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Extra info for Calculus of several variables

Sample text

Let P ∈ Prim A be a primitive ideal of the algebra A. Show first, that there exists a point t ∈ T such that P ∩ Z = Jt . Indeed, let π be the irreducible representation of the C ∗ -algebra A with ker π = P ; then C · I ⊂ π(Z) ⊂ π(A)′ = C · I. Thus C∼ = π(Z) = (Z + P )/P ∼ = Z/(Z ∩ P ), and therefore Z ∩ P is a maximal ideal of the algebra Z, thus equal to Jt for some point t ∈ T . It is easy to see that the conditions P ∩ Z = Jt and P ⊃ J(t) are equivalent for each primitive ideal P . 7, J(t) = P.

Denote by ∆ = ∆(C1 , . . , Cn ) ⊂ Rn the joint spectrum of the operators C1 , . . , Cn . Then the C ∗ -algebra RC is naturally isomorphic to C(∆), and the coordinate function tk , k = 1, n, (t = (t1 , . . , tn ) ∈ ∆ ⊂ Rn ) is the image of Ck under this isomorphism. 20) qk (t) = diag (0, . . , 0, where t = (t1 , . . , tn ) ∈ ∆. 1 , 0, . . 21) 30 Chapter 1. Preliminaries To describe the algebra S(∆) we introduce first a sort of “maximal” algebra. Let n n ∆n−1 = {t = (t1 , . . , tn ) ∈ R : tk ≥ 0, k = 1, n, tk = 1} k=1 be the standard (n − 1) - dimensional simplex.

Cn of the algebra A, such that n ck = e and C ∗ (c1 , . . , cn ) = A. k=1 Of course, there are many possibilities to construct such a family. For example one may take, for α ∈ [0, 1], ck = ck (α) = 1 1+α sk ), (e + 4n0 sk k = 1, n0 , n 0 1 ck = ck (α) = (e − cj ), n − n0 j=1 k = n0 + 1, n. Finally, the family of the following systems of self-adjoint idempotents (projections) satisfies all the necessary properties, 1/2 1/2 p = p(α) = (cj (α) · ck (α))nj,k=1 , qk = qk (α) = diag (0, . . , 0, where α ∈ [0, 1].