Banach algebra

Definition 1.

A Banach algebra $𝒜$ is a Banach space (over $ℂ$) with an multiplication law compatible with the norm which turns $𝒜$ into an algebra. Compatibility with the norm means that, for all $a,b\in 𝒜$, it is the case that the following product inequality holds:

$$\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$$

Definition 2.

A Banach *-algebra is a Banach algebra $𝒜$ with a map ${}^{*}:𝒜\to 𝒜$ which satisfies the following properties:

$a^{**}$$=$$a,$(1)

${(ab)}^{*}$$=$$b^{*}{a}^{*},$(2)

${(a+b)}^{*}$$=$$a^{*}+b^{*},$(3)

${(\lambda a)}^{*}$$=$$\overline{\lambda }{a}^{*}\mathit{\hspace{1em}}\forall \lambda \in ℂ,$(4)

$\parallel a^{*}\parallel$$=$$\parallel a\parallel ,$(5)

where $\overline{\lambda }$ is the complex conjugation of $\lambda$. In other words, the operator ${}^{*}$ is an involution.

Example 1

The algebra of bounded operators on a Banach space is a Banach algebrafor the operator norm.