Suppose $S$ is a set ,can we find a smallest nuclear $C^*$ algebra containing $S$
closed as off-topic by Jendrik Stelzner, YuiTo Cheng, Lee David Chung Lin, steven gregory, Cesareo May 13 at 7:24
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No. There are C$^*$-algebras that are not contained in any nuclear C$^*$-algebra. Any non-exact algebra would give the counterexample, as exactness passes to subalgebras.