| Summary: | <p>This thesis studies the existence and uniqueness of $G$-kernels on those C$^*$-algebras classified by the Elliott programme. We develop two obstructions to the possible $H^3$ invariants of a $G$-kernel. These obstructions arise from studying the unitary algebraic $K_1$ group and the topological $K_0$ group of a C$^*$-algebra. As a consequence of these obstructions, we show that any $G$-kernel on the Jiang-Su algebra has trivial $H^3$ invariant. Similarly, for finite groups $G$, any $G$-kernel on the Cuntz algebra $\mathcal{O}_\infty$ must have trivial $H^3$ invariant.</p>
<p>We construct multiple examples of $G$-kernels with non-trivial $H^3$ invariant and, under a UHF-absorption condition, we classify those $G$-kernels that have the Rokhlin property on both Kirchberg algebras satisfying the UCT and unital, separable, simple, nuclear, tracially AF C$^*$-algebras that satisfy the UCT. As a follow up to this classification, we study the structure of $G$-kernels with the Rokhlin property.</p>
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