A Gram-Schmidt method in Hilbert modules

Abstract

H*-algebras. An H*-algebra A over a field F (C or R) is a Banach algebra whose norm ∣.∣ is derived from an inner product 〈, 〉 and where for each A a two sided adjoint exists such that

$$ \begin{array}{*{20}{c}} {\left\langle {xy,z} \right\rangle = \left\langle {y,x*z} \right\rangle }\\ {\left\langle {yx,z} \right\rangle = \left\langle {y,zx*} \right\rangle } \end{array} $$

for all y and z (see [1]). A is called proper if the adjoint of each element is unique and it is simple if it has no nontrivial closed two-sided ideals. It is known that A has a unit if and only if it is finite dimensional. Two elements x and y are called doubly orthogonal if 〈x, y〉 = 0 and x*y = 0. If A has a unit, which we shall denote by 1, x*y = 0 implies 〈x, y〈 = 〈x1, y〉 = 〈1, x*y〉 = 0.