Geodesic
The global dimension theorem for non-unital and certain other separable $C^*$-algebras
Sbornik. Mathematics, Tome 186 (1995) no. 9, pp. 1223-1239

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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. 
@article{SM_1995_186_9_a0,
     author = {O. Yu. Aristov},
     title = {The global dimension theorem for non-unital and certain other separable $C^*$-algebras},
     journal = {Sbornik. Mathematics},
     pages = {1223--1239},
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     volume = {186},
     number = {9},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_9_a0/}
}
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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/