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on Spa es of Continuous Fun tions by Wolfgang Arendt and Philippe B enilan November 3, 1998 Dedi ated to Professor Herbert Amann on the o asion of his sixtieth birthday. Abstra t. Let IRN be open. It is shown that the Diri hlet Lapla ian generates a (holomorphi ) C0 semigroup on C0( ) if and only if is regular in the sense of Wiener. The same result remains true for ellipti operators in divergen e form.