We study BDF theory in the context of (not necessarily simple) purely infinite corona algebras. Let $\mathcal{B}$ be a nonunital separable simple C*-algebra with standard regularity properties for which $\mathcal{C}\left(\mathcal{B}\right)$ is purely infinite (though not necessarily simple). Let $\mathcal{A}$ be a separable nuclear C*-algebra. We prove a BDF Voiculescu decomposition theorem for maps from $\mathcal{A}$ to $\mathcal{C}\left(\mathcal{B}\right)$, and use this to prove that ${\mathbf{E}\mathbf{x}\mathbf{t}}_{\sigma }(\mathcal{A},\mathcal{B})$ is a group. Then, restricting to the case $\mathcal{A}=C\left(X\right)$, for some compact metric space X, and $\mathcal{B}$ has continuous scale, we provide characterizations of the neutral element for $\mathbf{E}\mathbf{x}\mathbf{t}\left(C\right(X),\mathcal{B})$.

我们在（不一定简单）纯无限电晕代数的背景下研究BDF理论。让$\mathcal{乙}$ 是具有标准正则性的非单位可分简单C *代数 $\mathcal{C}（\mathcal{乙}）$是纯粹无限的（尽管不一定简单）。让$\mathcal{一种}$是一个可分离的核C *代数。我们证明了BDF Voiculescu分解定理$\mathcal{一种}$ 至 $\mathcal{C}（\mathcal{乙}）$，并以此证明 ${\mathbf{Ë}\mathbf{X}\mathbf{Ť}}_{\sigma }（\mathcal{一种}，\mathcal{乙}）$是一个小组。然后，仅限于这种情况$\mathcal{一种}=C（X）$，对于某些紧凑的度量空间X，和$\mathcal{乙}$ 具有连续的尺度，我们提供中性元素的表征 $\mathbf{Ë}\mathbf{X}\mathbf{Ť}（C（X），\mathcal{乙}）$。