Soit $A$ une algèbre uniforme et soit $I$ un idéal fermé de $A$ tel que $A/I$ soit une algèbre isométriquement isomorphe à $C\left(X\right)$, il existe alors une sous-algèbre fermée $B\subset A$ telle que $A$ est isométriquement isomorphe à $I\oplus B$.

We consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x ↦ x̂ are: (i) Is $K_{\nu }=sup{\left|\right|(e-x)}^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,max\left|x̂\right|\le \nu$ bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is $C_{\delta }=sup\left|\right|x^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,min\left|x̂\right|\ge \delta$ bounded? Both questions are related to a “uniform spectral radius” of the algebra, $r_{\infty }\left(A\right)$, introduced by Björk. Question (i) has an affirmative answer if and only if $r_{\infty }\left(A\right)<1$, and this result is extended to more general nonlinear extremal problems...

A unital commutative Banach algebra is spectrally separable if for any two distinct non-zero multiplicative linear functionals φ and ψ on it there exist a and b in such that ab = 0 and φ(a)ψ(b) ≠ 0. Spectrally separable algebras are a special subclass of strongly harmonic algebras. We prove that a unital commutative Banach algebra is spectrally separable if there are enough elements in such that the corresponding multiplication operators on have the decomposition property (δ). On the other hand,...