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Abstract: We study Banach $\widetilde{\mathbb C}$-algebras, i.e., completeultra-pseudo-normed algebras over the ring $\widetilde{\mathbb C}$ of Colombeaugeneralized complex numbers. We develop a spectral theory in such algebras. Weshow by explicit examples that important parts of classical Banach algebratheory do not hold for general Banach $\widetilde{\mathbb C}$-algebras andindicate a particular class of Banach $\widetilde{\mathbb C}$-algebras thatovercomes these limitations to a large extent. We also investigate C*-algebrasover $\widetilde{\mathbb C}$.

Autor: Hans Vernaeve



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