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Introduction to the theory of operators

V. Metric Rings

Published online by Cambridge University Press:  24 October 2008

S. W. P. Steen
S. W. P. Steen
Affiliation:
Christ's CollegeCambridge
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Extract

This paper is a continuation of four others under the same title†. The paragraphs are numbered following on to those of the fourth paper of the series. In § XXIV we show that, if λ(A) is a linear functional, then there exists a resolution Eμ such that λ(A) = ∫μdr(AEμ), and if B = ∫μdEμ is bounded, then λ(A) = τ(AB)‡ for all A, where τ is the trace. This implies that τ(A) is a linear functional, and that the conjugate space ℒ, i.e. the space of the linear functionals, has a subset ℒ′ which is in (1, 1) correspondence with the original set of operators, and that in this correspondence the linear functional τ(A) is associated with the unit operator.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 36 , Issue 2 , April 1940 , pp. 139 - 149
Copyright
Copyright © Cambridge Philosophical Society 1940

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References

† Steen, , Proc. London Math. Soc. (2), 41 (1936), 361–92Google Scholar; 43 (1937), 529–43; 44 (1938), 398–411. Proc. Cambridge Phil. Soc. 35 (1939).Google Scholar

‡ We write throughout AB instead of A × B; no confusion will arise.

§ R.C. Circ. mat. Palermo, 22 (1906), 1.Google Scholar

∥ A referee has drawn my attention to the following papers: Nagumo, N., Jap. J. Math. 13 (1936), 61–80CrossRefGoogle Scholar; K., Yosida. Jap. J. Math. 13 (1936), 7–26.Google Scholar

† Omit the axioms for multiplication and for adjoint and we are left with the axioms for a Hilbert space. See Stone, , Linear transformations in a Hilbert space, Amer. Math. Soc. Coll. Publ. 15 (1932).Google Scholar

‡ see Banach, , Opérations linéaires (Warsaw, 1932), p. 54et seq.Google Scholar

§ Trans. Amer. Math. Soc. 41 (1937), 208–48.Google Scholar

† See § XXI, Cor. (xix), footnote §.

† This lemma also proves the existence of positive linear functionals.

† This excludes rings with nil potent elements, for which A = A*.

† von Neumann, and Jordan, , Annals of Math. 36 (1935), 719–23.Google Scholar

∥A∥ corresponds to the metric l.u.b. |f(x)| in the space of continuous functions; ρ(A) corresponds to the metric √f|f|2dx in L 2-space.

† Semi-order is only required for members of a given association. 2iA:B = AB − BA.

‡ Stone, l.c. Theorem 2·25.

§ Acta Litt. ac Sci. Szeged. 5 (1930), 23–54.Google Scholar

∥ Math. Ann. 110 (1934), 722–5.Google Scholar

† Note that A:B = O is essential.