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Let $H$ and $K$ be Hilbert spaces. Recall that a Ternary ring of operator(TRO) $V$ is a closed subspace of $B(H,K)$ such that $xy^{\ast}z \in V$ for all $x,y,z \in V$. I have recently started reading about TROs. I am interested in studying about representation theory of TRO. As I am beginner i am interested in learning basics of the same first.

What are some papers/books to read about the representation theory of TRO?

Thanks in advance.

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    $\begingroup$ Zettl's thesis is of course the main reference. I think David Blecher has written a lot about TRO's from the point of view of operator spaces. I have also considered some aspects of TRO's in relation to partial actions in "Twisted partial actions, a classification of regular C*-algebraic bundles", Proc. London Math. Soc., 74 (1997), 417-443. $\endgroup$
    – Ruy
    Commented Dec 16, 2019 at 1:36
  • $\begingroup$ The introduction to the paper "Local properties of Ternary Rings of Operators and their linking C*-algebras" by Kaur and Ruan (JFA 2002) has several references (including the 2 mentioned by Ruy) about TROs. My relationship with TROs was short and long ago but I don't remember seeing a systematic study of their representation theory anywhere. It is probably closely tied to the representation theory of the linking C*-algebras which already has a fully developed theory. $\endgroup$
    – Caleb Eckhardt
    Commented Dec 17, 2019 at 20:46
  • $\begingroup$ @CalebEckhardt: Do you know any paper where the relationship between representation of C*- algebras and its linking C*-algebras has been established? $\endgroup$
    – Math Lover
    Commented Jan 1, 2020 at 8:06
  • $\begingroup$ @MathLover In Section 2 of the Kaur-Ruan paper they outline, with reference to Hamana, how a TRO morphism gives rise to a C*-algebra morphism of the linking C*-algebras. $\endgroup$
    – Caleb Eckhardt
    Commented Jan 1, 2020 at 14:53
  • $\begingroup$ @CalebEckhardt: Thank you. I still don’t see how a morphism of C*- algebras gives rise to a TRO morphism. Any ideas? $\endgroup$
    – Math Lover
    Commented Jan 8, 2020 at 1:21

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