User piotr nowak - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:21:35Z http://mathoverflow.net/feeds/user/2412 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/126788/definition-of-a-uniformly-bounded-dual-of-a-group Definition of a uniformly bounded dual of a group Piotr Nowak 2013-04-07T16:35:57Z 2013-04-11T09:36:16Z <p>The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite functions, or equivalently, diagonal matrix coefficients, associated to the representations).</p> <blockquote> <p><strong>Question:</strong> How to define a "uniformly bounded dual" of $G$ using uniformly bounded representations on a Hilbert space and what topology should it be equipped with?</p> </blockquote> <p>Such a uniformly bounded dual is mentioned a few times in the literature, but usually vaguely defined to be "analogous to the unitary dual". In particular, there is hardly any explicit statement about what topology is used. </p> <p>I am interested in this since I would like to understand a series of results of M. Cowling from the early 1980s on a stronger version of Kazhdan's property (T). Cowling proved that for a simple Lie group $\Gamma$, the trivial representation is isolated in such a uniformly bounded dual if and only if the rank of $\Gamma$ is $\ge 2$.</p> <p>In the case of $Sp(n,1)$ Cowling constructs an explicit family of uniformly bounded representations that approximate the trivial representation. The best description of his construction I have found is in his abstract from an Oberwolfach workshop from 2001 on the geometrization of property (T) (see <a href="http://www.mfo.de/document/0128a/Report_29_01.ps" rel="nofollow">MFO report</a>, the following abstract is on page 3):</p> <p><img src="http://s20.postimg.org/z8aq62gm5/spn1.jpg" alt="alt text"></p> http://mathoverflow.net/questions/118616/cocycles-for-right-and-left-regular-representations-on-ell-2g Cocycles for right- and left- regular representations on $\ell_2(G)$ Piotr Nowak 2013-01-11T11:18:37Z 2013-01-12T10:24:56Z <p>Using the standard notation, let $\lambda_gf(x)=f(g^{-1}x)$ be the left regular representation and $\rho_gf(x)=f(xg)$ be the right regular one, acting on the space $V$ of complex-values functions on $G$.</p> <p>Let $b:G\to \ell_2(G)$ be a cocycle for $\rho$; i.e., $b_g=\rho_g f-f\in \ell_2(G)$ for some $f\in V$ and every $g\in G$.</p> <p>Question: Does $a_g=\lambda_g f-f$ belong to $\ell_2(G)$ for every $g\in G$? If not, what is the "smallest" subspace $W$ of $V$, $\ell_2(G)\subseteq W\subseteq V$ for which this happens?</p> <p>The most interesting case seems to be when $G$ is amenable.</p> <p>Edit: The metric interpretation is the following. Take all $f\in V$ whose discrete gradient with respect to the left-invariant metric is $\ell_2$-summable. Then $W$ is the space consisting of gradients of these $f$, with respect to the right-invariant metric. </p> http://mathoverflow.net/questions/43066/injectivity-for-bimodules-and-hochschild-cohomology Injectivity for bimodules and Hochschild cohomology Piotr Nowak 2010-10-21T16:13:46Z 2010-10-25T01:25:19Z <p>Let $A$ be a Banach algebra and let $X$ be an $A$-bimodule. Is there a notion of (relative) injectivity for $X$ which would imply that $\mathcal{H}^n(A,X)$ vanishes for all $n\ge 1$? Here $\mathcal{H}^n(A,X)$ denotes the continuous Hochschild cohomology of $A$ with coefficients in $X$ ($X$ can be assumed to be dual $A$-bimodule).</p> <p>I expected there is such a notion, but after reading books by Helemskii, Runde and a few other sources on cohomology of Banach algebras I can't seem to find a general statement of this type, even though versions of projectivity and injectivity are discussed there.</p> http://mathoverflow.net/questions/30709/when-can-closedness-of-the-range-of-an-operator-be-checked-on-a-positive-cone When can closedness of the range of an operator be checked on a positive cone? Piotr Nowak 2010-07-06T02:32:05Z 2010-07-06T22:15:25Z <p>Let $T:X\to Y$ be an operator between Banach spaces $X$ and $Y$. Assume that $X$ has a positive cone $C\subset X$, which generates $X$: every element of $X$ can be written as a difference of elements of the positive cone. </p> <p>Assume now that we can show $$k\Vert x\Vert\le \Vert Tx\Vert$$ for some $k>0$ and every $x\in C$. </p> <p>Question: Are there any natural, general conditions on $T$, $X$ or both that allow to conclude that the image of $X$ under $T$ is closed in $Y$ (the above inequality holds for all $x\in X$, possibly with a different $k$)? </p> <p>The motivating example is the following. Let $G=(V,E)$ be an infinite (oriented) graph and take $X=\ell_1(V)$, $Y=\ell_1(E)$. Let $T$ be the discrete gradient, $Tf(x,y)=f(y)-f(x)$. Then it is enough to check the inequality only for positive $f$, then use the triangle inequality.</p> <p>Update: Given Andreas' response I realized I should probably ask not for general conditions but any sufficient condition that would give the above property, in particular in special cases like the motivating example above..</p> http://mathoverflow.net/questions/27609/an-example-of-a-non-amenable-exact-group-without-free-subgroups/29863#29863 Answer by Piotr Nowak for An example of a non-amenable exact group without free subgroups. Piotr Nowak 2010-06-29T02:39:38Z 2010-06-29T02:39:38Z <p>Owen, I'm a bit late to the party, but I think the answer to your question is no", to the best of my knowledge. To phrase it properly, I believe it is not known whether any of the known counterexamples to von Neumann's conjecture is exact. </p> <p>Jon, one has to be careful with limits hyperbolic groups, for example Gromov's random groups which are not exact are such limits (they are lacunary hyperbolic, in the sense of Olshanskii, Osin and Sapir). </p> http://mathoverflow.net/questions/127610/from-positive-definite-function-to-folner-sequence-a-question-on-amenabilit Comment by Piotr Nowak Piotr Nowak 2013-04-15T12:08:54Z 2013-04-15T12:08:54Z You will find these proved in appendices G and C in the recent monograph by Bekka, de la Harpe and Valette (see <a href="http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf" rel="nofollow">perso.univ-rennes1.fr/bachir.bekka/&hellip;</a>). http://mathoverflow.net/questions/126788/definition-of-a-uniformly-bounded-dual-of-a-group Comment by Piotr Nowak Piotr Nowak 2013-04-11T09:08:54Z 2013-04-11T09:08:54Z @Mikael: I agree that Lafforgue's strong (T) generalizes this in some sense, but he is considering representations which are not necessarily uniformly bounded, but satisfy some growth conditions. Thus the fact you mention for hyperbolic groups should be, at least philosophically, weaker than what Cowling proved for $Sp(n,1)$ (provided the two properties are directly comparable). http://mathoverflow.net/questions/126788/definition-of-a-uniformly-bounded-dual-of-a-group Comment by Piotr Nowak Piotr Nowak 2013-04-11T05:32:14Z 2013-04-11T05:32:14Z @Mikael: in Cowling's result for $Sp(n,1)$, there is a family of representations $\{\pi_i\}$, such that $\pi_i\to 1$ and for each $i$ we have $\Vert \pi_i\Vert=\sup_{g\in G}\Vert \pi_i(g)\Vert&lt;\infty$, but apparently $\Vert \pi_i\Vert\to \infty$ as $\pi_i\to 1$. This last fact is mentioned in a paper of Julg on the Baum-Connes conjecture in the Comptes Rendus. http://mathoverflow.net/questions/126788/definition-of-a-uniformly-bounded-dual-of-a-group Comment by Piotr Nowak Piotr Nowak 2013-04-10T08:15:47Z 2013-04-10T08:15:47Z Thanks for the suggestion. I have tried the obvious alternate routes already, moreover, it seemed that MO is the natural place to ask such a question. Several other people also reference or use Cowling's results in their work. For instance, Julg has an approach to the Baum-Connes conjecture with coefficients for $Sp(n,1)$ and mentioned the &quot;u.b. dual&quot; explicitly; Shalom's unpublished proof that $Sp(n,1)$ admits a proper affine action with uniformly bounded linear part is also based on Cowling's result, as far as I know. However, the meaning of the term &quot;isolated&quot; is never defined precisely. http://mathoverflow.net/questions/126788/definition-of-a-uniformly-bounded-dual-of-a-group Comment by Piotr Nowak Piotr Nowak 2013-04-10T07:29:30Z 2013-04-10T07:29:30Z Understanding how nets converge to the trivial representation is pretty much what I am looking for. More specifically, I would like to know the meaning of the term &quot;isolated&quot; in the context of Cowling's results. One example of his results is that for $Sp(n,1)$ the trivial representation <i>is not</i> isolated within the uniformly bounded representations. Another is that for simple Lie groups of rank at least 2, the trivial representation <i>is</i> isolated within the uniformly bounded ones. Also, of course I meant irreducible in the definition of the unitary dual, I'll edit the question. http://mathoverflow.net/questions/126788/definition-of-a-uniformly-bounded-dual-of-a-group Comment by Piotr Nowak Piotr Nowak 2013-04-08T04:07:48Z 2013-04-08T04:07:48Z Yemon, I am only considering the Hilbert space. http://mathoverflow.net/questions/126788/definition-of-a-uniformly-bounded-dual-of-a-group Comment by Piotr Nowak Piotr Nowak 2013-04-07T18:26:03Z 2013-04-07T18:26:03Z Yves, no behavior of norms of the representations is assumed. In fact Cowling has proved that for $Sp(n,1)$ the trivial representation can be approximated by uniformly bounded ones, even though $Sp(n,1)$ has (T). However, the norms of these uniformly bounded representations blow up to infinity as one approaches the trivial one. http://mathoverflow.net/questions/118616/cocycles-for-right-and-left-regular-representations-on-ell-2g/118642#118642 Comment by Piotr Nowak Piotr Nowak 2013-01-11T21:59:02Z 2013-01-11T21:59:02Z By the way, the metric version is the following: if the gradient of $f$ is in $\ell_2(E)$, where $E$-edges of the Cayley graph, when $G$ has the right-invariant word length metric, then $W$ would denote the space of gradients of all such $f$ with respect to the left-invariant metric. http://mathoverflow.net/questions/118616/cocycles-for-right-and-left-regular-representations-on-ell-2g/118642#118642 Comment by Piotr Nowak Piotr Nowak 2013-01-11T20:50:50Z 2013-01-11T20:50:50Z Misha, I think $W$ can be much larger than $\ell_{\infty}(G)$ and in general $f$ does not have to be bounded, it just has to have $\ell_2$-summable gradient with respect to the left-invariant metric. More explicitly, consider a function on $\mathbb{Z}$, where $f(n)=f(n-1)+\frac{1}{n}$ for $n$ positive, and $0$ elsewhere. http://mathoverflow.net/questions/118616/cocycles-for-right-and-left-regular-representations-on-ell-2g/118642#118642 Comment by Piotr Nowak Piotr Nowak 2013-01-11T18:14:40Z 2013-01-11T18:14:40Z Agreed. Any candidates for $W$? http://mathoverflow.net/questions/118616/cocycles-for-right-and-left-regular-representations-on-ell-2g Comment by Piotr Nowak Piotr Nowak 2013-01-11T13:53:22Z 2013-01-11T13:53:22Z Misha, I want this to happen for all functions $f$ on $G$, for which $\rho_g f-f$ is a cocycle into $\ell_2(G)$. On $\mathbb{Z}$ the key is commutativity and that $\rho_{g}=\lambda_{g^{-1}}$. http://mathoverflow.net/questions/43964/folner-sequences-of-amenable-groups-of-exponential-growth/43972#43972 Comment by Piotr Nowak Piotr Nowak 2010-10-28T17:51:57Z 2010-10-28T17:51:57Z Indeed, the proof of the &quot;only if&quot; direction in lemma 2.2 is not clear. This is surprising, as far as I know it is considered to be an established fact. http://mathoverflow.net/questions/43964/folner-sequences-of-amenable-groups-of-exponential-growth/43972#43972 Comment by Piotr Nowak Piotr Nowak 2010-10-28T14:16:05Z 2010-10-28T14:16:05Z You're right, although it looks from the proof of lemma 2.2 in Pittet's paper that he actually needs a weaker statement, with &quot;=&quot; replaced by &quot;&gt;&quot; in lemma 2.1. http://mathoverflow.net/questions/43066/injectivity-for-bimodules-and-hochschild-cohomology/43457#43457 Comment by Piotr Nowak Piotr Nowak 2010-10-25T02:06:38Z 2010-10-25T02:06:38Z This is great, thanks for such a complete answer! http://mathoverflow.net/questions/43066/injectivity-for-bimodules-and-hochschild-cohomology Comment by Piotr Nowak Piotr Nowak 2010-10-23T14:24:02Z 2010-10-23T14:24:02Z @Yemon: you should definitely post it as an answer.