## Semi-simplicity of a proper weak $$H^*$$-algebra.(English)Zbl 0808.46080

A weak right $$H^*$$-algebra is a Banach algebra $$A$$ which is a Hilbert space and which has a dense subset $$D_ r$$ such that for each $$x$$ in $$A$$ there is $$x^ r$$ in $$D_ r$$ with $$(yx,z)= (y,zx^ r)$$ for all $$y$$, $$z$$ in $$A$$. The author shows that each proper weak right $$H^*$$-algebra is semi-simple. It follows that $$A$$, being a right complemented algebra, is a direct sum of simple weak right $$H^*$$-algebras, each of which is isometrically isomorphic to the algebra of all Hilbert-Schmidt operators on some Hilbert space.

### MSC:

 46K15 Hilbert algebras 46H15 Representations of topological algebras 46H20 Structure, classification of topological algebras 46K10 Representations of topological algebras with involution
Full Text: