Operator algebras over Cayley-Dickson numbers. Operator algebras, \(C^*\)-algebras, spectra, spectral measures
The Cayley-Dickson algebra \({\mathcal A}_r\) over the field \(\mathbb{R}\) is generated by \(2^r\) generators \(\{i_0,i_1, \dots, i_{2^r-1}\}\) such that \(i_0=1\), \(i_j^2=-1\) for each \(j=1, 2, \dots, 2^{r}-1\) and \(i_ji_k=-i_ki_j\) for each \(1 \leq k \neq j\leq 2^r-1\), where \(r \geq 1\). At the same time, the algebra \(A_{r+1}\) is formed from the algebra \(A_r\) with the help of the doubling procedure by generator \(i_{2^r}\), in particular, \({\mathcal A}_1\) is the field of complex numbers, \({\mathcal A}_2\) is the skew field of quaternions, \({\mathcal A}_3\) is the algebra of octonions, and \({\mathcal A}_4\) is the algebra of sedenions.NEWLINENEWLINEThe book under review is mainly devoted to operator algebras and their spectral theory over the Cayley-Dickson algebras \({\mathcal A}_r\). The book consists of a preface, 2 chapters, a bibliography that contains 64 references and a notation list as well as a subject index.NEWLINENEWLINEIn the first chapter, entitled ``Spectra of operators'', the author presents the non-commutative theory of \(\mathbb{R}\)-linear and additive operators in Banach spaces over the Cayley-Dickson algebras and investigates unbounded as well as bounded quasi-linear operators in Hilbert spaces over \({\mathcal A}_r\). He studies graded operators of projections and graded projection valued measures and establishes some theorems about spectral representations of projection valued graded measures of normal quasi-linear operators.NEWLINENEWLINEThe second chapter, entitled ``Algebras of operators'', is devoted to the study of general properties of \(C^*\)-algebras over \({\mathcal A}_r\). The author provides some Gelfand-Naimark-Segal, von Neumann and Kaplansky type theorems and describes the topological and algebraic irreducibility of actions of \(C^*\)-algebras of quasi-linear operators in Hilbert spaces over the Cayley-Dickson algebras \({\mathcal A}_r\). He also provides a theory of quasi-permutability of normal operators in Hilbert spaces over Cayley-Dickson algebras.
scientific article; zbMATH DE number 6090404