Strict positivity in dense subalgebras of $C^{*}$-algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T05:16:03Z http://mathoverflow.net/feeds/question/60317 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60317/strict-positivity-in-dense-subalgebras-of-c-algebras Strict positivity in dense subalgebras of $C^{*}$-algebras alterationx10 2011-04-01T20:15:25Z 2012-09-18T11:22:00Z <p>Let $A$ be a $C^{*}$-algebra, represented on a Hilbert space $H$, and $D$ a selfadjoint unbounded operator on $H$ (note that we do not impose that $D$ have compact resolvent). Let </p> <p>$\mathcal{A}:=${$a\in A : [D,a]\in B(H)$}</p> <p>and topologize by the spectral invariant norm $\|a\|_{1}:=\|a\|+\|[D,a] \|$. Let $\mathcal{I}$ be a two sided ideal in $\mathcal{A}$, and denote by $I$ its closure in $A$. Let $0\leq h\leq k\in\mathcal{A}$ be such that $h\mathcal{I}$ is dense in $\mathcal{I}$ (in its Banach algebra topology).</p> <p>Question: Is $k\mathcal{I}$ dense in $\mathcal{I}$?</p> <p>Note that the hypotheses imply that $h$ and hence $k$ are strictly positive $\mod I$ in $A$. Therefore both $hI$ and $kI$ are dense in $I$. Also, in case $h$ and $k$ commute, or when $h,k\in\mathcal{I}$, the answer to this question is yes.</p> http://mathoverflow.net/questions/60317/strict-positivity-in-dense-subalgebras-of-c-algebras/83032#83032 Answer by unknown (yahoo) for Strict positivity in dense subalgebras of $C^{*}$-algebras unknown (yahoo) 2011-12-09T06:04:26Z 2011-12-09T06:04:26Z <p>${\cal I}$ is dense in $I$, since $I$ is the closure of ${\cal I}$.</p> <p>Therefore $k{\cal I}$ is dense in $kI$, using continuity of multiplication.</p> <p>You note $kI$ is dense in $I$.</p> <p>Since density is transitive, $k{\cal I}$ is dense in $I$. Since ${\cal I} \subseteq I$, $k{\cal I}$ is also dense in ${\cal I}$. In the topology of $A$. Do you require density in the stronger topology of ${\cal A}$?</p>