Remarks on computing the Grothendieck rings of C*-algebras

TL;DR

This paper adapts Grothendieck's construction to C*-algebras, exploring conditions for nontrivial Grothendieck rings and analyzing rings over complex, real, and quaternionic fields.

Contribution

It introduces a novel application of Grothendieck rings to C*-algebras and investigates their properties across different base fields.

Findings

Identification of conditions for nontrivial Grothendieck rings in C*-algebras

Analysis of rings over $\

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Abstract

In this paper, we present a captivating construction by Grothendieck, originally formulated for algebraic varieties, and adapt it to the realm of C*-algebras. Our main objective is to investigate the conditions under which this particular class of C*-algebras possesses a nontrivial Grothendieck ring. To achieve this, we explore the existence of nontrivial characters, which significantly enriches our understanding of these algebras. In particular, we conduct a detailed study of rings of C*-algebras over $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{H}$.