KK-theory for some graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math>-algebras

Germain, Emmanuel; Sarr, Abdoulaye

doi:10.1016/j.crma.2015.02.012

Cited by 3 publications

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Relative nuclearity for C⁎-algebras and KK-equivalences of amalgamated free products

Hasegawa

2015

Journal of Functional Analysis

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We prove a relative analogue of equivalence between nuclearity and CPAP. In its proof, the notion of weak containment for C * -correspondences plays an important role. As an application we prove KK-equivalence between full and reduced amalgamated free products of C * -algebras under a strengthened variant of 'relative nuclearity'. 1 4 KEI HASEGAWA introduced by Jian and Sepideh [JS], as well as relative amenability for von Neumann algebras ([Po][AD3] [OP]). In §5, we see that the examples listed above are actually (strongly) nuclear. §6 is devoted to proving Weyl-von Neumann-Voiculescu type results. Applications in KK-theory are given in the last three sections. In §7 we prove that strong relative nuclearity implies relative K-nuclearity. The proof of Theorem C and Theorem D are given in §8. In the final section, we establish six term exact sequences in KK-theory of reduced amalgamated free products and a KK-equivalence result for HNN-extensions.Notation. We basically follow the notation of Brown and Ozawa's book [BO]. For a C * -algebra A we denote by 1 A the unit of the multiplier algebra M(A) of A. The symbols B(H) and K(H) stand for the set of bounded operators and the set of compact ones on a Hilbert space H, respectively. We use the symbol ⊙ to denote the algebraic tensor product over C. For C *algebras A and B we denote by A ⊗ B and A ⊗ max B the minimal and the maximal tensor products, respectively. For a von Neumann algebra M , we denote by M * the (unique) predual of M . The von Neumann algebraic tensor product of M and another von Neumann algebra N is denoted by M⊗ N . We denote by CP(

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