SCHWEITZER 1997 Problem 8

Let H be an infinite dimensional, separable, complex Hilbert space and denote $\cal B (\cal H)$ the $\cal H$ -algebra of its bounded linear operators. Consider the algebras $l_{\infty} ({\Bbb N}, \cal B (\cal H) ) =$ $\{ (a_n) | A_n \in\cal B (\cal H)$ $(n \in {\Bbb N}), \sup_n ||A_n|| <\infty \}$ $C(\beta {\Bbb N}, \cal B (\cal H) )$ = $\{ f: \beta {\Bbb N} \to \cal B (\cal H)|$ f is continuous $\}$ with pointwise operations and supremum norm. Show that these C*-algebras are not isometrically isomorphic. (Here, $\beta {\Bbb N}$ denotes the Stone-Cech compactification of the set of natural numbers.)