topologies on U(H) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:36:42Z http://mathoverflow.net/feeds/question/65669 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65669/topologies-on-uh topologies on U(H) André Henriques 2011-05-21T19:14:02Z 2011-05-21T19:46:11Z <p>There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space:<br> the <i>weak</i>, <i>strong</i>, <i>ultraweak</i> (also called &sigma;-<i>weak</i>), <i>ultrastrong</i> (also called &sigma;-<i>strong</i>), and some more...</p> <p>Luckily, the weak and strong topologies agree when restricted to $U(H)\subset B(H)$.<br> Similarly, the ultraweak and ultrastrong topologies agree on $U(H)$.</p> <blockquote> <p>Is it true that the weak and ultraweak topologies agree when restricted to $U(H)$?</p> </blockquote> <p><hr> <b>Definitions:</b><br> A generalized sequence $a_i$ is <i>weak</i>ly, <i>strong</i>ly, <i>ultraweak</i>ly, <i>ultrastrong</i>ly convergent if:<br> &bull; $\langle a_i\xi,\eta\rangle\to\langle a\xi,\eta\rangle\qquad \forall \xi,\eta\in H$ <br>&bull; $a_i\xi\to a\xi\qquad \forall \xi\in H$ <br>&bull; $\langle (a_i\otimes 1)\xi,\eta\rangle\to\langle (a\otimes 1)\xi,\eta\rangle\qquad \forall \xi,\eta\in H\otimes \ell^2(\mathbb N)$ <br>&bull; $(a_i\otimes 1)\xi\to (a\otimes 1)\xi\qquad \forall \xi\in H\otimes \ell^2(\mathbb N)$, <br> respectively.<br> Here, $H\otimes \ell^2(\mathbb N)$ denotes the Hilbert space tensor product of $H$ and $\ell^2(\mathbb N)$.</p> http://mathoverflow.net/questions/65669/topologies-on-uh/65670#65670 Answer by Alain Valette for topologies on U(H) Alain Valette 2011-05-21T19:46:11Z 2011-05-21T19:46:11Z <p>The weak and ultraweak topologies coincide on bounded subsets of $B(H)$: see section 3.5 in Pedersen's book "C*-algebras and their automorphism groups".</p>