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Published online by Cambridge University Press: 10 February 2020
In the remarkable paper where he proved the equivalence, Kirchberg studied more generally the pairs of C*-algebras(A,B) admitting only one C*-norm on their algebraic tensor product.We call such pairs "nuclear pairs''. A C*-algebra A istraditionally called nuclear if this holds for any C*-algebra B. Our exposition chooses as its cornerstone Kirchberg's theoremasserting the nuclearity of what is for us the "fundamental pair'', namely the pair (B,C)where B is the algebra B of bounded operators on Hilbert space and C isthe full group C*-algebra C of the free group with countably infinitely many generators. Our presentation leads us to highlight two properties of C*-algebras, the Weak Expectation Property (WEP) and the Local Lifting Property (LLP).
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