Let $H$ be a separable Hilbert space and let $A$ be a non-unital $C^*$-subalgebra in $B(H)$ such that the second dual $A^{**} = B(H).$ Does $A$ coincide with the ideal of compact operators $K(H)?$
Let $H$ be a separable Hilbert space and let $A$ be a non-unital $C^*$-subalgebra in $B(H)$ such that the second dual $A^{**} = B(H).$ Does $A$ coincide with the ideal of compact operators $K(H)?$