Remark

Tag 029U, permanent. Cite it as [Bourbaki, Tag 029U].

Théories spectrales, chapter I, § 4. Calcul fonctionnel holomorphe

If the radical of the algebra A is zero, one can omit the continuity condition in (CF1) (cf. prop. 9 of I, p. 40).

A holomorphic functional calculus on A is the data, for every integer $n\geqslant 1$ , of a holomorphic functional calculus in $n$ variables on A, satisfying:

(CF3) Whatever the integers $m$ and $n$ such that $1\leqslant m\leqslant n$ , and whatever $\boldsymbol{a}\in A^n$ and $f\in \mathscr{O}$ (Sp $^m(\pi_{m,n}(\boldsymbol{a}); A)$ , one has

\Theta_{\boldsymbol{a}}(\pi^*_{m,n}(f)) = \Theta_{\pi_{m,n}(\boldsymbol{a})}(f)

The object of this paragraph is to prove the following theorem:

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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