Banach $\widetilde{\mathbb C}$-algebras

Details Banach $\widetilde{\mathbb C}$-algebras Banach $\widetilde{\mathbb C}$-algebras

2008-11-11

arxiv.org/abs/0811.1742v1

Mathematics

Functional Analysis

36 pages

Scientific paper

We study Banach $\widetilde{\mathbb C}$-algebras, i.e., complete ultra-pseudo-normed algebras over the ring $\widetilde{\mathbb C}$ of Colombeau generalized complex numbers. We develop a spectral theory in such algebras. We show by explicit examples that important parts of classical Banach algebra theory do not hold for general Banach $\widetilde{\mathbb C}$-algebras and indicate a particular class of Banach $\widetilde{\mathbb C}$-algebras that overcomes these limitations to a large extent. We also investigate C*-algebras over $\widetilde{\mathbb C}$.

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