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In previous definition of $\mathrm{E}$-theory, separability of the$\mathrm{C}^*$-algebras is needed either to construct the composition productor to prove the long exact sequences. Considering the latter, the potentialfailure of the long exact sequences can be traced back to the fact that these$\mathrm{E}$-theory groups accommodate information about asymptotic processesin which one real parameter goes to infinity, but not about more complicatedasymptotics parametrized by directed sets. We propose a definition for$\mathrm{E}$-theory which also incorporates this additional information bygeneralizing the notion of asymptotic algebras. As a consequence, it not onlyhas all desirable products but also all long exact sequences, even fornon-separable $\mathrm{C}^*$-algebras. More precisely, our construction yieldsequivariant $\mathrm{E}$-theory for $\mathbb{Z}_2$-graded$G$-$\mathrm{C}^*$-algebras for arbitrary discrete groups $G$. We suspect thatour model for $\mathrm{E}$-theory could be the right entity to investigateindex theory on infinite dimensional manifolds.

Generalized asymptotic algebras and $\mathrm{E}$-theory for non-separable $\mathrm{C}^*$-algebras