We call a Banach algebra A,
whose norm is a Hilbert space norm, a twosided H^{∗}algebra if for each x ∈ A there
are elements x^{l}, x^{r} in A such that (xy,z) = (y,x^{l}z) and (yx,z) = (y,zx^{r}) for all
y,z ∈ A. A twosided H^{∗}algebra is called discrete is each right ideal R such that
{x^{r}∣x ∈ R} = {x^{l}∣x ∈ R} contains an idempotent e such that e^{r} = e^{l} = e. The
purpose of this paper is to obtain a structural characterization of those twosided
H^{∗}algebras M which consist of complex matrices x = (x_{ij}∣i,j ∈ J) (J is any index
set) for which
converges. Here t_{i} is real and 1 ≦ t_{i} ≦ a for all i ∈ J and some real a. The inner
product in M is
and
Then every algebra M is discrete simple and proper (Mx = 0 implies x = 0).
Conversely every discrete simple and proper twosided H^{∗}algebra is isomorphic to
some algebra M. An incidental result is that the radical of a twosided H^{∗}algebra is
the right (left) annihilator of the algebra.
