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PERIODIC
$k$-GRAPH ALGEBRAS REVISITED
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 99 / Issue 2 / October 2015
- Published online by Cambridge University Press:
- 20 April 2015, pp. 267-286
- Print publication:
- October 2015
-
- Article
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- You have access
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Let
$P$ be a finitely generated cancellative abelian monoid. A 
$P$-graph 
${\rm\Lambda}$ is a natural generalization of a 
$k$-graph. A pullback of 
${\rm\Lambda}$ is constructed by pulling it back over a given monoid morphism to 
$P$, while a pushout of 
${\rm\Lambda}$ is obtained by modding out its periodicity, which is deduced from a natural equivalence relation on 
${\rm\Lambda}$. One of our main results in this paper shows that, for some 
$k$-graphs 
${\rm\Lambda}$, 
${\rm\Lambda}$ is isomorphic to the pullback of its pushout via a natural quotient map, and that its graph 
$\text{C}^{\ast }$-algebra can be embedded into the tensor product of the graph 
$\text{C}^{\ast }$-algebra of its pushout and 
$\text{C}^{\ast }(\text{Per}\,{\rm\Lambda})$. As a consequence, in this case, the cycline algebra generated by the standard generators corresponding to equivalent pairs is a maximal abelian subalgebra, and there is a faithful conditional expectation from the graph 
$\text{C}^{\ast }$-algebra onto it.
