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The next lemma allows us to complete Cc(G)cc(H) to obtain the desired full Hilbert G* (iJ)-module X. 16. Suppose that AQ is a dense *-subalgebra of a C*-algebra A, and that Xo is a right A^-module. 1 (in (d), we require that (x , x) > 0 in the completion A). Then there is a Hilbert A-module X and a linear map q : XQ —> X such that q(Xo) is dense, q(x) • a = q(x • a) for all x G Xo; a G AQ, and (q(x) , q(y)) = (x , y) ; we call X the completion of the pre-inner product module XQ. Hilbert C*-Modules 16 Proof.

4 Induced Representations 37 category of nondegenerate representations of B and bounded intertwining operators to the corresponding category for A. The second says that induction is well-defined on ideals. This is important in applications, because it means our induction process is well-suited to non-type I problems, where one wishes to analyze spaces of (primitive) ideals rather than spaces of (irreducible) representations. 69. Suppose A acts nondegenerately as adjointable operators on a Hilbert B-module X, that iii : B —• B(Tii) are nondegenerate representations of B, and that T : Hi —> H2 is a bounded intertwining operator: T(7Ti(b)h) = 7r2(6)(T/i).

Finally, an e/3-argument shows that the limit / is *-strong continuous. 57. Let T be a locally compact space and /C = K,(H) the compact operators on a Hilbert space H. If m G Cb(T,B(H)*-s), then Lm(f)(t) := m(t)f(t) for f G C0(T, JC) defines a multiplier L m G M(Co(T,/C)) with \\Lm\\ — 11^1 loo- Furthermore, vn \—> Lm is a isomorphism of Cb{T, B(H)*-S) onto M(Co(T, JC)). The proof requires an easy Lemma which will be used many times. 58. Suppose a : Co(T, JC) —> Co(T, JC) is a bounded linear map satisfying a((/)f) = (j)a(f) for (j) G CQ{T) and f G Co(T, JC).

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