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For a digraph \(\Gamma \) with vertex set \(\Gamma ^0\) and edge set \(\Gamma ^1\) , there corresponds a \(C^*\) -algebra denoted by \(C^*(\Gamma )\) . In this paper we define a certain inverse subsemigroup \(W_\Gamma \) of the multiplicative semigroup of \(C^*(\Gamma )\) . Its elements are of the form \(S_eS_f^*\) where e, f are edges with common range and \(S_e,S_f\) are the corresponding partial isometries. We observe that \(W_\Gamma \) is canonically embedded into the Leavitt inverse semigroup \(LI(\Gamma )\) . It is known that the Leavitt path algebra \(L_K (\Gamma )\) over a field K is (von Neumann) regular if and only if \(\Gamma \) is acyclic. Here we prove that for any arbitrary column-finite digraph \(\Gamma \) , acyclic or not, the subalgebra of \(L_K(\Gamma )\) generated by \(W_\Gamma \) is always (von Neumann) regular and semisimple.

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On a Canonical Multiplicative Semigroup Related with the  \(C^*\) -Algebra of a Digraph

  • 会议论文

  • 发表日期：

    2025年

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  • Promit Mukherjee,
  • Sujit Kumar Sardar

摘要

For a digraph \(\Gamma \) with vertex set \(\Gamma ^0\) and edge set \(\Gamma ^1\) , there corresponds a \(C^*\) -algebra denoted by \(C^*(\Gamma )\) . In this paper we define a certain inverse subsemigroup \(W_\Gamma \) of the multiplicative semigroup of \(C^*(\Gamma )\) . Its elements are of the form \(S_eS_f^*\) where e, f are edges with common range and \(S_e,S_f\) are the corresponding partial isometries. We observe that \(W_\Gamma \) is canonically embedded into the Leavitt inverse semigroup \(LI(\Gamma )\) . It is known that the Leavitt path algebra \(L_K (\Gamma )\) over a field K is (von Neumann) regular if and only if \(\Gamma \) is acyclic. Here we prove that for any arbitrary column-finite digraph \(\Gamma \) , acyclic or not, the subalgebra of \(L_K(\Gamma )\) generated by \(W_\Gamma \) is always (von Neumann) regular and semisimple.