$K$-theory of Leavitt path algebras

Abstract

Let $E$ be a row-finite quiver and let $E_0$ be the set of vertices of $E$; consider the adjacency matrix $N_E^{\prime }=(n_{ij})\in {\mathbb{Z}}^{(E_0\times E_0)}$, $n_{ij}=#\{$ arrows from $i$ to $j\}$. Write $N_E^t$ and 1 for the matrices $\in \Z^{(E_0\times E_0\setminus\Sink(E))}$ which result from $N_E^{\prime t}$ and from the identity matrix after removing the columns corresponding to sinks. We consider the $K$-theory of the Leavitt algebra $L_R(E)=L_{\mathbb{Z}}(E)\otimes R$. We show that if $R$ is either a Noetherian regular ring or a stable $C^{\ast }$-algebra, then there is an exact sequence ($n\in \mathbb{Z}$) \[ K_n(R)^{(E_0\setminus\Sink(E))}\stackrel{1-N_E^t}{\longrightarrow} K_n(R)^{(E_0)}\to K_n(L_R(E))\to K_{n-1}(R)^{(E_0\setminus\Sink(E))} \] We also show that for general $R$, the obstruction for having a sequence as above is measured by twisted nil-$K$-groups. If we replace $K$-theory by homotopy algebraic $K$-theory, the obstructions dissapear, and we get, for every ring $R$, a long exact sequence \[ KH_n(R)^{(E_0\setminus\Sink(E))}\stackrel{1-N_E^t}{\longrightarrow}KH_n(R)^{(E_0)}\to KH_n(L_R(E))\to KH_{n-1}(R)^{(E_0\setminus\Sink(E))} \] We also compare, for a $C^{\ast }$-algebra $\fA$, the algebraic $K$-theory of $L_\fA(E)$ with the topological $K$-theory of the Cuntz-Krieger algebra $C^*_\fA(E)$. We show that the map \[ K_n(L_\fA(E))\to K^{\top}_n(C^*_\fA(E)) \] is an isomorphism if $\fA$ is stable and $n\in \mathbb{Z}$, and also if $\fA=\C$, $n\ge 0$, $E$ is finite with no sinks, and $\det (1-N_E^t)\ne 0$.

30 pages

References in corpus (2)

• Module theory over Leavitt path algebras and $K$-theory
• Vogel's notion of regularity for non-coherent rings

Cited by in corpus (14)

• Leavitt $R$-algebras over countable graphs embed into $L_{2,R}$
• Leavitt path algebras of separated graphs
• Homotopy classification of Leavitt path algebras
• Mixed quiver algebras
• Isomorphism and Morita equivalence of graph algebras
• How to categorify the ring of integers localized at two
• The Cuntz splice does not preserve $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• The graded Grothendieck group and the classification of Leavitt path algebras
• Graded $K$-Theory, Filtered $K$-theory and the classification of graph algebras
• Generalizations of Loday's assembly maps for Lawvere's algebraic theories
• A1-homotopy invariants of corner skew Laurent polynomial algebras
• Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras
• Leavitt path algebras: the first decade
• Lifting graph $C^{\ast }$-algebra maps to Leavitt path algebra maps