We say the singly generated
C∗-algebra, C∗(T1⊕T2), splits if C∗(T1⊕T2) = C∗(T1) ⊕C∗(T2). A necessary and
sufficient condition is derived for the splitting of C∗(T1⊕ T2) in terms of the
topological structure of the primitive ideal space of C∗(T1⊕T2). In particular, when
C∗(T1⊕ T2) is strongly amenable, the necessary and sufficient condition can be
simplified and does not depend on the topology of the primitive ideal space of
C∗(T1⊕ T2). Several applications of this theorem, such as the cases, among
others, where T1, T2 are compact operators, and C∗(T1), C∗(T2) have only
finite-dimensional irreducible representations, are discussed. For the splitting of the
W∗-algebra, W∗(T1⊕ T2), two equivalent conditions are derived which are
quite different in nature. It is also shown that W∗(T1⊕ T2) splits if either
W∗(ReT1⊕ReT2) or W∗(ImT1⊕ImT2) splits, but the converse is false. An
example is given to show that W∗(T1⊕ T2) splits whereas C∗(T1⊕ T2) does
not.
