Inductive limits of finite dimensional hermitian symmetric spaces and K-theory

TL;DR

This paper explores the use of K-theory to analyze inductive limits of finite dimensional hermitian symmetric spaces, demonstrating the power of homological methods over classical approaches and adapting invariants for better insights.

Contribution

It extends K-theory techniques to inductive limits of hermitian symmetric spaces, providing a homological perspective and refining invariants for improved analysis.

Findings

K-theory offers a powerful homological framework for symmetric spaces.

Behavior of morphisms between bounded symmetric domains is more complex than in C*-algebras.

Modified invariants using traces from co-root lattices enhance the analysis.

Abstract

K-Theory for hermitian symmetric spaces of non-compact type, as developed recently by the authors, allows to put Cartan's classification into a homological perspective. We apply this method to the case of inductive limits of finite dimensional hermitian symmetric spaces. This might be seen as an indication of how much more powerful the homological theory is in comparison to the more classical approach. When seen from high above, we follow the path laid out by a similar result in the theory of C*-algebras. Important is a clear picture of the behavior of morphisms between bounded symmetric domains of finite dimensions, which is more complex than in the C*-case, as well as an accessible K-theory. We furthermore have to slightly modify the invariant from our previous work. Roughly, we use traces left by co-root lattices on K-groups, instead of co-roots themselves, which had been used previously.