arXiv subjects

Mitch Hamidi

Publications and source records attributed to Mitch Hamidi.

3 recordsLinked to original sources

arXiv2025

Simplicity of Cuntz-Pimsner algebras of quantum graphs

Let $\mathcal{G}$ be a quantum graph without quantum sources and $E_\mathcal{G}$ be the quantum edge correspondence for $\mathcal{G}.$ Our main results include sufficient conditions for simplicity of the Cuntz-Pimsner algebra $\mathcal{O}_{E_\mathcal{G}}$ in terms of $\mathcal{G}$ and for defining a surjection from the quantum Cuntz-Krieger algebra $\mathcal{O}(\mathcal{G})$ onto a particular relative Cuntz-Pimsner algebra for $E_\mathcal{G}$. As an application of these two results, we give the first example of a quantum graph with distinct quantum Cuntz-Krieger and local quantum Cuntz-Krieger algebras. We also characterize simplicity of $\mathcal{O}_{E_\mathcal{G}}$ for some fundamental examples of quantum graphs, including rank-one quantum graphs on a single full matrix algebra, complete quantum graphs, and trivial quantum graphs. Along the way, we provide an equivalent condition for minimality of $E_\mathcal{G}$ and sufficient conditions for aperiodicity of $E_\mathcal{G}$ in terms of the underlying quantum graph $\mathcal{G}$.

math.OA↗

arXiv2024

Admissibility of C*-Covers for Operator Algebra Dynamical Systems

We characterize when a C*-cover admits a C*-dynamical extension of dynamics on an operator algebra in terms of the boundary ideal structure for the operator algebra in its maximal representation and show that the C*-covers that admit such an extension form a complete lattice. We study dynamical systems arising from groups acting via inner automorphisms in a C*-cover and produce an example of a C*-cover that admits no extension of dynamics on a finite-dimensional non-self-adjoint operator algebra. We construct a partial action on a class of C*-covers that recovers the crossed product of an operator algebra as a subalgebra of the partial crossed product, even when the C*-cover admits no dynamical extension.

math.OA↗

arXiv2022

Quantum edge correspondences and quantum Cuntz-Krieger algebras

Given a quantum graph $\mathcal{G}=(B,\psi,A)$, we define a C*-correspondence $E_\mathcal{G}$ over the noncommutative vertex C*-algebra $B$, called the quantum edge correspondence. For a classical graph $\mathcal{G}$, $E_\mathcal{G}$ is the usual graph correspondence spanned by the edges of $\mathcal{G}$. When the quantum adjacency matrix $A\colon B\to B$ is completely positive, we show that $E_\mathcal{G}$ is faithful if and only if $\ker(A)$ does not contain a central summand of $B$. In this case, we show that the Cuntz-Pimsner algebra $\mathcal{O}_{E_\mathcal{G}}$ is isomorphic to a quotient of the quantum Cuntz-Krieger algebra $\mathcal{O}(\mathcal{G})$ defined by Brannan, Eifler, Voigt, and Weber. Moreover, the kernel of the quotient map is shown to be generated by "localized" versions of the quantum Cuntz-Krieger relations, and $\mathcal{O}_{E_\mathcal{G}}$ is shown to be the universal object associated to these local relations. We study in detail some concrete examples and make connections with the theory of Exel crossed products.

math.OA↗