribution of multiplicities of weights in $\mathrm{Sym}^n(V)$.

I've seen many allusions to the fact that the analogous statement is true for general compact Lie groups $K$ (i.e., one pushes further forward to a positive Weyl chamber and compares with the irrep distribution), e.g., in Allen Knutson's reply to my last question and in the appendix of Guillemin-Prato's 1990 paper, but could not find an explicit statement of this in the literature.

Do you know whether this statement is true at all, and do you maybe even have a reference?

Example in Guillemin-Sternberg's Convexity Paper

2011-07-18T12:30:11Z

At the end of the introduction of Guillemin-Sternberg's 1982 Inventiones paper "Convexity Properties of the Moment Mapping", there is a nice picture of a non-Abelian moment polytope attributed to Heckman. Is this example worked out somewhere in more detail? Also, are you aware of other illustrations of the intricacies of the non-Abelian convexity theorem?

Decomposability of positive maps

2011-02-16T09:10:33Z

By results of Størmer and Woronowicz, every positive map $\Phi \colon \mathcal{M}_{d \times d} \rightarrow \mathcal{M}_{d' \times d'}$ for $dd' \leq 6$ can be decomposed as a convex combination

$$\Phi = p \phi + (1-p) ~ T \circ \psi$$

where $\phi$, $\psi$ are completely positive maps and $T$ is the transposition map.

For higher dimensions, this is in general false. Does there however (for fixed $d$, $d'$) exist a finite set of positive maps $(P_i)$ such that every general positive map $\Phi$ is a convex combination

$$\Phi = \sum p_i P_i \circ \phi_i$$

where the $\phi_i$ are suitably chosen completely positive maps?

What morphisms / Morita equivalences induce the 2-periodicity isomorphisms of KK-theory?

2009-11-18T10:27:14Z

In Kasparov's paper, the canonical isomorphisms $KK_* \rightarrow KK_{*+2k}$ are defined rather implicitely (by tensoring and stabilization).

Are there morphisms of $C^*$-algebras which induce them (e.g. I've heard that the morphism $\varphi: \mathbb{C} \rightarrow \mathbb{C}_2$ sending 1 to $1 + i e_1 e_2$ induces the iso $KK_* \rightarrow KK_{*+2}$ ), and how to see that?

Similarly, a graded irreducible representation of $\mathbb{C}_{2k}$ gives a Morita equivalence between $\mathbb{C}$ and $\mathbb{C}_{2k}$ . Does it induce the periodicity isomorphism, and if yes, which of the two gradings should one use? (relationship to the above question: $\varphi$ is easily seen to be the $KK$-inverse of the standard graded irreducible representation $W$ of $\mathbb{C}_2$ where the complex volume element implements the grading, since then $\varphi(1)$ projects onto the even part of $W$)?

Answer by Michael for Cobordisms of bundles?

2010-01-17T14:37:57Z

See Daccach and Pergher, Splitting vector bundles up to cobordism, 1985.

Answer by Michael for Integration in equivariant K-theory

2009-12-18T22:51:41Z

I don't know about the homotopy-theoretic picture (and would like to learn more about it), but maybe the following also helps: If I understand you correctly, the integration maps of K-theory are what is often called the Gysin/shriek maps (the case you mention would be the Gysin map induced by the zero section of a trivial G-vector bundle). I think they only exist if the map is K-oriented; in that case they can be implemented by right multiplication with functorial KK-elements. The canonical reference for the non-equivariant case would be Connes-Skandalis and for the equivariant case Kasparov-Skandalis.

Answer by Michael for Spectra of $C^*$ algebras

2009-11-11T14:12:29Z

The Gelfand representation also works for non-unital commutative C^*-algebras. In this case, it establishes a category equivalence to the category of locally compact Hausdorff spaces with proper maps (implemented by C_0(.) and the spectrum). Hence Matthew's comment, the spectrum of C_0(X) is just X.

Answer by Michael for What is the conceptual significance of supercommutativity?

2009-11-07T00:05:11Z

This is not a satisfying answer to your question, but one observes that the exterior algebra and Clifford algebras have this kind of commutativity, so it certainly arises "naturally".

Answer by Michael for Atiyah-Singer index theorem

2009-10-19T11:21:48Z

Ålthough written from the K-homology point of view, the book "Analytic K-homology" by Higson and Roe should be quite useful (both for basics about elliptic differential operators and index theory; iirc they sketch the proof of the index theorem for Spin^c manifolds).

Answer by Michael for What m minimizes E(|m-X|^3) for a random variable X?

2009-10-18T17:46:47Z

E(|X-EX|^k) is called the k-th central (or centered) moment of the random variable X.

Comment by Michael

2012-01-15T09:37:05Z

For an easier proof, observe that the partial transpose $\mathbf 1 \otimes T$ of $a$ is not positive semidefinite for small $\varepsilon$ (while it is obviously positive semidefinite for every matrix that is decomposable in the above sense).

Comment by Michael

2011-09-04T22:01:22Z

Dear Allen, thank you very much for your reply.

Comment by Michael

2011-09-02T21:48:59Z

(The third term should be $w$ times the complex conjugate of $w$; there seems to be a MathJax rendering bug.)

Comment by Michael

2011-09-02T21:47:20Z

$||w||^2 = ||w^*w|| = ||\overline{w}w|| \leq ||I_n||$, unless I misunderstand the question.

Comment by Michael

2011-06-17T22:40:37Z

(I felt that $||f(A)|| = f(||A||)$ could be seen easily in the $L^2$-picture.)

Comment by Michael

2011-06-16T22:22:04Z

Check the identity for a multiplication operator on some $L^2$.

Comment by Michael

2011-03-21T16:12:05Z

The Clifford action on $S \otimes E$ is defined by tensoring $c$ with the identity on $E$. The Dirac operator is defined using the "tensor product connection" $\nabla_S \otimes 1 + 1 \otimes \nabla_E$ where $\nabla_E$ is a connection on the line bundle $E$ compatible with the metric.

Comment by Michael

2011-02-19T23:29:12Z

Check out math.yale.edu/~mh644/….

Comment by Michael

2011-02-16T17:40:36Z

Stefan, I have seen it described as an open problem in the book "Geometry of Quantum States" (and should have mentioned that in my question), but was wondering whether there were any more recent (partial) results (e.g. for the already interesting case $d = d' = 3$).

Comment by Michael

2011-02-16T17:37:19Z

Jon, the results can be found in the articles "Positive maps of low dimensional matrix algebras" (Woronowicz) and "Positive linear maps of operator algebras" (Størmer).

Comment by Michael

2011-02-16T10:00:19Z

Thank you for your suggestion. A linear map $\Phi$ of $C^*$-algebras is positive if sends the positive cone to the positive cone. It is $k$-positive if $\Phi$ tensored with the identity map on $k \times k$-matrices is positive. It is completely positive if is $k$-positive for all $k > 0$.

Comment by Michael

2011-01-12T20:22:22Z

Another distribution on nonnegative matrices which people use is the Wishart distribution. Its eigenvalues are then asymptotically distributed according to the Marčenko–Pastur distribution.

Comment by Michael

2010-05-10T23:55:57Z

For 3 -> 1, extend the induced map R^k --> End(V) using the universal property of Cl(R^k).

Comment by Michael

2010-02-17T10:49:57Z

The answer is that you can choose such a Morita quivalence in $KK(C_2,0)$ and then exterior Kasparov multiplication with this element implements 2-periodicity.

Comment by Michael

2010-01-24T14:06:31Z

Start with en.wikipedia.org/wiki/Master_equation