This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces, ∑dc spaces, where an infinite dimensional Banach space X is called a ∑dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton. It shows that two strongly irreducible operators T1 and T2 on a ∑dc space are similar if and only if theK0-group of the commutant algebra of the direct sum T1 GT2 is isomorphic to the group of integers Z. On a ∑dc space X, it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in （∑SI）（X）, it further gives a necessary and sufficient condition that two operators in （∑SI）（X） are similar by using the ordered K0-groups. It also proves that every operator in （∑SI）（X） has a unique （SI） decomposition up to similarity on a ∑dc space X, where （∑SI）（X） denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.
