Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0102213

A functorial approach to the C*-algebras of a graph

Jack Spielberg
View original source ↗Read source document ↗Export metadata ↓

Abstract

A functor from the category of directed trees with inclusions to the category of commutative C*-algebras with injective *-homomorphisms is constructed. This is used to define a functor from the category of directed graphs with inclusions to the category of C*-algebras with injective *-homomorphisms. The resulting C*-algebras are identified as Toeplitz graph algebras. Graph algebras are proved to have inductive limit decompositions over any family of subgraphs with union equal to the whole graph. The construction is used to prove various structural properties of graph algebras.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jack Spielberg. 2001-10-05. A functorial approach to the C*-algebras of a graph. https://arxiv.org/abs/math/0102213

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Set-theoretic absoluteness for analysts and separable \cstar-algebras without Choice

Basic theory of separable C*-algebras can be developed without the Axiom of Choice, and it does not depend on the Continuum Hypothesis, Martin's Axiom, and other standard set-theoretic assumptions. This can be proved in two ways. First, by showing that the standard proofs do not require Choice. Second, by utilizing set-theoretic absoluteness theorems. We provide an introduction to projective complexity and absoluteness for analysts. We also give some limiting examples consistent with ZF, such as a commutative \cstar-algebra concretely represented on a Hilbert space but not isomorphic to $C(X)$ for any compact Hausdorff space $X$ and whose state space is not compact and has no extreme points.

math.OA↗
arXiv2026

Schur Multipliers with Unequal Operator and Completely Bounded Norms on $S_p$, $1<p\ne 2<\infty $

For every $1<p\neq2<\infty$, we exhibit an explicit Schur multiplier on $S_p$ whose operator norm is strictly smaller than its completely bounded norm. More precisely, for each such $p$, we construct a finitely supported Schur symbol $m_p$ such that \[ \|M_{m_p}\|_{p\to p} < \|M_{m_p}\|_{\mathrm{cb},p}. \] This answers the question raised by Lafforgue and de la Salle in 2011 after their Conjecture~1.8 and gives an affirmative answer to Statement~2 in Section~5 of Caspers and Wildschut (2019). In particular, it disproves Statement~3 of Caspers and Wildschut (2019) throughout the same range of exponents.

math.OA↗
arXiv2026

Stationary states on a $C^*$-algebra for an inner action

We study stationary states for actions of countable discrete groups on unital separable $C^*$-algebras. We prove that the stationary state space associated with an inner action of a subgroup of the unitary group that generates the algebra is a Choquet simplex. We also give a locality criterion covering Bernoulli shifts. The simplex structure yields a canonical decomposition of stationary states into tracial and purely nontracial parts. For inner actions, we characterize extreme stationary states by factoriality of their GNS von Neumann algebras. We further show that a stationary state is tracial if and only if its GNS von Neumann algebra is finite, and that it is purely nontracial if and only if this algebra is of type~$\mathrm{III}$. As applications, for $2\leq d\leq\infty$ the stationary state simplex of $C^*(\mathbb{F}_d)$ has a Poulsen face, while for a nontrivial property~$(T)$ group the stationary state simplex of $C^*(Γ)$ is not Poulsen. Finally, we give a sufficient spectral-gap condition for $S_μ(A)$ to be a Bauer simplex and construct a family $(A_d,Γ_d,μ_d)$ satisfying this condition.

math.OA↗