Structure theorem for infinite-dimensional semisimple algebras?

[Jamie]

Hello. What structure theorems are known for infinite-dimensional
semisimple algebras? I am especially interested in the case that they
are defined over a field of characteristic 0, and in the case that the
algebra carries a positive involution.

My understanding is that such algebras are not necessarily cartesian
products of finite-dimensional simple algebras in a canonical way. If
I am correct here, is there some natural property by which we can
identify those semisimple algebras which are indeed infinite direct
sums of finite-dimensional simple algebras?

Thanks,
Jamie.

[Mariano Suárez-Alvarez]

What do you mean, exactly?

A ring is semisimple iff it is a product of
a finite number of simple artinian rings, and a
ring is simple artinian iff it is a matrix algebra
over a division ring.

In particular, a semisimple ring cannot be an infinite
product of finite dimensional simple algebras...

-- m