Centers of Cuntz-Krieger \(C^\ast\)-algebras. (English) Zbl 1377.46047
The central topic of this paper is to determine the center of a graph \(C^*\)-algebra \(C^*(E)\) over a finite graph \(E\); here, the authors use the term Cuntz-Krieger \(C^*\)-algebra to name graph \(C^*\)-algebras over row-finite graphs, in contrast with the common use of this term in the literature to describe graph \(C^*\)-algebras over finite graphs with neither sources nor sinks (see, e.g., [S. E. Arklint and E. Ruiz, Trans. Am. Math. Soc. 367, No. 11, 7595–7612 (2015; Zbl 1343.46052)]).
To deal with this problem, the authors choose a different approach to that of J. Cuntz [Commun. Math. Phys. 57, 173–185 (1977; Zbl 0399.46045)] (where the description relies directly on properties of the underlying graph), or that of B. Steinberg [Adv. Math. 223, No. 2, 689–727 (2010; Zbl 1188.22003)] (where the description is given in terms of “functions” defined over the graph groupoid), by following a strategy similar to that used in their joint work [Bull. Math. Sci. 6, No. 1, 145–161 (2016; Zbl 1344.16004)].
In order to state the main result in the paper under review, we will need to fix some definitions. Given a graph \(E\), we will denote \(V:=E^0\) the set of vertices of \(E\). Then:
Definition 1.3. Let \(W\subset V\) be a nonempty subset. We say that a path \(p=e_1e_2\dots e_n\), with \(e_i\in E^1\) for every \(1\leq i\leq n\), is an arrival path in \(W\) if \(r(p)\in W\) and \(\{s(e_i)\}_{i=1}^n \cap W=\emptyset\). Let Arr(\(W\)) denote the set of arrival paths in \(W\).
Definition 1.4. A hereditary set \(W\subseteq V\) is called finitary if \(|\operatorname{Arr}(C) | < \infty\).
If \(W\) is a hereditary finitary subset of \(V\) then \(e(W):=\sum_{p\in \operatorname{Arr}(C) }pp^*\). Also, if \(C\) is a cycle with no exits in \(E\), we will denote by \(Z(C)\) the center of the subalgebra \(\text{CK}(C)\) of \(\text{CK}(E)\).
With that in mind, we can state the main result of the paper:
Theorem 1.5. Let \(E\) be a finite graph. Then the center \(Z(\text{CK}(E))\) is spanned by:
Corollary 1.6. \(Z(\text{CK}(E))\) is the closure of \(Z(L_{\mathbb{C}}(E))\).
To deal with this problem, the authors choose a different approach to that of J. Cuntz [Commun. Math. Phys. 57, 173–185 (1977; Zbl 0399.46045)] (where the description relies directly on properties of the underlying graph), or that of B. Steinberg [Adv. Math. 223, No. 2, 689–727 (2010; Zbl 1188.22003)] (where the description is given in terms of “functions” defined over the graph groupoid), by following a strategy similar to that used in their joint work [Bull. Math. Sci. 6, No. 1, 145–161 (2016; Zbl 1344.16004)].
In order to state the main result in the paper under review, we will need to fix some definitions. Given a graph \(E\), we will denote \(V:=E^0\) the set of vertices of \(E\). Then:
Definition 1.3. Let \(W\subset V\) be a nonempty subset. We say that a path \(p=e_1e_2\dots e_n\), with \(e_i\in E^1\) for every \(1\leq i\leq n\), is an arrival path in \(W\) if \(r(p)\in W\) and \(\{s(e_i)\}_{i=1}^n \cap W=\emptyset\). Let Arr(\(W\)) denote the set of arrival paths in \(W\).
Definition 1.4. A hereditary set \(W\subseteq V\) is called finitary if \(|\operatorname{Arr}(C) | < \infty\).
If \(W\) is a hereditary finitary subset of \(V\) then \(e(W):=\sum_{p\in \operatorname{Arr}(C) }pp^*\). Also, if \(C\) is a cycle with no exits in \(E\), we will denote by \(Z(C)\) the center of the subalgebra \(\text{CK}(C)\) of \(\text{CK}(E)\).
With that in mind, we can state the main result of the paper:
Theorem 1.5. Let \(E\) be a finite graph. Then the center \(Z(\text{CK}(E))\) is spanned by:
- (i)
- \(e(W)\), where \(W\) runs over all nonempty hereditary finitary subsets \(W\) of \(V\),
- (ii)
- subspaces \(\left\{ \sum_{p\in \operatorname{Arr}(C)}pzp^* : z\in Z(C)\right\}\), where \(C\) runs over all cycles with no exits \(C\) such that \(V(C)\) (its set of vertices) is finitary.
Corollary 1.6. \(Z(\text{CK}(E))\) is the closure of \(Z(L_{\mathbb{C}}(E))\).
Reviewer: Enrique Pardo Espino (Cádiz)
MSC:
| 46L55 | Noncommutative dynamical systems |
| 16U70 | Center, normalizer (invariant elements) (associative rings and algebras) |
References:
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