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B is quite clearly unique given that its restriction to V should be f. 3. 0 In order to formulate a standard functorial consequence of this universal mapping property, let V' be another real inner product space and let g : V --+ V' be an isometric linear map.

C[V] admits a unique even central state: its trace T, of which any continuous central linear functional is a scalar multiple. c to be a continuous central linear functional. 7 to deduce that a = a(l)T on C(M). 13 now imply that a = a(l)T on C(V). Finally, we deduce that a = a(l)T on C[V] by continuity. 0 Let us temporarily denote by Zr the set of all a E C[V] with the property that T(ba) = 0 whenever bE C[V]: thus Zr = {a E C[V] : T(C[V]ยท a) = a}. Since T is a central linear functional, Zr C C[V] is a bilateral ideal; since T is continuous, Zr C C[V] is closed.

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Differential Algebra by Joseph Fels Ritt


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