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Let $H$ be an infinite dimensional separable Hilbert space. The set $K(H)$ of all compact operators is a non-unital nuclear $C^*$-algebra which has no tracial states and the multiplier algebra of $K(H)$ is also traceless.

My question: do there exist other concrete non-unital nuclear $C^*$-algebras without tracial states such that their multiplier algebras are also traceless?

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    $\begingroup$ Try to think about what happens if you take any such example and tensor it with a C*-algebra. $\endgroup$
    – Jamie Gabe
    Commented Jan 10, 2022 at 13:43
  • $\begingroup$ You can try twisted tensors also like ergodic group actions on measured space for example. $\endgroup$
    – InfiniteLooper
    Commented Jan 10, 2022 at 13:44
  • $\begingroup$ @Jamie Gabe，you mean $K(H)\otimes A$? $\endgroup$
    – math112358
    Commented Jan 11, 2022 at 16:22
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    $\begingroup$ I think you should try to work this out on your own. What would happen if $K(H)\otimes A$ or its multiplier algebra had a tracial state? Or if $B$ is such a nuclear $C^\ast$-algebra, what would happen if $A\otimes B$ or its multiplier algebra had a tracial state? $\endgroup$
    – Jamie Gabe
    Commented Jan 11, 2022 at 16:47
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    $\begingroup$ Hmmm, what seems harder to come by is examples of traceless A where M(A) has tracial states. I can only think of a very convoluted example. $\endgroup$
    – Leonel Robert
    Commented Jan 12, 2022 at 18:40

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