Relative Cuntz-Pimsner algebras: a complete description of their lattice of gauge-invariant ideals

Abstract: We give a new, systematic approach to the gauge-invariant uniqueness theorem describing all relative Cuntz-Pimsner algebras, and whence revealing a complete description of their gauge-invariant ideal lattice.

For this we start with a swift introduction to C*-correspondences, in particular drawing a comparison to Fell bundles.

Continuing, we provide a slightly deeper analysis of covariances as well as their relation to kernels and quotients. With these observations at hand, we introduce the relevant reduction leading us to a suitable parametrization of relative Cuntz-Pimsner algebras, and so revealing a complete description of their gauge-invariant ideal lattice. Our parametrization is a heuristic analog of Katsura's originally obtained one.

With this at hand, we arrive at the gauge-invariant uniqueness theorem, for all arbitrary gauge-equivariant representations.

From here we move on to the analysis part of the program. We compute the covariances in the case of the Fock representation and its quotients. As a result, we derive that the parametrization of relative Cuntz-Pimsner algebras introduced above is also classifying. In other words, we obtain a complete and intrinsic picture of the lattice of quotients, and equivalently of their lattice of gauge-invariant ideals.

If time permits, we finish off with the next chapter on their induced Fell bundles and dilations, as already investigated by Schweizer.

functional analysisgroup theoryoperator algebrasquantum algebra

Audience: researchers in the topic

---

Groups, Operators, and Banach Algebras Webinar

Series comments: This is an online seminar series for early career researchers working in group theory, operator theory/operator algebra, and Banach algebras. To be added to our mailing list and receive links to our meetings please email us at gobaseminar@gmail.com.

---