![]() Contrast this with the purely algebraic situation described in IV.3.12. At the same time, by 9.4, every closed stable subspace has a stable orthogonal complement. ![]() RemarkĪs we have seen above, a *-representation may have no irreducible stable subspaces. However, roughly speaking, if W is a “small” neighborhood of a point ϕ in Â, we can say that the vectors ξ in range( P( W)) will be those for which T aξ, is “nearly” equal to ξ( a) ξ ( a ∈ A). In the general case, as we have seen, no such description is possible. In the special case where  is discrete, the ” W-subspace” would be simply the Hilbert direct sum of the ϕ-subspaces for which ϕ ∈ W. Thus, if W is a Borel subset of Â, range( P( W)) can be regarded as the ” W-subspace” of X( T) (wth respect to T). Nevertheless, for an arbitrary (non-degenerate) *-representation T of A, the spectral measure P of T provides a very appropriate generalization of the purely algebraic notion, encountered in IV.2.12, of decomposing a completely reducible representation into its σ-subspaces. Thus T is as far as possible from being completely reducible (in the *-representational sense). By 10.11 this implies that T has no one-dimensional subrepresentations, and hence, as we shall see in 14.3, no closed irreducible T - stable subspaces at all. Then in general there will exist a non-zero (numerical) regular Borel measure μ on  assigning zero measure to every one-element set and from this by 10.12 one can construct a non-degenerate *-representation T of A whose spectral measure assigns zero measure to every one-element set. Let us now drop the assumption that  is discrete. We shall make a further study of “*-representational complete reducibility” in §14. ![]() X ( T ) = ∑ ϕ ∈ A ^ Y ϕ algebraically), but in the analogous *-representational sense expressed by (10) and (11). We have shown that every *-representation of A is “completely reducible,” not in the strict algebraic sense of IV.2.1 (which would require that Notice that Y ϕ is the ϕ-subspace of X( T) in the strict algebraic sense of IV.2.11. Together, (11) and (12) assert that T is a Hilbert direct sum of one-dimensional *-representations.
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