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

We study Banach $\tilde{\mathbb{C}}$-algebras, i.e., complete ultra-pseudo-normed algebras over the ring $\tilde{\mathbb{C}}$ 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 $\tilde{\mathbb{C}}$-algebras and indicate a particular class of Banach $\tilde{\mathbb{C}}$-algebras that overcomes these limitations to a large extent. We also investigate C*-algebras over $\tilde{\mathbb{C}}$.