By Jean-Pierre Serre

ISBN-10: 3540070281

ISBN-13: 9783540070283

This version reproduces the 2d corrected printing of the 3rd variation of the now vintage notes by means of Professor Serre, lengthy tested as one of many regular introductory texts on neighborhood algebra. Referring for history notions to Bourbaki's "Commutative Algebra" (English version Springer-Verlag 1988), the e-book focusses at the numerous size theories and theorems on mulitplicities of intersections with the Cartan-Eilenberg functor Tor because the imperative idea. the most effects are the decomposition theorems, theorems of Cohen-Seidenberg, the normalisation of jewelry of polynomials, measurement (in the feel of Krull) and attribute polynomials (in the experience of Hilbert-Samuel).

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**Example text**

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).

### Algebre Locale, Multiplicites. Cours au College de France, 1957 - 1958 by Jean-Pierre Serre

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