Algebras of operators in Banach spaces over the quaternion skew field and the octonion algebra

Meta-Invariant Operators over Cayley-Dickson Algebras and Spectra

“…An operator T is r  -meta-invariant, consequently, a von Neumann subalgebra . In accordance with Theorem 2.22 [12] we have that an algebra r A  is isomorphic with   , r C   for some extremely disconnected compact Hausdorff topological space  .…”

“… . Preliminary results on operator theory and operator algebras over quaternions and octonions were published in [12]. This articles develops further operator theory already over general Cayley-Dickson algebras.…”

“…At the same time for a symmetrical operator, the inclusion   T   R is satisfied and due to the relations given in §2.23 [18] …”

“… is bounded and it is defined everywhere and this case was considered in Theorem 2.24 [18]. Therefore, the mapping h is the homeomorphism…”

“…Definitions and notations of the preceding work [12] are used below. is formed from the algebra r  with the help of the doubling procedure by generator 2 r i , in particular, 0 : R  is the real field, 1  C  coincides with the field of complex numbers, 2  H  is the skew field of quaternions, 3  is the algebra of octonions, 4  is the algebra of sedenions.…”