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Proof. Regard A as a subalgebra of the algebra Q(A) constructed above. Set ˜ = 0 (r = 1, . . , n) . J= a ˜ = (ai ) ∈ Q(A) : xr a Then J is a closed ideal in Q(A) and, by the preceding lemma, J = {0}. Define the operator T : J → J by T (ai ) = (x0 ai ). Let λ ∈ ∂σ B(J) (T ). 28, there exist u˜j ∈ J (j ∈ N) such that u ˜j Q(A) = 1 and limj→∞ (T − λ)˜ uj Q(A) = limj→∞ (x0 − λ)˜ uj Q(A) = 0. 3. Approximate point spectrum in commutative Banach algebras 29 For every k ∈ N there exists j ∈ N such that (x0 − λ)˜ uj Q(A) < k −1 .

Since the integral is defined as a limit of Riemann’s sums and ϕ is continuous and multiplicative, we have ϕ(f (x)) = 1 2πi f (z) z − ϕ(x) −1 dz = f (ϕ(x)). Γ The second statement follows from the definition of the Gelfand transform. In commutative Banach algebras it is possible to introduce the notion of spectrum for n-tuples of elements. Definition 14. Let A be a commutative Banach algebra, x1 , . . , xn ∈ A. The spectrum σ(x1 , . . , xn ) is the set σ(x1 , . . , xn ) = (ϕ(x1 ) . . ϕ(xn )) : ϕ ∈ M(A) .

It remains to show the submultiplicativity of ||| · |||. We have |||b1 ||| · |||b2 ||| = ≥ inf (|a1 | + k 2 b1 − a1 ) · (|a2 | + k 2 b2 − a2 ) inf |a1 | · |a2 | + k 2 a1 ,a2 ∈A a1 ,a2 ∈A + b 1 − a1 ≥ inf a1 ,a2 ∈A a1 · b 2 − a2 + b 1 − a1 · a2 · b 2 − a2 |a1 a2 | + k 2 b1 b2 − a1 a1 ≥ |||b1 b2 |||. 27 (ii), a topological divisor of zero is singular (= non-invertible) in any extension B ⊃ A. For commutative Banach algebras the opposite statement is also true. Definition 2. An element x in a commutative Banach algebra A is called permanently singular if it is singular in each commutative Banach algebra B ⊃ A.

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