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For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.

The subject for investigation in this note is concerned with strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras, or $L_\infty[1]$-algebras. We extract an intrinsic character, the Atiyah class, when a smaller $L_\infty[1]$-algebra $A$ is extended to a bigger one $L$. In fact, we establish that given such an SH Lie pair $(L,A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[\alpha^E]$, living in $H^2_{\mathrm{CE}}(A, A^\perp\otimes End(E))$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. In particular, the quotient space $L/A$ is a canonical $A$-module and there associates the Atiyah class $[\alpha^{L/A}]\in H^2_{\mathrm{CE}}(A, Hom(L/A\otimes L/A,L/A))$. The nature of such classes is that they induce on $H^\bullet_{\mathrm{CE}}(A,L/A[-2])$ a Lie algebra structure, and on $H^\bullet_{\mathrm{CE}}(A,E[-2])$ a Lie algebra module structure. An alternative point of view is that the process of taking Atiyah classes defines a functor from the category of $A$-modules to the category of $H^\bullet_{\mathrm{CE}}(A,L/A[-2])$-modules. Moreover, Atiyah classes are invariant under gauge equivalent $A$-compatible infinitesimal deformations of $L$.

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A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarily-semisimple or rigid braided tensor categories C of modules for the fixed-point vertex operator subalgebra V^G of a vertex operator (super)algebra V with finite automorphism group G. The main results are that every V^G-module in C with a unital and associative V-action is a direct sum of g-twisted V-modules for possibly several g∈G, that the category of all such twisted V-modules is a braided G-crossed (super)category, and that the G-equivariantization of this braided G-crossed (super)category is braided tensor equivalent to the original category C of V^G-modules. This generalizes results of Kirillov and Müger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether V^G is strongly rational if V is strongly rational. We show that V^G is indeed strongly rational if V is strongly rational, G is any finite automorphism group, and V^G is C_2-cofinite.