Ideals in smooth subalgebras of C*-algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:41:59Z http://mathoverflow.net/feeds/question/58751 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58751/ideals-in-smooth-subalgebras-of-c-algebras Ideals in smooth subalgebras of C*-algebras alterationx10 2011-03-17T13:42:31Z 2011-03-17T19:38:57Z <p>Let $B$ be a $C^{*}$-algebra and $\mathcal{B}$ a dense *-subalgebra stable under holomorphic functional calculus and $C^{1}$-functional calculus for selfadjoint elements. Also, $\mathcal{B}$ is a Banach algebra in a norm <code>$\|\cdot\|_{1},$</code> satisfying </p> <p><code>$\|\cdot\|\leq\|\cdot\|_{1}$</code>. </p> <p>Also, there is a countable bounded approximate unit <code>$u_{n}$</code> for <code>$\mathcal{B}$</code> which is a contractive, increasing approximate unit for $B$. Let $\mathcal{I}$ be a closed two sided ideal in $\mathcal{B}$, and denote by $I$ its closure in $B$. </p> <p>Is it true that $\mathcal{I}=I \cap \mathcal{B}$ ?</p> <p>The pertinent examples are Lipschitz functions on the circle and on the real line, both with norm <code>$\|f\|_{1}=\|f\|+\|\partial f\|$</code>.</p> http://mathoverflow.net/questions/58751/ideals-in-smooth-subalgebras-of-c-algebras/58778#58778 Answer by Yemon Choi for Ideals in smooth subalgebras of C*-algebras Yemon Choi 2011-03-17T19:38:57Z 2011-03-17T19:38:57Z <p>$\newcommand{\norm}[1]{\Vert#1\Vert}$</p> <p>In general, I think the answer to your question is no. Take ${\mathcal B}=C^1[-1,1]$ with the norm <code>$\norm{f}= \norm{f}_\infty+\norm{f'}_\infty$</code> and let ${\mathcal I}$ be the closed ideal consisting of those $C^1$-functions which vanish at $x=0$ and whose 1st derivative vanishes at $x=0$. Then $I\cap {\mathcal B}$ contains the function $f(x)=x$ which is evidently not in ${\mathcal I}$.</p> <hr> <p>[Some general remarks follow, in a rambling style owing to lack of sleep. I may try to edit these later.]</p> <p>In the commutative unital setting, taking $B=C(X)$, we know what the closed ideals of $B$ are (they are precisely the "kernels" of closed subsets of $X$, in the language of hulls and kernels).</p> <p>If your subalgebra ${\mathcal B}$ also has maximal ideal space (homeo to) $X$, then your question is related to -- perhaps is equivalent to, I have not thought in detail -- the following one:</p> <p><strong>Can I find a closed two sided ideal in ${\mathcal B}$ which is not the kernel of its hull?</strong></p> <p>Without your restrictions on stability-under-func-calc, this kind of question has been much studied for commutative examples, and I think also for certain noncommutative examples related to group algebras.</p> <p>For little Lipschitz algebras (on the circle) the answer is no -- this ought to be in a <a href="http://www.ams.org/journals/tran/1964-111-02/S0002-9947-1964-0161177-1/" rel="nofollow">paper of Sherbert from the 1970s</a> -- so I expect the answer to your original question is "yes". (For the "big" Lipschitz algebras my suspicion is that the counter-example I gave for $C^1[-1,1]$ would also work.)</p>