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Publishing type：Research paper (scientific journal)   Publisher：European Mathematical Society - EMS - Publishing House GmbH  

We show that a not necessarily closed ideal in a C^{*} -algebra is semiprime if and only if it is idempotent, if and only if it is closed under square roots of positive elements. Among other things, it follows that prime and semiprime ideals in C^{*} -algebras are automatically self-adjoint. To prove the above, we isolate and study a particular class of ideals, which we call Dixmier ideals . As it turns out, there is a rich theory of powers and roots for Dixmier ideals. We show that every ideal in a C^{*} -algebra is squeezed by Dixmier ideals from inside and outside tightly in a suitable sense, from which we are able to deduce information about the ideal in the middle.

DOI： 10.4171/jems/1699

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Publishing type：Research paper (scientific journal)   Publisher：Theta Foundation  

DOI： 10.7900/jot.2023ian21.241

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Publishing type：Research paper (scientific journal)   Publisher：MathDoc/Centre Mersenne  

In this short note, quantum subgroups in finite free products of the Pontryagin duals of free unitary quantum groups are classified. They correspond to pairs of a subgroup and a subset of the free group such that is -invariant, containing , and connected in the Cayley graph of .

DOI： 10.5802/crmath.641

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Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.aim.2024.109848

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Publishing type：Research paper (scientific journal)   Publisher：Oxford University Press (OUP)  

Abstract

We classify discrete quantum subgroups in the quantum double of the $q$-deformation of a compact semisimple Lie group, regarded as the complexification. We also record their classifications in some variants of quantum groups. Along the way, we show that quantum doubles of non-Kac-type compact quantum groups do not admit the quantum analog of lattices considered by Brannan–Chirvasitu–Viselter.

DOI： 10.1093/imrn/rnad117

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