Vol. 84, No. 1, 1979

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The splitting of operator algebras

Sze-Kai Tsui and Steve Wright

Vol. 84 (1979), No. 1, 201–215

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(Re T1 Re T2) or W(Im T1 Im T2) splits, but the converse is false. An example is given to show that W(T1 T2) splits whereas C(T1 T2) does not.

Mathematical Subject Classification 2000
Primary: 46L05
Secondary: 46L10, 47C15
Received: 9 August 1978
Revised: 3 January 1979
Published: 1 September 1979
Sze-Kai Tsui
Steve Wright