\(C^\ast\)-algebras generalizing both relative Cuntz-Pimsner and Doplicher-Roberts algebras (Q2839382)

Modifying the terminology of \textit{P. Ghez}, \textit{R. Lima} and \textit{J. E. Roberts} [Pac. J. Math. 120, 79--109 (1985; Zbl 0609.46033)], the author first gives some useful preliminaries on \(C^*\)-precategories and characterizes ideals in \(C^*\)-precategories via ideals in \(C^*\)-algebras. Generalizing the representations of \(C^*\)-correspondences, he provides a definition of a right tensor representation as a representation of a \(C^*\)-precategory. He presents three different definitions of the \(C^*\)-algebra \(\mathcal{O}_\mathcal{T}(\mathcal{K},J)\) of an ideal \(\mathcal{K}\) in a right tensor \(C^*\)-precategory \(\mathcal{T}\) relative to an ideal \(\mathcal{J}\) and establishes their equivalence. He studies the ideal structure of \(\mathcal{O}_\mathcal{T}(\mathcal{K},J)\) as a structure theorem stating that the ideal \(\mathcal{O}(\mathcal{N})\) in \(\mathcal{O}_\mathcal{T}(\mathcal{K},\mathcal{J})\) generated by an invariant ideal \(\mathcal{N}\) in \(\mathcal{T}\) can be naturally identified as \(\mathcal{O}_\mathcal{T}(\mathcal{K}\cap\mathcal{N},\mathcal{J}\cap\mathcal{N})\), and the quotient \(\mathcal{O}_\mathcal{T}(\mathcal{K},\mathcal{J})/\mathcal{O}(\mathcal{N})\) as \(\mathcal{O}_{\mathcal{T}/\mathcal{N}}(\mathcal{K}/\mathcal{N},\mathcal{J}/\mathcal{N})\). As an application of the structure theorem, he obtains procedures of reduction of relations defining \(\mathcal{O}_\mathcal{T}(\mathcal{K},J)\); see [\textit{B. K. Kwaśniewski} and \textit{A. V. Lebedev}, ``Relative Cuntz-Pimsner algebras, partial isometric crossed products and reduction of relations'', \url{arXiv:0704.3811}]. An analogous gauge-invariant uniqueness theorem extending the corresponding theorems for relative Cuntz-Pimsner algebras is given; cf.\ [\textit{N. J. Fowler}, \textit{P. S. Muhly} and \textit{I. Raeburn}, Indiana Univ. Math. J. 52, No. 3, 569--605 (2003; Zbl 1034.46054)]. The author also investigates some conditions assuring that algebras of type \(\mathcal{O}_\mathcal{T}(\mathcal{K},J)\) are embedded into another via universal representations; see [\textit{S. Doplicher}, \textit{C. Pinzari} and \textit{R. Zuccante}, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 1, No. 2, 263--281 (1998; Zbl 0916.46053)], and introduces an analogue of relative Doplicher-Roberts algebras \(\mathcal{D}R(J,X)\). He then describes representations of \(\mathcal{D}R(J,X)\); see [Fowler, Muhly and Raeburn, loc. cit.].