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Generalized quasidiagonality for extensions. (English) Zbl 1425.46040

The authors suggest a generalization for quasidiagonality of \(C^*\)-algebras that avoids the use of projections.
Let \(B\) be a \(\sigma\)-unital non-unital \(C^*\)-algebra. A sequence \(\{b_n\}_{n\in\mathbb N}\) of norm 1 positive elements of \(B\) is a system of quasidiagonal units if \(b_nb_m=0\) for \(n\neq m\) and \(\sum_{n\in\mathbb N}x_n\) is strictly convergent in the multiplier algebra \(M(B)\) for any bounded sequence \(x_n\in b_nBb_n\). Further, \(x\in M(B)\) is generalized block-diagonal if \(x=\sum_{n\in\mathbb N} x_n\) for some \(x_n\in b_nBb_n\), and a \(C^*\)-subalgebra \(A\subset M(B)\) is generalized quasidiagonal if any \(a\in A\) can be written as \(a=x+b\), where \(x\) is generalized block-diagonal and \(b\in B\).
It is shown that some results on quasidiagonal \(C^*\)-algebras hold also for the generalized ones. E.g., if \(B\) is simple, then a pointwise-norm limit of generalized quasidiagonal extensions is generalized quasidiagonal.
Some \(K\)-theory conditions are given that provide generalized quasidiagonality of extensions of a commutative \(C^*\)-algebra by a continuous-scale \(C^*\)-algebra.

MSC:

46L05 General theory of \(C^*\)-algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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References:

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