Left--right Transfer for $C4^{\ast}$-Rings Beyond the Regular Case

TL;DR

This paper investigates conditions under which left--right symmetry of the strongly C4*-ring condition holds beyond regularity, introducing transfer mechanisms, obstruction patterns, and permanence properties within a Morita-type framework.

Contribution

It develops new transfer principles for C4*-rings beyond regularity, identifies structural obstructions, and establishes permanence results, advancing the classification of left--right symmetry.

Findings

Established sufficient conditions for left--right transfer beyond regularity.

Identified structural obstructions preventing transfer.

Formulated transfer principles that persist under matrix and corner operations.

Abstract

The first structural fact is that regularity is sufficient for left--right symmetry of the strongly \(C4^{\ast}\) condition. It is not necessary for the definition itself and is too strong for classification. The problem is therefore to determine which weaker hypotheses still force right \(C4^{\ast}\)-type conditions to pass to the left side, and which obstructions prevent such transfer. We study right \(C4^{\ast}\)-rings, strongly right \(C4^{\ast}\)-rings, and right semi-weak-CS \(C4^{\ast}\)-rings under hypotheses strictly weaker than regularity. The method does not repeat the regular decomposition argument. Instead, it isolates a transfer mechanism based on orthogonal decomposition, corner control, and summand-square-free separation. This yields sufficient conditions for left--right transfer beyond the regular case and also identifies necessary obstruction patterns. In particular, we determine settings in which one-sided \(C4^{\ast}\)-behavior forces two-sided \(C4^{\ast}\)-behavior, and settings in which the implication fails. A second aim is exact separation. We show that failure of transfer is structural, not accidental: it is encoded by persistent one-sided square-free phenomena and by the breakdown of compatible corner decompositions. This produces counterexample criteria rather than isolated examples. A third aim is permanence. The transfer principles are formulated so as to persist under matrix and full-corner passage whenever the relevant hypotheses are stable. Thus the symmetry problem is placed in a Morita-type framework, although full Morita invariance is not asserted for every \(C4^{\ast}\)-condition. The paper therefore divides the symmetry problem into transfer, obstruction, and permanence, and thereby sharpens the regular theory, delimits its range, and gives a classification scheme for the left--right behavior of \(C4^{\ast}\)-type rings.