Banach Lecture 5

Banach5

Excerpts

Lemma I.7.1 B is unital and J is a proper ideal.  Then \mbox{dist}(1,J) \ge 1.

notes: the r defined and used in the proof is not necessarily the spectral radius, but surely at least as big so r can be used as in the proof.

Question: why does proof of corollary 2 show continuity?

 

B is unital for the rest of the lecture.

Thm I.7.2 (Gelfand – Mazur) Upto iso, \mathbb{C} is the only normed division algebra over \mathbb{C}.

notes: is a -c is not invertible in the completed space, it is certainly not invertible in the smaller space.

Thm I.7.1 Let a \in B.  It is invertible iff its not contained in a maximal ideal.  Also sp(a) = \{\phi(a) : \phi \in M(B)\} hence there exists multiplicative linear funtionals (since the spectrum is non empty).

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