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The book is devoted to operator algebras and their spectral theory over the CayleyDickson algebras. In the first chapter noncommutative theory of Rlinear and additive operators in Banach spaces over the CayleyDickson algebras is presented. Unbounded, as well as bounded quasilinear operators in the Hilbert spaces X over the CayleyDickson 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 quasilinear operators, which can be unbounded, are proved. More general properties of C*algebras over the CayleyDickson algebras are given in Chapter 2. For them analogs of theorems like GelfandNaimarkSegal'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 quasilinear operators in the Hilbert space over the CayleyDickson algebras is described.
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