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representation q of an involutive Banach algebra 5 over a measure space (F, u). As shown by several authors in [4], [8], [9], [13] etc., this decomposition cannot be regarded as a decomposition of the unitary equivalence class of 9o into the unitary equivalence classes of q (ry) except for some fairly nice cases, whereas this decompositon is determined only up to unitary equivalence. For instance, some representations can be decomposed in two ways that have no common components as in [8] and some two representations of quite different types can be decomposed into the direct integrals of the same components as in [13]. Therefore it comes into considerations what determines the unitary equivalence relation among the components {gyp( ry); ry EF of the decomposition

tohoku mathematical journal/tohoku mathematical journal. first series/tohoku mathematical journal. second series

A complement to ``On the unitary equivalence among the components of decompositions of representations of involutive Banach algebras and the associated diagonal algebras''