A liminal C^{∗}-algebra which isn’t postliminal.

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Let A denote Toeplitz algebra. K(H^{2}) is liminal. \frac{A}{K(H^{2})} is ∗-isomorphic to C(T), so it is abelian and therefore liminal. Hence A is postliminal .But identity representation of A on H^{2} is irreducible and not finite dimensional, so A isn’t liminal. For details see [Mur, Example 5.6.4]. Ref.

[Mur] G.J. Murphy, C^{∗}-algebras and operator theory, Academic Press, 1990.