Abstract
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.
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Acknowledgements
A part of this work was done during the discussion meeting on ‘Noncommutative Convexity’ held during February 2019 at Kerala School of Mathematics (KSoM), Kozhikode, India. The authors wish to thank Dr. P Shankar and S Pramod for useful discussions. The first author is thankful to the Department of Atomic Energy, Government of India for NBHM Ph.D. Fellowship. The third author is thankful to the Council for Scientific and Industrial Research (CSIR), Government of India for a research fellowship.
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Communicated by B V Rajarama Bhat.
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Arunkumar, C.S., Shabna, A.M., Syamkrishnan, M.S. et al. Extreme states on operator spaces in ternary rings of operators. Proc Math Sci 131, 44 (2021). https://doi.org/10.1007/s12044-021-00639-2
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DOI: https://doi.org/10.1007/s12044-021-00639-2