Let $(X, Ω, µ)$ be a measure space. For $ϕ ∈ L^∞(X, Ω, µ)$, set $π(ϕ)f := ϕf$ for $f ∈ L^2(X, Ω, µ)$. Let $π(ϕ) ∈ B(L^2(X, Ω, µ))$, and that the map $ϕ → π(ϕ)$ yields a representation $π: L^∞(X, Ω, µ) → B(L^2(X, Ω, µ))$ of the $C^∗-algebra$ $L^∞(X, Ω, µ)$ on $L^2(X, Ω, µ)$. Suppose that $µ$ is σ-finite. (a) Show that π is a cyclic representation of $L^∞(X, Ω, µ)$. (b) Determine the cyclic vectors for π.
I tried to solve this problem by relying on the fact $L^∞(X, Ω, µ)$ is dense in $L^2(X, Ω, µ)$. Also I want to show that: there is at least $e ∈ H$ Hilbert space such that $cl[π(A)e]=H$. But I am struggling with that, so I need help to solve this problem. Thank you
Note that because $\mu$ is $\sigma$-finite that the constant one function $1_X:X\to \mathbb{C}$ is contained in both $L^{\infty}(X,\Omega,\mu)$ and $L^{2}(X,\Omega,\mu)$. As you mentioned $L^{\infty}(X,\Omega,\mu)$ is dense in $L^{2}(X,\Omega,\mu)$ so $L^{\infty}(X,\Omega,\mu)\cdot 1_X=L^{\infty}(X,\Omega,\mu)$ which is dense in $L^{2}(X,\Omega,\mu)$.
For the second part the following condition is necessary and sufficient; $h$ is cyclic if and only if $h$ is nonzero a.e.
$"\implies"$ Suppose that $h\in L^{2}(X,\Omega,\mu)$ is a cyclic vector. Then there exists a sequence of functions $(f_n)_n$ in $L^{\infty}(X,\Omega,\mu)$ such that $f_nh\to 1_X$ in $L^{2}(X,\Omega,\mu)$, implying that $h$ is nonzero a.e.
$"\impliedby"$ If $h$ is nonzero a.e. $1/h$ can be defined a.e. and $h\cdot 1/h=1_X$ in $L^{2}(X,\Omega,\mu)$. As a result $L^{\infty}(X,\Omega,\mu)h\subset L^{\infty}(X,\Omega,\mu)\cdot 1/h\cdot h=L^{\infty}(X,\Omega,\mu)\cdot 1_X$ which is dense in $L^{2}(X,\Omega,\mu)$