By Emilio Prieto Sáez, Alberto A. Álvarez López

En los capítulos que comprende este texto se exponen los instrumentos matemáticos básicos del Álgebra Lineal, así como una introducción a las sucesiones de números reales. Incluye un tomo con problemas resueltos.

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Suppose that B has an approximute urzit. For each w E Y 4 3 . 9 0 (e being the unit of G ) let p ( w ) be the multiplier of Be obtained by restricting w t o E,. Then p : We(B)+ W ( E , ) is a *-isomorphism (onto all of %‘-(Be)). Proof. It is obvious that p is a *-homomorphism. The preceding proposition states that it is one-to-one and onto W(B,J. 9. 14). A multiplier u of W will be called unitary if u*u = uu* = 4 and llullo I 1. (a) is a group under the multiplication multipliers of 99 by @(a).

7) To do this, we first notice that, for any bounded linear endomorphism F of A, IlFll = sup{IIF(a)bll:a, b E A, llall I1, llbll 5 1 ) = sup{llbF(a)ll:a,hE A, (8) Ilall 5 1, llhll 5 1). Indeed: Clearly IlFll rnajorizes the two suprema in (8). Now,given E > 0, choose a so that llall = 1 and IIF(a)(l > IlFll - E ; and put b = IIF(u)II-~F(u)*. Then llhll = 1 and IIF(a)bII = IIF(a)JJ> - E. Thus the first supremum in (8) equals (JFII. Similarly the second supremum equals 11Fl1. So (8) is proved. If u = ( A , p ) E %‘-(A),it follows from (8) that 11/41 = suP{IlmbII: Ilall, llbll 5 whence = {Ilal(h)ll: IblL llbll 5 1) = 11~11, I> 780 VIII.

A multiplier u of W will be called unitary if u*u = uu* = 4 and llullo I 1. (a) is a group under the multiplication multipliers of 99 by @(a). operation of W(B), the inverse in @(B) being involution. The map, zO:%(A3) --t G sending each unitary multiplier into its order is a group homomorphism. # of order x-that is, if the group homomorphism no of the last paragraph is onto G. Definition. Let u be a unitary multiplier of A3 of order x. The left action of u is norm-decreasing and its inverse is the norm-decreasing left action of u*.