Silambarasan C | Searcharxiv

arXiv2026

Regular irreducible inclusions of simple $C^*$-algebras and crossed product structure

We study regular irreducible inclusions $B\subset A$ of simple unital $C^*$-algebras admitting a conditional expectation. We introduce a generalized notion of quasi-basis extending Watatani's framework and show that such inclusions admit a unitary orthonormal generalized quasi-basis. As a consequence, we prove that every regular irreducible inclusion in this setting is canonically isomorphic to a reduced twisted crossed product of $B$ by its Weyl group. This extends earlier crossed product characterizations beyond the finite-index setting.

math.OA↗

arXiv2026

Unitary normalizers in finite-dimensional inclusions

We study regular inclusions of finite-dimensional von Neumann algebras from a matrix-theoretic perspective. To this end, we introduce a new combinatorial invariant of an inclusion, called the normalizer matrix, which encodes the structure of the normalizer purely at the level of the inclusion matrix. Using this invariant, we obtain a complete characterization of regular inclusions of finite-dimensional von Neumann algebras. As consequences, we show that every regular inclusion decomposes into finite direct sums and tensor products of basic building blocks, and that regular inclusions are necessarily of depth two. We further investigate the existence of unitary orthonormal bases in the sense of Pimsner-Popa and prove that, under a natural spectral condition, regularity is equivalent to the existence of a unitary orthonormal basis contained in the normalizer. These results provide a unified description of regularity, unitary bases, and depth through the normalizer matrix formalism.

math.OA↗