A C^{∗}-algebra A in which the closed unit ball of A^{+} isn’t the closed convex hull of the projections of A. (Note that the closed unit ball of positive elements of each hereditary C^{∗}-algebra A of a von Neumann algebra is the closed convex hull of its projections ).

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The only projections of C([0, 1]) are 0 and 1. So the closed convex hull of C([0, 1]) is {f |∃c ∈ [0, 1]; f = c}, not equal to (C([0, 1]))_{1}^{+}.