Extreme states on operator spaces in ternary rings of operators

• Published in: Proceedings of the Indian Academy of Sciences. Mathematical sciences Vol. 131; no. 2
• Main Authors: Arunkumar, C S, Shabna, A M, Syamkrishnan, M S, Vijayarajan, A K
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.10.2021; Springer Nature B.V
• Subjects: Extreme values; Mathematics; Mathematics and Statistics; Operators (mathematics); Operator spaces; 46L52; 47L07; 46L07; rectangular operator states; 47L25; ternary rings of operators; boundary representations
• ISSN: 0253-4142; 0973-7685
• DOI: 10.1007/s12044-021-00639-2

An extension result for rectangular operator extreme states on operator spaces in ternary rings of operators is proved. It is established that for operator spaces in rectangular matrix spaces extreme states are conjugates of the inclusion map implemented by isometries or unitaries. Further, a characterisation of operator spaces of matrices for which the inclusion map is an extreme state is deduced. In the context of operator spaces, a version of Arveson’s boundary theorem is proved. We also show that for any TRO extreme state on an operator space, the corresponding Paulsen map can be extended to a pure unital completely positive (UCP) map on the C ∗ -algebra generated by the Paulsen system.