Some new Hilbert algebras

On semi-inner product algebras

“…Introduction. R. Keown [5] introduced some new classes of commutative Hilbert algebras which is some sense are generalizations of the algebras studied by W. Ambrose [1]. The essential difference between the works of Keown and Ambrose is that the latter doe not obtain the decomposition of the algebra into orthogonal subspaces each of which is a minimal left ideal.…”

“…For example, we show that for any generalized s.i.p, algebra A and for an idempotent e, eAe is a division algebra. For definitions we follow Keown [5] and Husain [2].2. We recall some of the definitions from [4] and [6].A complex (real) vector space X is called a complex (real) s.i.p.…”

“…For example, we show that for any generalized s.i.p, algebra A and for an idempotent e, eAe is a division algebra. For definitions we follow Keown [5] and Husain [2].…”

“…(The same is true for left ideal also:) :. As in Keown [5] we take all the algebras to be commutative and semi-simple in the sequel. Furthermore we assume that the norm in the s.i.p, space (complete and continuous) satisfies the inequality Lemma 1 can be used to adopt Keown's proof of (c" [5], Lemma 2.1) of the following result for the Hilbert regular algebras to generalized s.i.p, regular algebras with appropriate changes.…”