arXiv subjects

Nikolaos Koutsonikos-Kouloumpis

Publications and source records attributed to Nikolaos Koutsonikos-Kouloumpis.

4 recordsLinked to original sources

arXiv2026

Morita equivalence for quantum graphs

We introduce an operator-algebraic framework for Morita equivalence of quantum graphs based on $Δ$-equivalence of operator systems introduced by Eleftherakis, Kakariadis and Todorov. Adopting the perspective of Weaver, we view quantum graphs as quantum relations, that is, operator systems endowed with a bimodule structure over the commutant of a von Neumann algebra. Within this framework, we show that two irreducibly acting quantum graphs are Morita equivalent if and only if they are both full pullbacks of a common quantum graph. This extends a result of Eleftherakis, Kakariadis and Todorov for graph operator systems to the quantum graph setting. In passing we construct a true-twin reduction analogue for an irreducibly acting quantum graph. We further characterise the case where we have simultaneous TRO-equivalence of the quantum graphs and their associated algebras, thus giving a second, stronger notion of Morita equivalence. In the special case of noncommutative graphs, corresponding to the zero-error quantum communication setting, the two notions coincide and we obtain a characterisation in terms of strong co-homomorphisms of noncommutative graphs. Finally, we show that connectivity, the independence number, Shannon capacity, quantum complexity and subcomplexity, Haemers bound, and the Lovász number are invariant under Morita equivalence.

math.OA↗

arXiv2025

Morita equivalence and stable isomorphism via unitary operators

We define $Δ$-equivalence for dual operator systems and prove that it is an equivalence relation. We show that weak TRO-equivalence of dual operator spaces induces a stable isomorphism between them which is given by multiplication with unitary operators, and in the special case of dual operator systems it is a unitary equivalence. We prove an analogous result for strong TRO-equivalence of operator spaces and for operator systems. Lastly, we show that $Δ$-equivalent dual operator spaces, considered as bimodules over their left and right adjointable multiplier algebras, have TRO-equivalent normal CES representations.

math.OA↗

arXiv2025

Unital operator spaces and discrete groups

We introduce the trivial intersection property for concrete operator spaces and we show that a unital space with this property has no nontrivial boundary ideals. We provide various examples of such spaces, among which are strongly reflexive masa bimodules and completely distributive CSL algebras. We show that unital operator spaces acting on $\ell^2(Γ)$ for any set $Γ$, that contain the masa $\ell^\infty(Γ)$, possess the trivial intersection property, and we use this to prove that a unital surjective complete isometry between such spaces is a unitary equivalence. Then, we apply these results to $w^*$-closed $\ell^\infty(G)$-bimodules acting on $\ell^2(G)$ for a group $G$ and we relate them to algebraic properties of $G$.

math.OA↗

arXiv2024

The product formula for regularized Fredholm determinants: two new proofs

For an $m$-summable operator $A$ in a separable Hilbert space the higher regularized Fredholm determinant $\det\nolimits_m(I+A)$ generalizes the classical Fredholm determinant. Recently, Britz et al presented a proof of a product formula \[ \det\nolimits_m\bigl( (I+A)\cdot(I+B) \bigr) = \det\nolimits_m (I+A) \cdot \det\nolimits_m (I+B) \cdot \exp\operatorname{Tr}\bigl({X_m(A,B)}\bigr), \] where $X_m(A,B)$ is an explicit polynomial in $A,B$ with values in the trace class operators. If $m=1$ then $X_1(A,B)=0$, hence the formula generalizes the classical determinant product formula. One of the purposes of this note is to present two very simple alternative proofs of the formula. The first proof is a priori analytic and makes use of the fact that $z\mapsto \det\nolimits_m(I+zA)$ is holomorphic, while the second proof is completely algebraic. The algebraic proof has, in our opinion, some interesting aspects in its own about the trace and commutators. Secondly, we extend the above mentioned formula to several factors \[ \det\nolimits_m\Bigl( \prod_{l=1}^r (I+A_l) \Bigr) =\left( \prod_{l=1}^r \det\nolimits_m (I+A_l) \right) \cdot \exp\operatorname{Tr}\bigl({X_{m,r}(A_r,\ldots,A_r)}\bigr). \] The latter is more than just a straightforward generalization as we will gain more insights into the combinatorics behind it. Also we will present an algebraized version of the analytic proof in the language of formal power series. The upshot is that the two identities are just combinatorial in nature.

math.SP↗