Semi-simplicity of a proper weak H*-algebra

Abstract

A weak right H*-algebra is a Banach algebra A which is a Hilbert space and which has a dense subset Dr with the property that for each x in Dr there exists xr such that (yx,z)=(y,zxr) for all y, z in A. It is shown that a proper (each xr is unique) weak right H*-algebra is semi-simple. Also there is an example of weak right H*-algebra which is not a left H*-algebra.

Keywords

• Hilbert algebra
• H*-algebra
• weak right H*-algebra
• weak left H*-algebra
• complemented algebra
• right complemented algebra
• left complemented algebra.