Abstract
By a function of support a numerical range is defined for elements of complex Banach algebras. The geometric character of the definition allows to proof some results about the numerical range in a very natural way.- In the second part holomorphic perturbations of the numerical range and the spectrum are treated as perturbations of the function of support to get a maximum principle for the numerical range and the convex hull of the spectrum, whereas there is in general no maximum principle for the spectrum. This answers a question raised by A.Brown and R.G.Douglas.
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