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Jan 6, 2019 at 15:48 comment added Nik Weaver (Indeed, any nonzero element of $B$ will not be in the kernel of some finite dimensional irrep.)
Jan 6, 2019 at 13:43 comment added Nik Weaver If $B$ is not zero then it has to contain some element which is nonzero on some factor. Evaluation on that factor will not be zero on $B$.
Jan 6, 2019 at 6:36 comment added math112358 If $\phi$ is a finite representation of $A$,$B$ is a $C^*$ subalgebra of $A$,I think that there is a possibility that $\phi_{B}=0$
Jan 6, 2019 at 6:16 vote accept math112358
Jan 6, 2019 at 5:21 history answered Nik Weaver CC BY-SA 4.0