A non-liminal (CCR) C^{∗}-algebra.

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Let H be an infinite dimensional Hilbert space. Then faithful irreducible representation

B(H) ⟶ B(H)  (B(H)′ = C1)
T ⟼ T

together with B(H) ≠ K(H), shows that B(H) isn’t liminal.

Comment. Another example may be found in CW22.