Assume that $R$ is a unital ring or a complex or real (Banach or $C^{*}$) algebra.
We define a relation $M$ on $R$ as follows: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for some}\;\; x,y\in R$$ It is a reflexive and symmetric (but not transitive ) relation. We define an equivalent relation $\simeq$ on $R$ as follows:
$a\simeq b$ if there are $p_{i}\in R,\;i=0,1,\ldots,n$ with $$\begin{cases}a=p_{0},\;\; b=p_{n},&\\ p_{i} \; M\; p_{i+1}\end{cases}$$
The space of nilpotent elements of $R$, denoted by $N(R)$, is a saturated subset of $R$ while the space of idempotent elements is not necessarily a saturated subset.
Notation: $M_{n}(R)$ is the space of $n\times n$ matrices with entries in $R$.
The natural mapping $M_{n}(R) \to M_{n+1} (R)$ with $A \mapsto A\oplus 0$ sends nilpotent elements to nilpotent elements. Moreover the above equivalent relation is preserved under this map.
We consider $\bigcup_{n=1}^{\infty} N(M_{n}(R))$, the union of all nilpotent matrices of all size. The equivalent relation $\simeq$ has a natural extension to the later space: $A\simeq B$ if there are natural numbers $k,p$ such that $A\oplus 0_{k} \simeq B\oplus 0_{p}$. The later are the zero matrices of size $k,p$, respectively.
This enable us to equip $\bigcup_{n=1}^{\infty} N(M_{n}(R))/\simeq$ to an Abelian semi group structure, via the usual matrix-sum.(Note that for two nilpotent elements $a,b$ with $ab=ba=0$, $a+b$ is again a nilpotent element. On the other hand every two elements of the above quotient space have two representation $A,B$ with $AB=BA=0$). Because, for every $a\in R$ we have $\begin{pmatrix} a&0\\0&0 \end{pmatrix} \simeq \begin{pmatrix} 0&0\\0&a \end{pmatrix}$.
The Grothendick group of this semi group is denoted by $NK(R)$.
Questions:
What is an example of a $C^{*}$ algebra $A$ for which $NK(A)$ is a non trivial group? Is there a commutative $C^{*}$ algebra $A$ with nontrivial $NK(A)$.
Note 1 The mapping $A\mapsto NK(A)$ is realy a functor on the category of rings or algebra. according to Gelfand Naimark duality this could be counted as a functor on the category of compact Hausdorff topological space
Note 2: This post is inspired by the construction in algebraic K theory and the following two posts
