Geodesic
The global dimension theorem for non-unital and certain other separable $C^*$-algebras
Sbornik. Mathematics, Tome 186 (1995) no. 9, pp. 1223-1239 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this article we prove that the global homological dimension of a separable $C^*$-algebra containing a bi-ideal of finite codimension that cannot be complemented as a subalgebra is at least 2. As a consequence we also obtain this bound for the global dimension of separable $C^*$-algebras without an identity and for finite-dimensional separable $\operatorname{GCR}$-algebras. 
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O. Yu. Aristov. The global dimension theorem for non-unital and certain other separable $C^*$-algebras. Sbornik. Mathematics, Tome 186 (1995) no. 9, pp. 1223-1239. http://geodesic.mathdoc.fr/item/SM_1995_186_9_a0/

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