[Paper Review] Purely atomic representations of higher-rank graph C*-algebras

Carla Farsi, Elizabeth Gillaspy+3 more|arXiv (Cornell University)|Jun 10, 2018

Advanced Operator Algebra Research43 references3 citations

TL;DR

This paper develops the theory of purely atomic and permutative representations for $C^*$-algebras of row-finite, source-free higher-rank graphs. It establishes unitary equivalence criteria for purely atomic representations, characterizes irreducibility via projection-valued measures, and proves that purely atomic representations supported on aperiodic orbits decompose into direct sums of permutative representations with injective encoding maps—extending results from Cuntz algebras to the higher-rank setting.

ABSTRACT

We study purely atomic representations of C*-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we give necessary and sufficient conditions for a purely atomic representation to be irreducible in terms of the associated projection valued measure. We also investigate the relationship between purely atomic representations, monic representations and permutative representations, and we describe when a purely atomic representation admits a decomposition consisting of permutative representations.

Motivation & Objective

• To develop a comprehensive theory of purely atomic representations for $C^*$-algebras of row-finite, source-free higher-rank graphs.
• To characterize when two purely atomic representations are unitarily equivalent.
• To determine necessary and sufficient conditions for a purely atomic representation to be irreducible using its associated projection-valued measure.
• To clarify the relationship between purely atomic, monic, and permutative representations in the context of $k$-graphs.
• To establish conditions under which a purely atomic representation decomposes into a direct sum of permutative representations.

Proposed method

• The paper uses projection-valued measures $P$ on the infinite path space $\Lambda^\infty$ associated with each representation $\pi$ of $C^*(\Lambda)$, which encode the spectral structure of the representation.
• It characterizes purely atomic representations via the support of the projection-valued measure $P$, focusing on representations supported on countable sets of infinite paths.
• The authors analyze the structure of permutative representations by constructing orthonormal bases indexed by paths and their shifts, using the action of the shift map $\sigma^j$ on infinite paths.
• They define encoding maps $E: I \to \Lambda^\infty$ from orthonormal bases to infinite paths, and show that injectivity of $E$ on subspaces implies permutative structure.
• The decomposition of representations is achieved by decomposing the support of $P$ into orbits of aperiodic paths, leveraging the uniqueness of factorizations in $k$-graphs.
• The proof of decomposition relies on constructing orthonormal bases $\{e_{\gamma,\ell}\}$ for each cyclic subspace $\mathcal{H}_\ell$, indexed by $\ell \in \mathcal{J}$, and showing that $\mathcal{H} = \bigoplus_{\ell \in \mathcal{J}} \mathcal{H}_\ell$.

Experimental results

Research questions

• RQ1 When are two purely atomic representations of a higher-rank graph $C^*$-algebra unitarily equivalent?
• RQ2 What conditions on the projection-valued measure ensure that a purely atomic representation is irreducible?
• RQ3 How do purely atomic representations relate to monic and permutative representations in the $k$-graph setting?
• RQ4 Under what conditions can a purely atomic representation be decomposed into a direct sum of permutative representations?
• RQ5 When does a purely atomic representation supported on a single orbit of an aperiodic path become permutative with injective encoding?

Key findings

• Two purely atomic representations of $C^*(\Lambda)$ are unitarily equivalent if and only if their associated projection-valued measures are equivalent as measures on $\Lambda^\infty$.
• A purely atomic representation is irreducible if and only if its projection-valued measure is supported on a single orbit of an aperiodic path.
• Every purely atomic representation supported on a single orbit of an aperiodic path is permutative and admits a decomposition into a direct sum of permutative representations with injective encoding maps.
• The decomposition of a purely atomic representation into permutative components is achieved by indexing the orthonormal basis via the orbit structure of an aperiodic path $\omega$ and the shift action $\sigma^j(\omega)$.
• The encoding map $E: I \to \Lambda^\infty$ is injective when restricted to each cyclic subspace $\mathcal{H}_\ell$, which ensures the permutative structure of the representation.
• The construction of orthonormal bases $\{e_{\gamma,\ell}\}$ via $t_a t^*_{\omega(0,j)} e_{\omega,\ell}$ confirms that the representation is unitarily equivalent to a permutative representation with injective encoding.