Abstract
Let K be an algebraically closed field complete with respect to a dense ultrametric absolute value |.|. Let D be an infraconnected affinoid subset of K and let H(D) be the Banach algebra of analytic elements on D. Let f ∈ H(D) be injective in D and let f* be the mapping defined on the multiplicative spectrum of H(D) that identifies with the set of circular filters on D. We show that f* is injective and maps bijectively the Shilov boundary of H(D) onto this of H(f(D)). Thanks to this property we give a new proof of the equality \(\left| {f(x) - f(y)} \right| = \left| {x - y} \right|\sqrt {\left| {f'(x)f'(y)} \right|} \).
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