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Abstract: Let $E$ be a row-finite quiver and let $E 0$ be the set of vertices of $E$;consider the adjacency matrix $N- E=n {ij}\in\Z^{E 0\times E 0}$,$n {ij}=#\{$ arrows from $i$ to $j\}$. Write $N^t E$ and 1 for the matrices$\in \Z^{E 0\times E 0\setminus\SinkE}$ which result from $N-^t E$ and fromthe identity matrix after removing the columns corresponding to sinks. Weconsider the $K$-theory of the Leavitt algebra $L RE=L \ZE\otimes R$. Weshow that if $R$ is either a Noetherian regular ring or a stable $C^*$-algebra,then there is an exact sequence $n\in\Z$ \K nR^{E 0\setminus\SinkE}\stackrel{1-N E^t}{\longrightarrow}K nR^{E 0}\to K nL RE\to K {n-1}R^{E 0\setminus\SinkE} \ We alsoshow that for general $R$, the obstruction for having a sequence as above ismeasured by twisted nil-$K$-groups. If we replace $K$-theory by homotopyalgebraic $K$-theory, the obstructions dissapear, and we get, for every ring$R$, a long exact sequence \KH nR^{E 0\setminus\SinkE}\stackrel{1-N E^t}{\longrightarrow}KH nR^{E 0}\toKH nL RE\to KH {n-1}R^{E 0\setminus\SinkE} \ We also compare, for a$C^*$-algebra $\fA$, the algebraic $K$-theory of $L \fAE$ with thetopological $K$-theory of the Cuntz-Krieger algebra $C^* \fAE$. We show thatthe map \ K nL \fAE\to K^{\top} nC^* \fAE \ is an isomorphism if$\fA$ is stable and $n\in\Z$, and also if $\fA=\C$, $n\ge 0$, $E$ is finitewith no sinks, and $\det1-N E^t e 0$.

Author: Pere Ara, Miquel Brustenga, Guillermo Cortiñas

Source: https://arxiv.org/

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