Homology for operator algebras. IV. On the regular classification of limits of 4-cycle algebras
According with the terminology and notations from the research notes ``Limit algebras'' [Pitman Research Notes in Math. 278, London (1993)] of the second author, the subject of the present paper is related to the 4-cycle algebras, that is a CSL-algebra whose reduced digraph is a 4-cycle, and to the regular direct systems of 4-cycle algebras with rigid embeddings. Namely, the regular classification of rigid systems \({\mathcal U}\) of general 4-cycle algebras in terms of the \(K_0\) and \(H_1\) groups, the \(K_0\) scale and so-called the joint scale \(\Sigma {\mathcal U}\) (a proper subset of the product scale \(\Sigma_0{\mathcal U}\otimes \Sigma_1{\mathcal U}\), where \(\Sigma_1\) is a natural symmetric scale in \(H_1\)) is given. Also, the problem of the lifting of \(K_0\oplus H_1\) isomorphisms to algebra isomorphisms is discussed and a number of obstructions to this is considered (congruence compatibility, sign compatibility and \(H_0H_1\) coupling class obstructions). A detailed discussion of the realization of \(K_0{\mathcal U}\oplus H_1{\mathcal U}\) and the product scale as a scaled group in \(\mathbb{Q}^2\oplus \mathbb{Q}^2\oplus \mathbb{Q}\) is given. It is analyzed also how \(K_0H_1\) isomorphisms are realized. Such realizations give a better appreciation of the range of the classifying invariants as well as insight into automorphisms of systems. The analysis of the lifting of realized \(K_0H_1\) isomorphisms reveals important subclasses of systems such as extreme type systems, even and odd systems, \(hr\)-asymmetric systems and decoupled systems. The efficacy of scales in \(K_0\oplus H_1\) for classification of systems suggest to the authors, the utility of the developing of a general theory of scaled partial isometry homology and its relationship with \(K\)-theory.
scientific article; zbMATH DE number 1080236
scientific article