The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra
with C*-condition
(C*)![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X0001124X/resource/name/S0008414X0001124X_eqn1.gif?pub-status=live)
is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.
At the same time they conjectured that the C*-condition can be replaced by the B*-condition.
(B*)![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X0001124X/resource/name/S0008414X0001124X_eqn2.gif?pub-status=live)
In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.