## Abstract

Let A be a unital complex Banach algebra and a ∈ A. For an interval K, we consider the Banach algebra (C(K), * _{n}), with the J^{n}-multiplication * _{n}. THEOREM 1. The element a is of class C^{n} if and only if there exists a continuous representation U of the Banach algebra (C(K), * _{n}) on A such that Ul = a^{n}/n!; U is uniquely determined and precisely the “weak representation of a on C(K).” A new perspective to the universal model provided by that representation is obtained as a consequence of a “non-commutative” Taylor formula for the analytical operational calculus. This formula is also used to obtain a perturbation result for semigroups of operators (Theorem 4).

Original language | English |
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Pages (from-to) | 139-152 |

Number of pages | 14 |

Journal | Journal of Functional Analysis |

Volume | 113 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1993 |

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