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Isometries of non-commutative Lp-spaces

  • F. J. Yeadon (a1)

The spaces Lp(, φ) for 1 ≤ p ≤ ∞, where φ is a faithful semifinite normal trace on a von Neumann algebra , are defined in (10),(2),(14). The problem of determining the general form of an isometry of one such space into another has been studied in (i), (6), (9), (12), (5). Our main result, Theorem 2, is a characterization of such isometries for 1 ≤ p ≤ ∞, ≠ 2. The method of proof is based on that of (7), where isometries between Lp function spaces are characterized. The main step in the proof is Theorem 1, which gives the conditions under which equality holds in Clarkson's inequality.

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(5) R. V. Kadison Isometries of operator algebras. Ann. Math. (2) 54 (1951), 325338.

(6) A. Katavolos Isometries of non-commutative Lp-spaces. Canad. J. Math. 28 (1976), 11801186.

(7) J. Lamperti On the isometries of certain function spaces. Pacific J. Math. 8 (1958), 459466.

(8) C. A. McCarthy cp. Israel J. Math. 5 (1967), 249271.

(9) B. Russo Isometries of Lp-spaces associated with finite von Neumann algebras. Bull. Amer. Math. Soc. 74 (1968), 228232.

(10) I. E. Segal A non-commutative extension of abstract integration. Ann. Math. (2), 57 (1953), 401457.

(11) E. Størmer On the Jordan structure of C*-algebras. Trans. Amer. Math. Soc. 120 (1965), 438447.

(12) P. K. Tam Isometries of Lp-spaces associated with semifinite von Neumann algebras. Trans. Amer. Math. Soc. 254 (1979), 339354.

(13) J. Tomiyama On the projection of norm one in W*-algebras. Proc. Japan Acad. 33 (1957), 608612.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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