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Completions of Leavitt path algebras. (English) Zbl 1344.16004

The authors define a class of topologies on a Leavitt path algebra \(L=L_K(E)\) over a field \(K\). As usual \(E\) will denote a graph \(E=(E^0,E^1,r,s)\). By using the notion of a “specialization” \(\gamma\colon E^0\setminus\{\text{sinks}\}\to E^1\), the edges in the image of \(\gamma\) are called “special”. The set \(B(\gamma)\) is defined as the one whose elements are \(pq^*\) with \(p,q\in\text{path}(E)\), \(r(p)=r(q)\) and the restriction: if the last edge of \(p\) agrees with the last edge of \(q\), then this edge is not special. This set \(B(\gamma)\) is known to be a basis of \(L\). A path is said to be special if all its edges are. Also, for an arbitrary path \(p=e_1\cdots e_n\), let \(i\) be the minimal integer such that \(e_{i+1}\cdots e_n\) is special (and if \(e_n\) is not special then \(i:=n\)), then \(n-i\) is called the special degree of \(p\) and the notation \(\text{sd}(p)=n-i\) is used. Now for non-negative integers \(n,s,d\) denote by \(V_{n,s,d}\) the subspace of \(L\) generated by all products \(pq^*\) where \(p,q\in\text{path}(E)\) with \(l(p)+l(q)\geq n\), \(\text{sd}(p)+\text{sd}(q)\leq s\), \(\deg(pq^*)\leq d\). Next, for a positive \(k\) define \(V_k:=\sum V_{n,s,d}\) extended to all triples satisfying \(n\geq k(s+d+1)\). These subspaces satisfy certain nice multiplicative relations that make them suitable to define a topology in which they are a basis of neighborhoods of \(0\). This topology in compatible with the algebra structure of \(L\) so that we may consider the completion \(\overline L\). Denote by \(\overline L_i\) the completion of the \(i\)th-homogeneous component of \(L\) and by \(\widehat L\) the sum \(\widehat L=\sum_{i\in\mathbb Z}\overline L_i\).
The paper is devoted to the study of this graded completion \(\widehat L\). It is proved that, if the graph is finite, the algebra \(\widehat L\) has a simpler structure than that of the Leavitt path algebra \(L\). In fact \(\widehat L\) is semisimple in the sense that it can be written as a sum of minimal graded ideals (each one of which is a topologically simple algebra).
It is worth to mention that, as the authors point out in the paper, the graded completion opens the possibility of defining certain infinite sums which in \(L\) has no sense, but that can be formally defined in \(\widehat L\). These infinite sums are very relevant in the study of the center of a Leavitt path algebra (on the other hand, a description of the center of \(L_R(E)\) where \(R\) is a (commutative) ring, and \(E\) a graph without finiteness conditions, is given in [L. O. Clark et al., Using the Steinberg algebra model to determine the center of any Leavitt path algebra. arXiv:1604.01079]).

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16S88 Leavitt path algebras
16W70 Filtered associative rings; filtrational and graded techniques
16W80 Topological and ordered rings and modules
22A22 Topological groupoids (including differentiable and Lie groupoids)
16G20 Representations of quivers and partially ordered sets
46L05 General theory of \(C^*\)-algebras
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References:

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