Arnold Neumaier
Apr15-04, 11:32 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>John Baez wrote:\n\n> Maybe I can summarize and clarify a bit. Any von Neumann algebra\n> has a nice topology on it called the weak-* topology. A state on\n> a von Neumann algebra is "normal" if it\'s continuous with respect\n> to this topology.\n\n> Now for two facts:\n>\n> Theorem: If A is a von Neumann algebra contained in the\n> algebra B(H) of all bounded operators on a Hilbert space H,\n> a state on A is normal if and only if it comes from a density\n> matrix on H.\n\nI suppose this is Bratteli/Robinson\'s Theorem 2.4.21.\n\n\n> This theorem should make us feel happy, since it means that to\n> some extent we can ignore states that don\'t come from density\n> matrices in this way.\n>\n> Theorem: if A is a type III von Neumann algebra, A has no\n> normal pure states.\n\nWhere is this proved?\n\n> This theorem should make us feel shocked, since it means that\n> pure states on this sort of von Neumann algebra are "unphysical" -\n> if you believe that only normal states are "physical", and you\n> believe that type III von Neumann algebras are "physical" (which\n> is actually debatable).\n\n\nArnold Neumaier\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>John Baez wrote:
> Maybe I can summarize and clarify a bit. Any von Neumann algebra
> has a nice topology on it called the weak-* topology. A state on
> a von Neumann algebra is "normal" if it's continuous with respect
> to this topology.
> Now for two facts:
>
> Theorem: If A is a von Neumann algebra contained in the
> algebra B(H) of all bounded operators on a Hilbert space H,
> a state on A is normal if and only if it comes from a density
> matrix on H.
I suppose this is Bratteli/Robinson's Theorem 2.4.21.
> This theorem should make us feel happy, since it means that to
> some extent we can ignore states that don't come from density
> matrices in this way.
>
> Theorem: if A is a type III von Neumann algebra, A has no
> normal pure states.
Where is this proved?
> This theorem should make us feel shocked, since it means that
> pure states on this sort of von Neumann algebra are "unphysical" -
> if you believe that only normal states are "physical", and you
> believe that type III von Neumann algebras are "physical" (which
> is actually debatable).
Arnold Neumaier
> Maybe I can summarize and clarify a bit. Any von Neumann algebra
> has a nice topology on it called the weak-* topology. A state on
> a von Neumann algebra is "normal" if it's continuous with respect
> to this topology.
> Now for two facts:
>
> Theorem: If A is a von Neumann algebra contained in the
> algebra B(H) of all bounded operators on a Hilbert space H,
> a state on A is normal if and only if it comes from a density
> matrix on H.
I suppose this is Bratteli/Robinson's Theorem 2.4.21.
> This theorem should make us feel happy, since it means that to
> some extent we can ignore states that don't come from density
> matrices in this way.
>
> Theorem: if A is a type III von Neumann algebra, A has no
> normal pure states.
Where is this proved?
> This theorem should make us feel shocked, since it means that
> pure states on this sort of von Neumann algebra are "unphysical" -
> if you believe that only normal states are "physical", and you
> believe that type III von Neumann algebras are "physical" (which
> is actually debatable).
Arnold Neumaier