
02074199
j
2007f.00384
Kulkarni, S.H.
A very simple and elementary proof of a theorem of Ingelstam.
Am. Math. Mon. 111, No. 1, 5458 (2004).
2004
Mathematical Association of America (MAA), Washington, DC
EN
I95
real Banach algebra
Hilbert space
Zbl 0122.35003
doi:10.2307/4145018
The theorem in question states that a unital Banach algebra over the field $\Bbb R$ whose underlying Banach space is a (real) Hilbert space is isomorphic to one of the algebras $\Bbb R$, $\Bbb C$ or $\Bbb H$ (=~quaternions), cf.\ [{\it I.~Ingelstam}, Math. Scand. 11, 2232 (1962; Zbl 0122.35003)]. The paper under review certainly lives up to its title since no Banach algebra theory and barely any Hilbert space theory is used; incidentally, completeness turns out to be inessential for the theorem. The only criticism I have is that the author's reasoning for Fact~1 only works if the set $S$ is assumed orthonormal and that the name Schwarz in CauchySchwarz inequality is misspelt two thirds of the time.
Dirk Werner (Berlin)