Let $G$ be a metrizable compact group, $A$ a separable ${{\text{C}}^{*}}$-algebra, and $\alpha :G\,\to \,\text{Aut}\left( A \right)$ a strongly continuous action. Provided that $\alpha $ satisfies the continuous Rokhlin property, we show that the property of satisfying the $\text{UCT}$ in $E$-theory passes from $A$ to the crossed product ${{\text{C}}^{*}}$-algebra $\mathcal{A}{{\rtimes }_{\alpha }}\,G$ and the fixed point algebra ${{A}^{\alpha }}$. This extends a similar result by Gardella for $KK$-theory in the case of unital ${{\text{C}}^{*}}$-algebras but with a shorter and less technical proof. For circle actions on separable unital ${{\text{C}}^{*}}$-algebras with the continuous Rokhlin property, we establish a connection between the $E$-theory equivalence class of $A$ and that of its fixed point algebra ${{A}^{\alpha }}$.

We study the E-theory group E[0, 1](A, B) for a class of C*-algebras over the unit interval with finitely many singular points, called elementary C[0, 1]-algebras. We use results on E-theory over non-Hausdorff spaces to describe E[0, 1](A, B) where A is a sky-scraper algebra. Then we compute E[0, 1](A, B) for two elementary C[0, 1]-algebras in the case where the fibers A(x) and B(y) of A and B are such that E1 (A(x), B(y)) = 0 for all x, y ∈ [0, 1]. This result applies whenever the fibers satisfy the UCT, their K0-groups are free of finite rank and their K1-groups are zero. In that case we show that E[0, 1](A, B) is isomorphic to Hom(0(A), 0(B)), the group of morphisms of the K-theory sheaves of A and B. As an application, we give a streamlined partially new proof of a classification result due to the first author and Elliott.

Let $A$ and $B$ be C*-algebras, with $A$ separable and $B$$\sigma$-unital and stable. It is shown that there are natural isomorphisms

$$ E(A,B) = KK(SA,Q(B)) = [SA,Q(B) \otimes K], $$

where $SA=C_0(0, 1) \otimes A$, $[\cdot, \cdot]$ denotes the set of homotopy classes of *-homomorphisms, $Q(B) = M(B) / B$ is the generalized Calkin algebra, and $K$ denotes the C*-algebra of compact operators of an infinite-dimensional separable Hilbert space.