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Let c and q/ be positive linear functionals on a *-algebra V . When the closed *-representations 7ro and 7, of _v generated by the GNSconstruction for 0 and w are self-adjoint, we shall show that to is selfadjoint if and only if ? c 1(7ry+>,); and there exists a self-adjoint extension p of 7ro+, such that p(,v)', = V+V(W)w if and only if 7r0+,(X )') is a von Neumann algebra.

Selfadjointness of the *-representation generated by the sum of two positive linear functionals