On the Imbedding of a Right Complemented Algebra into Ambrose's 77*-algebra

Let A be a Banach algebra with a Hilbert space norm (norm defined by a scalar product). We shall call A a right complemented algebra if it has the property that the orthogonal complement of a right ideal is again a right ideal. This notion was introduced in the author's doctoral thesis [5]. It was proved that under certain additional assumptions every right complemented algebra is left complemented. We shall prove this theorem for a general right complemented algebra. We shall also show that the most general simple right (left) complemented algebra is of the following form. Example. Let a be a (possibly unbounded) self-adjoint linear operator with domain dense in a Hilbert space 77 and the range being a subset of H. Let A be the algebra of all linear operators a of the Hilbert Schmidt type on 77 such that \aa\ < °°, where | | is the trace norm of an operator: |a|2 = tr (a*a). Then A is a right (as well as left) complemented algebra in the scalar product (a, b) = [aa, ba] = tr (aa(ba)*). We shall use the following terminology (see [5]). A Banach algebra shall be called simple if it is semi-simple and has no proper two-sided ideals except those which are dense in whole algebra. We shall say that xl is the left adjoint of x if (xy, z) = (y, x'z) holds for all y, z in the algebra. A left projection e is a left self-adjoint (nonzero) idempotent; a primitive left projection is a left projection which cannot be written as a sum of two doubly orthogonal left projections (compare with W. Ambrose [l]). The orthogonal complement of an ideal 7 will be denoted by 7P. We have proved in [5 ] that every simple right complemented algebra has a primitive left projection. So we begin by proving: