positive element in a Banach $*$- algebra

Question

By definition, $a$ is positive in C*-algebra $A$ if $\sigma(a) \subset \Bbb R^+$. I would like to know the definition of a positive element in a Banach $*$-algebra. I think it's the same as the definition of a positive element in a C*-algebra, but searching in net, I could not find a definition for that. Please regard me. Thanks.

Answer 1

Consider the algebra $M_n \oplus M_n$, where $M_n$ is the complex $n \times n$ matrix algebra. Equip with with the involution $(x, y)^{\ast} := (\bar y^T, \bar x^T)$.

Self-adjoint elements are exactly of the form $(x, \bar{x}^T)$ for some matrix $x$, so their spectra are not necessarily real. Since $x$ here is an arbitrary matrix, easy $2 \times 2$ examples show that the sum of self-adjoint elements with positive spectra doesn't necessarily have positive (or even real) spectrum. Moreover, every self-adjoint element $(x, \bar{x}^T)$ can be represented as $(x, \bar{x}^T) = (x, 1) (x, 1)^\ast = (x, 1)^\ast (x,1)$, so representability in such a form is not equivalent to being self-adjoint and having real spectrum.

This example is supposed to convince you that the standard notions of positive element are not even remotely as well-behaved in general $\ast$-algebras as in $C^{\ast}$-algebras, which is probably the reason why you don't see such a definition in the literature.