Questions about special $C^*$-subalgebras and ideals. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:00:57Z http://mathoverflow.net/feeds/question/90561 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90561/questions-about-special-c-subalgebras-and-ideals Questions about special $C^*$-subalgebras and ideals. Aviv 2012-03-08T10:17:55Z 2012-05-08T16:15:45Z <p>Let $A$ be a $C^*$-algebra and $I$ be a two side closed (essential) ideal of $A$. Suppose that $p \in A\backslash I$ is a non trivial projection. Let $B=pIp$. My questions are:</p> <p>(1) Is $B$ a $C^*$-subalgebra of $I$?</p> <p>(2) If (1) is correct, then, is $B$ unital?</p> <p>(3) If both (1) and (2) are right, then, what is the "<strong>unit</strong>" of $B$? is it the projection "p"?</p> <p>Special case of this may be: $A=M(I)$, the multiplier algebra of $C^*$-algebra $I$. Hope any comments for these.</p> http://mathoverflow.net/questions/90561/questions-about-special-c-subalgebras-and-ideals/90572#90572 Answer by Ulrich Pennig for Questions about special $C^*$-subalgebras and ideals. Ulrich Pennig 2012-03-08T12:41:11Z 2012-03-08T12:41:11Z <p>$B$ does not have to be unital. Think of the case $A = M(I)$. Then $p =1$ is a reasonable projection in $A \backslash I$. In this case $B= I$. Since a unit $1$ in $B$ has to satisfy $1 = p\cdot 1\cdot p = p^2 = p$, $p$ is the only choice you have. Therefore $B$ is never unital for $p \in A \backslash I$.</p> http://mathoverflow.net/questions/90561/questions-about-special-c-subalgebras-and-ideals/90627#90627 Answer by Martin Argerami for Questions about special $C^*$-subalgebras and ideals. Martin Argerami 2012-03-08T21:03:53Z 2012-03-08T21:03:53Z <p>It is true that $B$ is a C$^*$-subalgebra. But it doesn't have to be unital. Consider for example $A=M_2(\ell^\infty(\mathbb{N}))$, $I=M_2(c_0(\mathbb{N}))$, and $$p=\begin{bmatrix}1&amp;0 \\ 0&amp;0\end{bmatrix}.$$ Then $pIp$ is $c_0(\mathbb{N})$, which is not unital. </p>