We characterise the ternary rings of operators possessing a completely isometric representation whose range consists of normalisers or semi-normalisers between the ranges of some * -representations of two fixed C * -algebras. We give some corollaries of these results. 2000 Mathematics Subject Classification. 46L07, 46L08. It is easy to see that the TRO U ⊆ B(H 1 , H 2 ) that is generated by T consists of normalisers of B into A; thus each normaliser of B into A belongs to a TRO each of whose

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... ments normalises B into A. As a matter of fact, if A and B are reflexive algebras, this holds also for the semi-normalisers of B into A [8]; that is, the operators T ∈ B(H 1 , H 2 ) satisfying only the first of the relations (1). Both operator algebras and TRO's can be viewed abstractly in the setting of Operator Space Theory (see e.g. [1], [6] and [7]). Suppose that A and B are abstract operator algebras. It is natural to ask for an abstract characterisation of the TRO's X possessing a completely isometric representation as a space of normalisers between (the ranges of) some completely isometric representations of A and B. In the present note we provide such a characterisation in the case in which A and B are C * -algebras. Concrete representations of abstract operator spaces, algebras, modules, etc. are of basic importance in Operator Space Theory. See [1] and [6]. Our results establish such representations in a new situation and point out a direction to consider normalisers of https://www.cambridge.org/core/terms. https://doi.