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Suppose $S$ is a set ,can we find a smallest nuclear $C^*$ algebra containing $S$

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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.

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  • $\begingroup$ If A is a nuclear C* algebra,does there exist a general method to construct a smallest nuclear C^* algebra B which contains A? $\endgroup$ – math112358 May 19 at 18:34
  • $\begingroup$ That would be $A$. $\endgroup$ – Martin Argerami May 19 at 19:57

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