Publications

Preprints

2024

1. 

   Arveson’s hyperrigidity conjecture is false

   Boris Bilich, and Adam Dor-On

   2024

   Abs arXiv

   Arveson’s hyperrigidity conjecture predicts that if the non-commutative Choquet boundary of a separable operator system S is the entire spectrum of its generated C*-algebra B then S is hyperrigid in B. We provide a counterexample to the conjecture with a C*-algebra B of type I generated by a single operator.

2. 

   Equivariant homotopy classification of graph C*-algebras

   Boris Bilich, Adam Dor-On, and Efren Ruiz

   2024

   Abs arXiv

   We show that shift equivalence of essential adjacency matrices coincides with gauge-equivariant homotopy equivalence of their stabilized graph C*-algebras. This provide the first equivalent formulation of shift equivalence of essential matrices in terms of gauge actions on graph C*-algebras. Our proof uses bicategory theory for C*-bimodules developed by Meyer and Sehnem, allowing us to avoid the use of K-theory classification of C*-algebras.

3. 

   Shift equivalence relations through the lens of C*-correspondences

   Boris Bilich, Adam Dor-On, and Efren Ruiz

   2024

   Abs

   We continue the study of shift equivalence relations from the perspective of C*-bimodule theory. We study emerging shift equivalence relations following work of the second-named author with Carlsen and Eilers, both in terms of adjacency matrices and in terms of their C*-correspondences, and orient them when possible. In particular, we show that if two regular C*-correspondences are strong shift equivalent, then the intermediary C*-correspondences realizing the equivalence may be chosen to be regular. This result provides the final missing piece in answering a question of Muhly, Pask and Tomforde, and is used to confirm a conjecture of Kakariadis and Katsoulis on shift equivalence of C*-correspondences.

Journal Articles

2025

1. 

   Maximality of correspondence representations

   Boris Bilich

   Advances in Mathematics, 2025

   Abs arXiv

   In this paper, we fully characterize maximal representations of a C*-correspondence. This strengthens several earlier results. We demonstrate the criterion with diverse examples. We also describe the noncommutative Choquet boundary and provide additional counterexamples to Arveson’s hyperrigidity conjecture following the counterexample recently found by the author and Adam Dor-On.

2024

1. 

   Ideal structure of Nica-Toeplitz algebras

   Boris Bilich

   Integral Equations Operator Theory, 2024

   arXiv Website

2022

1. 

   Classification of noncommutative monoid structures on normal affine surfaces

   Boris Bilich

   Proc. Amer. Math. Soc., 2022

   arXiv

2. 

   Taylor spectrum for modules over Lie algebras

   B. Bilich

   Funct. Anal. Appl., 2022

   Translation of Funktsional. Anal. i Prilozhen. 56 (2022), no. 3, 3–15

   arXiv