$K_0$ groups of noncommutative $\mathbb{R}^{2n}$

TL;DR

This paper proves that the K_0 groups of noncommutative Euclidean spaces are isomorphic to the integers for all positive dimensions and explores the smooth case, linking to binomial number sequences.

Contribution

It establishes the K_0 group structure for noncommutative R^{2n} and initiates the calculation for the smooth case, introducing new sequence problems.

Findings

K_0 groups of noncommutative R^{2n} are isomorphic to Z for all n

Approach to smooth case calculation reveals connections to binomial number sequences

Provides foundational results for noncommutative geometry in Euclidean spaces

Abstract

In this paper we show that the $K_0$ groups of noncommutative $\mathbb{R}^{2n}$ are $\mathbb{Z}$ for $\forall n\in\mathbb{N}^*$ and make an approach to the calculation of the smooth case, which will bring many new sequence problems relating to binomial numbers.