Vol. 16, No. 2, 1966

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On two-sided H∗-algebras

P. P. Saworotnow

Vol. 16 (1966), No. 2, 365–370

We call a Banach algebra A, whose norm is a Hilbert space norm, a two-sided H-algebra if for each x A there are elements xl, xr in A such that (xy,z) = (y,xlz) and (yx,z) = (y,zxr) for all y,z A. A two-sided H-algebra is called discrete is each right ideal R such that {xrx R} = {xlx R} contains an idempotent e such that er = el = e. The purpose of this paper is to obtain a structural characterization of those two-sided H-algebras M which consist of complex matrices x = (xiji,j J) (J is any index set) for which


converges. Here ti is real and 1 ti a for all i J and some real a. The inner product in M is

(x,y) =   tixijyijtj


 r         --    l        --
xij = (ti∕tj)xji, xij = (tj∕ti)xji.

Then every algebra M is discrete simple and proper (Mx = 0 implies x = 0). Conversely every discrete simple and proper two-sided H-algebra is isomorphic to some algebra M. An incidental result is that the radical of a two-sided H-algebra is the right (left) annihilator of the algebra.

Mathematical Subject Classification
Primary: 46.60
Received: 2 July 1964
Published: 1 February 1966
P. P. Saworotnow