Note on a Theorem of Fuglede and Putnam

1. An involution in a ring A is a mapping a—^a* (a(Ei;A) such that a**=a, (a+b)*=a*+b*, (ab)* = b*a*. An element a&A is (1) normal if a*a=aa*, (2) self-adjoint if a*=a, (3) unitary if a*a=aa* = l (1= unity element of A). We say that "Fuglede's theorem holds in A" incase the relations a(E.A, a normal, b^A, ba=ab, imply ba* = a*b; briefly, A is an FT-ring. It follows from a theorem of B. Fuglede that the ring A of all bounded operators in a Hilbert space (hence any adjoint-containing subring thereof) is an FT-ring [3, Theorem I]. For this ring, C. R. Putnam obtained the following generalization [9, Lemma]: if ai, a2 are normal, and bai=a2b, then ba*=a*b. A ring with involution, in which the latter theorem holds, will be called a PT-ring. We denote by An the ring of all nXn matrices x = (a(j), a,j^A, provided with the "conjugate-transpose" involution x* = (af,).