Interpolating norms for orthogonal and spin Lie algebras

by R. F. Streater; Symposia Mathematica, XIV, 173-179, 1974.

ABSTRACT

The real Lie algebra so(H)_0 of real anti-symmetric finite rank operators on a real Hilbert space H may be furnished with the norms ||X||_p=tr(X^tX)^{p/2}^{1/p}, where 1 leq p < infinity. The real Lie algebra spin(H)_0, which is isomorphic to so(H)_0, is furnished with the norms ||Y||_p=omega(Y^*Y)^{p/2}^{1/p}, where omega is the unique central state on the Clifford algebra over H, in which spin(H)_0 is embedded. Let phi:so(H)_0 mapsto spin(H)_0 be the natural isomorphism. We prove analogues of the Hausdorff-Young inequalities

||X||_{p'}\geq 4.2^{-1/p}||phi(X)||_p for 2 leq p leq infinity
and
||X||_{p'} leq 4.2^{-1/p}||phi(X)||_p for 2 leq p' leq infinity
where 1/p+1/p'=1 and ||.||_infinity denotes the operator norm. Equality holds if p=infinity or 2; and for any other p, if and only if rank X=0 or 2.

This work was presented at the conference on geometry and mathematical physics, Rome, 1973. Some of my expenses were paid by the Italian Mathematical Society, but I could not receive them until I submitted a manuscript. I had not anticipated this, and had not brought enough money to pay the hotel; so I spent the entire conference in the office of the conference, typing up the paper on an old type-writer.

Jaak Peetre has confirmed some of these results by a more direct method.