$L^{\mathrm{\infty }}(X,\mu )$

Let $X$ be a nonempty set and $𝒜$ be a $\sigma$-algebra on $X$. Also, let $\mu$ be a non-negative measure defined on $𝒜$.Two complex valued functions $f$ and $g$ are said to be equal almost everywhere on $X$(denoted as $f=g$ a.e. if $\mu \{x\in X:f(x)\ne g(x)\}=0.$ The relation of being equal almost everywhere on $X$ defines an equivalence relation.It is a common practice in the integration theory to denote the equivalence class containing $f$ by $f$ itself.It is easy to see that if $f_1,f_2$ are equivalent and $g_1,g_2$ are equivalent, then $f_1+g_1,f_2+g_2$ are equivalent, and $f_1{g}_1,f_2{g}_2$ are equivalent.This naturally defines addition and multiplication among the equivalent classes of such functions.For a measureable$f:X\to ℂ$, we define

$${\parallel f\parallel }_{\text{ess}}=\inf \{M>0:\mu \{x:|f(x)|>M\}=0\},$$

called the essential supremum of $|f|$ on $X$.Now we define,

Here the elements of $L^{\mathrm{\infty }}(X,\mu )$ are equivalence classes.

Properties of $L^{\mathrm{\infty }}(X,\mu )$

1. The space $L^{\mathrm{\infty }}(X,\mu )$ is a normed linear space with thenorm ${\parallel \cdot \parallel }_{\text{ess}}$. Also, the metric defined bythe norm is complete, making $L^{\mathrm{\infty }}(X,\mu )$, a Banach space.

2. $L^{\mathrm{\infty }}(X,\mu )$ is the dual of $L^1(X,\mu )$ if $X$ is $\sigma$-finite.

3. $L^{\mathrm{\infty }}(X,\mu )$ is closed under pointwise multiplication, andwith this multiplication it becomes an algebra.Further, $L^{\mathrm{\infty }}(X,\mu )$ is also a $C^{*}$-algebra (http://planetmath.org/CAlgebra) with the involution defined by $f^{*}(x)=\overline{f(x)}$. Since this $C^{*}$-algebra is also a dual of some Banach space, it is called von Neumann algebra.