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Schaefer and Zhang have recently obtained an analogue, for sequentially order continuous functionals on      C ( K ) C(K)      , of a much earlier theorem of Dixmier. In this note it is shown that the Schaefer-Zhang Theorem has a natural generalisation to non-commutative      C ∗  C^*      -algebras. These results are obtained as consequences of our main theorem which is concerned with affine functions on compact convex sets.

A Dixmier-Schaefer-Zhang theorem for operator algebras