On almost commutative unital complex normed Q-algebras
Abstract
We show that a unital complex normed Q-algebra (A,∥.∥) in which the spectral radius satisfies:
ρA(x) = inf{p(x) : p ∈ Eun(A), p ≤ ∥.∥},
where Eun(A) denotes the set of all algebra-norms p on A equivalent to the given algebra-norm ∥.∥ such that p(e) = 1, is commutative modulo its Jacobson radical. The same conclusion is obtained if (A, ∥.∥) satisfies:
ρAb(xy) ≤ ρAb(x) ∥y∥ for every x ∈ Ab, y ∈ A, where Ab is the completion of (A, ∥.∥).
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