How is direct sum of von Neumann algebras defined?

Question

Consider the following fragment from Murphy's '$C^*$-algebras and operator theory'

How is $\bigoplus_\lambda A_\lambda$ defined? Is it $\{(a_\lambda)_\lambda\in \prod_\lambda A_\lambda: \sum_\lambda \Vert a_\lambda \Vert^2 < \infty\}$? Or $\{(a_\lambda)_\lambda\in \prod_\lambda A_\lambda: \sup_\lambda \Vert a_\lambda \Vert^2 < \infty\}$?

Answer 1

Digging through the book, it appears that Murphy intends to define the direct sum of von Neumann algebras to be $$\oplus_\lambda A_\lambda=\{(a_\lambda):\sup_\lambda\|a_\lambda\|<\infty\}.$$ His definition of direct sum of Banach algebras can be found in Exercise 1.1.

Answer 2

Note that we have natural injective $*$-morphisms

$$\bigoplus_\lambda A_\lambda \hookrightarrow \bigoplus_\lambda B(H_\lambda) \hookrightarrow B\left(\bigoplus_\lambda H_\lambda\right)$$

and this allows us to see $\bigoplus_\lambda A_\lambda$ as a $*$-subalgebra of $B\left(\bigoplus_\lambda H_\lambda\right)$. The exercise is then to show that $\bigoplus_\lambda A_\lambda$ (or rather its image under the composition above) is strongly closed in $B\left(\bigoplus_\lambda H_\lambda\right)$.