I learn that in a Banach algebra over F (R or C), an element a is positive if and only if a=b*b, and equivalently a is positive if a = a* and spectrum of a is a subset of R of non negative elements ([0, +\infny). My questions are:
1) If F=R, that is real Banach algebra, what can we say about it's positive elements?
2) Do we still need involution on the real Banach algebra to have a positive elements?
3) Could you please give an example to justify the answer to Q2.
Consider the example: Take any nilpotent operator J, i.e. J^2 = 0. Then
{c J : c element of F} is a Banach algebra over F with norm || cJ || := |c| and involution (cJ)^* := c* J, where c* is a conjugate of c in the field F. This algebra does not contain any positive element, apart from 0 (which is
"positive" in any Banach algebra).
Thank you for your answer, Detlev Buchholz. However, I did not grasp your conclusion regarding the questions. Are you trying to say that one can get positive elements from any Banach algebra without imposing any additional condition on it?
My example shows: The zero element may always be regarded as a "positive" element in any Banach algebra. Since whatever involution * you may invent, you have 0* = 0 and 0*0 = 0. Also, the spectrum of 0 is {0}.
Good response professor, I am grateful for that. To be more precise my question is: can one get positive elements of a real Banach algebra without imposing it to be involutive?
Final comment: one can talk about the spectrum of an operator in any Banach algebra (extending it by the unit operator 1, if necessary). In my
example, with F real, any operator (a1 + bJ) has an inverse in the algebra
if a =/= 0, i.e. it has spectrum {0} and hence is "positive". No involution is needed for that conclusion.
Exercise: In a real Banach algebra there can exist, however, operators which have empty spectrum. Consider for example the real Banach algebra generated by J with J^2 = -1.
In a real unital Banach algebra $A$, there are two ways to define the spectrum of an element $x$. Each has its drawbacks and its avantages. The one consists of $Sp(x) = \{ \lambda \in \mathbb{R} : x - \lambda \text{ is not invertible in} A\}$ and the other consists of the spectrum of $x$ as an element of the complexification of $A$. The first one may be empty while the second is a non empty compact subset of $\mathbb{C}$.
Your statement that an element a is positive iff it has the form a=b*b only makes sense if you are in a (Banach) algebra with involution. And I suppose this is only true if you are in a C*-algebra.
Consider for instance the disc algebra, that is, the Banach algebra of all continuous functions on the unit disc that are analytic on its interior. This is a Banach algebra with the involution f*(z):=\overline{f(\overline{z})}. In this algebra the function z (identity) is self adjoint and then z^2 is of the form b*b, but its spectrum is not positive.
Thank you all for your contributions, in fact these discussions impact more on my knowledge of operator theory.
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