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Antisymmetry and contractive representations of function algebras
Abstract
In the present paper the antisymmetry of the image of a function algebra under its contractive representation is characterized. A complete solution of this problem is obtained for subnormal contractive representations. Some applications, in particular, to the von Neumann functional calculus, are given.
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The theory of the numerical range of a linear operator on an arbitrary normed space had its beginnings around 1960, and during the 1970s the subject has developed and expanded rapidly. This book presents a self-contained exposition of the subject as a whole. The authors develop various applications, in particular to the study of Banach algebras where the numerical range provides an important link between the algebraic and metric structures.
Preface.- Introduction. The Krein-Milman theorem as an integral representation theorem.- Application of the Krein-Milman theorem to completely monotonic functions.- Choquet's theorem: The metrizable case.- The Choquet-Bishop-de Leeuw existence theorem.- Applications to Rainwater's and Haydon's theorems.- A new setting: The Choquet boundary.- Applications of the Choquet boundary to resolvents.- The Choquet boundary for uniform algebras.- The Choquet boundary and approximation theory.- Uniqueness of representing measures.- Properties of the resultant map.- Application to invariant and ergodic measures.- A method for extending the representation theorems: Caps.- A different method for extending the representation theorems.- Orderings and dilations of measures.- Additional Topics.- References.- Index of symbols.- Index.
- F Riesz
- B Sz
- Nagy
F. Riesz and B. Sz.-Nagy, Functional analysis, Ungar, New York, 1955.
Let F be as in §2 We show that Pe c X.By Proposition 5 and the assumptions
- Proof
Proof. Let F be as in §2. We show that Pe c X.By Proposition 5 and the
assumptions, all the operators S E B are convexoid, hence r(S) = v(S),
Spec B), then ta = t ° <¡p with some r E Spec B, hence X = tx(z)
- X If
If X E rp^-(Spec B), then ta = t ° <¡p with some r E Spec B, hence X = tx(z)
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H) is a contraction (|| T|| < 1), then the closed unit disc D is a spectral set for T (the von Neumann inequality
- T E L If
If T E L(H) is a contraction (|| T|| < 1), then the closed unit disc D is a
spectral set for T (the von Neumann inequality; see [7, 4.3], [9, Theorem A, p.
H) is a subnormal contraction, which is not any scalar multiple of I, then the following conditions are equivalent: (a) <pT(P(D)) is antisymmetric, (b) spÄ T contains Y, (c) q>T is an isometry
- T E L If
If T E L(H) is a subnormal contraction, which is not any
scalar multiple of I, then the following conditions are equivalent:
(a) <pT(P(D)) is antisymmetric,
(b) spÄ T contains Y,
(c) q>T is an isometry.
P(D)); by the Krein-Milman theorem, P = K(P(D)) and by Proposition 3, <pT is isometric. (c)=>(a) follows from the antisymmetry of
- Ch
Ch(P(D)); by the Krein-Milman theorem, P = K(P(D)) and by Proposition
3, <pT is isometric. (c)=>(a) follows from the antisymmetry of P(D) and
Antisymmetric operator algebras, I, Ann
- F Riesz
- B Sz
- Nagy
F. Riesz and B. Sz.-Nagy, Functional analysis, Ungar, New York, 1955.
10. W. Szymanski, Antisymmetric operator algebras, I, Ann. Polon. Math, (to appear).