Certain invertible operator-block matrices induced by $$C^{*}$$-algebras and scaled hypercomplex numbers

Abstract

Abstract The main purposes of this paper are (i) to enlarge scaled hypercomplex structures to operator-valued cases, where the operators are taken from a $$C^{*}$$ C ∗ -subalgebra of an operator algebra on a separable Hilbert space, (ii) to characterize the invertibility conditions on the operator-valued scaled-hypercomplex structures of (i), (iii) to study relations between the invertibility of scaled hypercomplex numbers, and that of operator-valued cases of (ii), and (iv) to confirm our invertibility of (ii) and (iii) are equivalent to the general invertibility of $$\left( 2\times 2\right) $$ 2 × 2 -block operator matrices.