Operator algebras over Cayley-Dickson numbers

The book is devoted to operator algebras and their spectral theory over the Cayley-Dickson algebras. In the first chapter non-commutative theory of R-linear and additive operators in Banach spaces over the Cayley-Dickson algebras is presented. Unbounded, as well as bounded quasi-linear operators in the Hilbert spaces X over the Cayley-Dickson algebras are studied. There are defined and investigated also graded operators of projections and graded projection valued measures. Theorems about spectral representations of projection valued graded measures of normal quasi-linear operators, which can be unbounded, are proved. More general properties of C*-algebras over the Cayley-Dickson algebras are given in Chapter 2. For them analogs of theorems like Gelfand-Naimark-Segal's, von Neuman's, Kaplansky's and so on are proved. Then a topological and algebraic irreducibility of the action of a C*-algebra of quasi-linear operators in the Hilbert space over the Cayley-Dickson algebras is described.