What is an example of a simple $C^{*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
8
Take compact operators on a non-separable Hilbert space. For each such an operator you'll find a rank-one projection with range in the kernel of that operator.
I guess that by some Löwenheim–Skolem-type argument, this gives you a separable example too. (As observed by Andreas Thom.)
answered Nov 29, 2014 at 11:21
-
7$\begingroup$ There is no separable example. If $(a_n)_n$ is a dense sequence, then $\sum_n a_n^*a_n/(2^n \|a_n\|^2)$ is a non-zero divisor. $\endgroup$ Commented Nov 29, 2014 at 11:50
-
$\begingroup$ Oh yes, you are of course right. Thank you for spotting that. $\endgroup$ Commented Nov 29, 2014 at 12:56
-
$\begingroup$ @TomekKania Thank you for your interesting answer. Is this example a universal example in the sense that: every $Z^{*}$ algebra has a faithful or irreducible representation in some $\mathcal{K}(H)$? For example consider the $C^{*}$ subalgebra of $B(H)$ generated by all countable rank operators. According to your example it is again a $Z^{*}$ algebra. Can we embedd it in compact operators ? Can we represent it irreducibly in compact operators? $\endgroup$ Commented Nov 29, 2014 at 15:15
-
1$\begingroup$ No, not even as a Banach space. $\endgroup$ Commented Nov 29, 2014 at 17:04
-
2$\begingroup$ The operators with separable range contain a copy of $\ell_\infty$ as a subspace. This is not the case for $K(H)$; dual of each separable subspace of $K(H)$ is separable. $\endgroup$ Commented Nov 30, 2014 at 13:14