By Jerry Marsden

ISBN-10: 091409811X

ISBN-13: 9780914098119

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**Additional info for Applications of global analysis in mathematical physics**

**Example text**

31) 1. The power series of z --+ (a(x) - z) - 1 converges for all X E X(A) provided that Z > sUPxeX(A) la(x) I = lIall. As we see, if a is a normal element of a C* algebra, then the radius of convergence of the series for (a - z) -1 is exactly lIall. ) The spectrum of a is precisely the image of X(A) under a. 2. Continuous functions f(a) are defined on the range of the Gel'fand isomorphism asf(a(x», and they exist for all normal a in any C* algebra. More specifically, a Hermitian element can be decomposed into a positive and a negative part, and unique square roots can be taken of positive elements.

Hence, for all X E X(d), x(a) E Sp(a), and thus Ix(a) I ~ Ilall· 2. Since x(a*a) = x(a)*x(a) = Ix(a)1 2 ~ 0 and XCI) = 1, every character is a state, which automatically makes the mapping X: d ~ C continuous. Indeed, every X is a pure state, since no convex combination a l Xl + a2 X2'O < ai < 1, a 1 + a2 = 1, can be multiplicative: (alXl + a2X2)(a 2) is at the same time (aIXl(a) + a2xz

2) are algebras when multiplication is defined componentwise for vectors, pointwise for functions, and in the usual way for matrices. These multiplication rules make all of them Abelian except for the matrices. The spaces LP, p < 00, are not generally algebras; for examples, x - I/Z E L 1([0, 1], dx) but X-I £t L 1([0, 1], dx). The spaces [P are algebras, but they have no unit if p < 00. The space [0 == {(VI' VZ, ••• ) E [00: limi Ivd = O} is a subalgebra of [00 without a unit. , it is n- 1(0).

### Applications of global analysis in mathematical physics by Jerry Marsden

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