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On isometric embedding ℓ p m → S ∞ and unique operator space structure
Author(s) -
Ray Samya Kumar
Publication year - 2020
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12336
Subject(s) - mathematics , commutative property , banach space , hilbert space , operator (biology) , embedding , operator space , mathematical proof , finite rank operator , pure mathematics , space (punctuation) , norm (philosophy) , compact operator , discrete mathematics , algebra over a field , combinatorics , computer science , biochemistry , chemistry , repressor , artificial intelligence , transcription factor , gene , operating system , extension (predicate logic) , geometry , law , political science , programming language
We study existence of linear isometric embedding of ℓ p m into S ∞ , for 1 ⩽ p < ∞ , and unique operator space structure on two‐dimensional Banach spaces. For p ∈ ( 2 , ∞ ) ∪ { 1 } , we show that indeed ℓ p 2 does not embed isometrically into S ∞ . This verifies a guess of Pisier and broadly generalizes the main result of Gupta and Reza ( Houston J. Math . 44 (2018) 1205–1212). We also show that S 1 m does not embed isometrically into S p n for all 1 < p < ∞ and m ⩾ 2 . As a consequence, we establish non‐commutative analogue of some of the results of Lyubich and Shatalova ( Algebra i Analiz 16 (2004) 15–32). We also show that ( C 2 , ∥ . ∥ B p , q) does not embed isometrically into S ∞ for 2 < p , q < ∞ . The main ingredients in our proofs are notions of Birkhoff–James orthogonality and norm‐parallelism for operators on Hilbert spaces. These enable us to deploy ‘infinite descent’ type of arguments to obtain contradictions. Our approach is new even in the commutative case. We prove that ( C 2 , ∥ . ∥ B p , q) does not have unique operator space structure whenever ( p , q ) ∈ ( 1 , ∞ ) × [ 1 , ∞ ) ∪ [ 1 , ∞ ) × ( 1 , ∞ ) by showing that they do not have Property P or two summing property. In view of Misra, Pal and Varughese ( J. Operator Theory 82 (2019) 23–47), this produces genuinely new examples of two‐dimensional Banach spaces without unique operator space structure, providing a partial answer to a question of Paulsen. In this case, we derive our result by transferring the problem to real case and applying known results of Arias, Figiel, Johnson and Schechtman ( Trans. Amer. Math. Soc . 347 (1995) 3835–3857).