Connectivity of the set of invertible elements of a complex Banach algebra
نویسنده
چکیده
Proof. Denote by 1 the unit element of B. Let a ∈ B×. We show that 1 and a can be connected by a path in B×. Let A be the subalgebra of B generated by 1 and a. That is A = {P (a) : P (t) ∈ C[t]}. Because A× ⊂ B×, it suffices to find a path in A× that connects 1 with a. First, we determine all multiplicative functionals of A. Since A is a linear subspace of B, it is finite dimensional. Then there exists n ∈ N such that 1, a, a, ..., a are linearly dependent. Assume n is the smallest number with this property. Then there exists a monic polynomial q(t) ∈ C[t] with degree n such that q(a) = 0. Let λ1, λ2, ..., λm ∈ C be the distinct roots of q(t). A multiplicative functional φ of A is determined by φ(a) because
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