As is well known, a complex $m \\times m$ matrix $A$ is a
\ncommutator (i.e., there are matrices $B$ and $C$ of the same dimensions as
\n$A$ such that $A=[B,C]=BC-CB$) if and only if $A$ has zero trace. If
\n$\\|\\cdot\\|$ is the operator norm from $\\ell_2^m$ to itself
\nand $|\\cdot|$ any ideal norm on $m\\times m$ matrices then clearly for any
\n$A,B,C$ as above $|A|\\le 2\\|B\\||C|$.<\/p>\n
Does the converse hold? That is, if $A$ has zero trace are there $m\\times
\nm$ matrices $B$ and $C$ such that $A=[B,C]$ and $\\|B\\||C|\\le K|A|$ for some
\nabsolute constant $K$? If not, what is the behavior of the best $K$ as a
\nfunction of $m$?<\/p>\n
The talk will concentrate on two recent results on this problem. The first
\nis a couple of years old result of Johnson, Ozawa and myself which gives
\nsome partial answers to this problem for the most interesting case of
\n$|\\cdot|=\\|\\cdot\\|$. The second is a more recent result of Angel and myself
\nwhich solves the problem for $|\\cdot|=$ the Hilbert–Schmidt norm.<\/p>\n","protected":false},"excerpt":{"rendered":"
As is well known, a complex $m \\times m$ matrix $A$ is a commutator (i.e., there are matrices $B$ and $C$ of the same dimensions as $A$ such that $A=[B,C]=BC-CB$) if and only if $A$ has zero trace. If $\\|\\cdot\\|$…
Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-458","post","type-post","status-publish","format-standard","hentry","category-colloquium"],"_links":{"self":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/458","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=458"}],"version-history":[{"count":1,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/458\/revisions"}],"predecessor-version":[{"id":459,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/458\/revisions\/459"}],"wp:attachment":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=458"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=458"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=458"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}