Real \(H^ *\)-algebras
The authors extend to real \(H^*\)-algebras the structure theory obtained by Ambrose for complex \(H^*\)-algebras. Such an algebra A is expressible as a direct sum \(A=Z\oplus Z^{\perp}\), where Z is the radical of A and the orthogonal complement \(Z^{\perp}\) is semi-simple (Theorem 1). A semi-simple \(H^*\)-algebra A is a direct sum of simple \(H^*\)-algebras each of which is a minimal two-sided ideal in A (Theorem 2). The main result of the paper (Theorem 4) states that a real simple \(H^*\)-algebra is \(semi\)-H\({}^*\)-isomorphic to a full Hilbert-Scmidt matrix \(H^*\)- algebra \(M_{\Omega}(F)\), where F is either the field of real numbers, or field of complex numbers or the division algebra of real quaternions. In the above two simple \(H^*\)-algebras A and A' are said to be \(semi\)- H\({}^*\)-isomorphic if there is an algebraic *-isomorphism \(\phi\) of A onto A', and a positive constant \(\rho\) such that \(<\phi (x),\phi (y)>=\rho <x,y>\) for all x,y in A. For a field F as in Theorem 4 and a set \(\Omega\) of indices, \(M_{\Omega}(F)\) is the algebra of all matrices \(a=(a_{i,j}),a_{i,j}\in F,i,j\in \Omega\) such that \(\sum_{i,j}\| a_{i,j}\|^ 2<\infty\).
scientific article; zbMATH DE number 3969403