0
$\begingroup$

It follows immediately from Gelfand duality that the involution in a commutative unital real C* algebra is the identity. Is there a direct proof from the axioms of C* algebras?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

Is that really true? Consider the set of real $2\times 2$ matrices of the form $\left[\begin{matrix}a&b\cr -b&a\end{matrix}\right]$. That's a commutative unital algebra which is stable under transpose, so a real C*-algebra according to the definition I know, but the involution is the transpose operation, which is not the identity here.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .