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Marius Ionescu

Publications and source records attributed to Marius Ionescu.

26 records · Page 2Linked to original sources
arXiv2009

Remarks on the Ideal Structure of Fell Bundle C*-Algebras

We show that if $p:\B\to G$ is a Fell bundle over a locally compact groupoid $G$ and that $A=Γ_{0}(G^{(0)};\B)$ is the \cs-algebra sitting over $G^{(0)}$, then there is a continuous $G$-action on $\Prim A$ that reduces to the usual action when $\B$ comes from a dynamical system. As an application, we show that if $I$ is a $G$-invariant ideal in $A$, then there is a short exact sequence of \cs-algebras \xymatrix{0\ar[r]&\cs(G,\BI)\ar[r] &\cs(G,\B)\ar[r]&\cs(G,\BqI)\ar[r]&0,} where $\cs(G,\B)$ is the Fell bundle \cs-algebra and $\BI$ and $\BqI$ are naturally defined Fell bundles corresponding to $I$ and $A/I$, respectively. Of course this exact sequence reduces to the usual one for \cs-dynamical systems.

math.OA↗
arXiv2009

The resolvent kernel for PCF self-similar fractals

For the Laplacian $Δ$ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, and also with Neumann boundary conditions. That is, we construct a symmetric function $G^{(λ)}$ which solves $(λ\mathbb{I} - Δ)^{-1} f(x) = \int G^{(λ)}(x,y) f(y) dμ(y)$. The method is similar to Kigami's construction of the Green kernel in \cite[\S3.5]{Kig01} and is expressed as a sum of scaled and "translated" copies of a certain function $ψ^{(λ)}$ which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket $SG_3$.

math.AP↗
arXiv2008

The Generalized Effros-Hahn Conjecture for Groupoids

The generalized Effros-Hahn conjecture for groupoid C*-algebras says that, if G is amenable, then every primitive ideal of the groupoid C*-algebra C*(G) is induced from a stability group. We prove that the conjecture is valid for all second countable amenable locally compact Hausdorff groupoids. Our results are a sharpening of previous work of Jean Renault and depend significantly on his results.

math.OA↗
arXiv2005

$C^{\ast}$-Algebras associated with Mauldin-Williams Graphs

A Mauldin-Williams graph $\mathcal{M}$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a $C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ with a Mauldin-Williams graph $\mathcal{M}$ and the invariant set $K$ laying emphasis on the singular points. We assume that the underlying graph $G$ has no sinks and no sources. If $\mathcal{M}$ satisfies the open set condition in $K$ and $G$ is irreducible and is not a cyclic permutation, then the associated $C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ is simple and purely infinite. We calculate the K-groups for some examples including the inflation rule of the Penrose tilings.

math.OA↗
arXiv2004

Mauldin-Williams graphs, Morita Equivalence and Isomorphisms

We describe a method for associating some non-self-adjoint algebras to Mauldin-Williams graphs and we study the Morita equivalence and isomorphism of these algebras. We also investigate the relationship between the Morita equivalence and isomorphism class of the C^{\ast} -correspondences associated with Mauldin-Williams graphs and the dynamical properties of the Mauldin-Williams graphs.

math.OA↗
arXiv2004

Operator Algebras and Mauldin Williams Graphs

We describe a method for associating a $C^{*}$-correspondence to a Mauldin-Williams graph and show that the Cuntz-Pimsner algebra of this $C^{*}$-correspondence is isomorphic to the $C^{*}$-algebra of the underlying graph. In addition, we analyze certain ideals of these $C^{*}$-algebras. We also investigate Mauldin-Williams graphs and fractal $C^{*}$-algebras in the context of the Rieffel metric. This generalizes the work of Pinzari, Watatani and Yonetani. Our main result here is a {}``no go'' theorem showing that such algebras must come from the commutative setting.

math.OA↗