The authors obtain a number of interesting results on independence in the context of

${C}^{*}$- and

${W}^{*}$-algebras, complementing and generalizing the results of the second-named author in Rev. Math. Phys. 2, 201-247 (1990;

Zbl 0743.46079). It is shown that, for

${C}^{*}$-subalgebras

$A$,

$B$ of a

${C}^{*}$-algebra

$C$, their

${C}^{*}$-independence is equivalent to the multiplicativity of the norm: for all

$a\in A$ and

$b\in B$,

$\parallel ab\parallel =\parallel a\parallel \parallel b\parallel $, even if the subalgebras do not commute. For a pair of commuting von Neumann subalgebras of a von Neumann algebra, the following notions of independence coincide:

${C}^{*}$-independence,

${W}^{*}$-independence, strict locality in both

${C}^{*}$- and

${W}^{*}$-algebra sense, Schlieder’s condition and the existence of a product extension. Some counterexamples are given and the question of kinematical independence is discussed. The results are further complemented by those of

*J. Hamhalter* in Ann. Inst. Henri Poincaré, Phys. Theor. 67, No. 4, 447-462 (1997).