Operator Algebras and Self-Similar Actions

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Operator algebras are algebras generated by a family of continuous linear transformations on Hilbert spaces. They are originally from quantum physics. They are now closely related to other areas, such as quantum computing, quantum information, signal processing, representation theory and differential geometry. They are playing increasingly important roles in modern mathematics. Intuitively speaking, an object is called self-similar if its parts are similar to the whole. So self-similarities are naturally related to symmetries appearing in fractal geometry and dynamics. Usually, they are encoded by semigroups/inverse semigroups/groupoids induced from self-similarities, which are isomorphisms between the parts of given self-similar objects on different scales. Self-similar actions have been attracting a great deal of attention in the fields of geometric group theory and operator algebras. My proposal mainly focuses on the interplay between operator algebras and self-similar actions. This has deep connections with a lot of hot and interesting topics, such as higher-rank graph algebras, self-similar actions, and the Yang-Baxter equation. To be more precise, given a higher-rank graph and a group, suppose that the group acts on the higher-rank graph self-similarly. To this self-similar action, one then can associate some important operator algebras, such as the Toeplitz type C*-algebra and the Cuntz-Pimsner-type C*-algebra. In particular, the Cuntz-Pimsner type C*-algebra, by definition, is the universal C*-algebra with the following properties: There are a Cuntz-Krieger representation of the higher-rank graph and a unitary representation of the group, such that these two representations are naturally compatible with respect to the given self-similar higher-rank graph action. Even in the case of rank-1 graph actions, those C*-algebras include many well-known and important classes, such as Katsura algebras (which exhaust all Kirchberg algebras) and Nekrashevych C*-algebras (which are first operator algebras essentially induced from self-similar actions). It is also a fact that higher-rank graph algebras are much more complicated than rank-1 graph algebras. Therefore, it would be no surprise at all that those operator algebras coming from self-similar higher-rank graph actions embrace more interesting, important classes of operator algebras, and are also much more sophisticated than those from self-similar graph actions. On one hand, the case of rank-1 graph self-similar actions will give us some intuitions and motivations to our development of the higher-rank case. On the other hand, what we will gain from working on the higher-rank case will also give us a better understanding of the rank-1 case. My proposal is an initiative to the study of self-similar higher-rank graph actions, and will open a lot of interesting and important topics which are worth studying in the future.