The normed space numerical index of 𝐶*-algebras
| Autor: | Tadashi Huruya |
|---|---|
| Rok vydání: | 1977 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 63:289-290 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1977-0438138-2 |
| Popis: | Given a complex C*-algebra X, we prove that the normed space numerical index n(X) of X is 1 or I according as X is commutative or not commutative. Let X be a normed linear space, X* its dual space, and B(X) the normed algebra of all bounded linear operators on X with operator norm. Given T E B(X), the numerical range V(T), and the numerical radius v(T) of Tare defined by V(T) = {f(Tx): x E X,f E X*,f(x) = llxll = Ilifi = 1}, v(T) = sup{ IXI: X E V(T)). The numerical index n(X) of the space X is defined by n(X) = inf{v(T): T E B(X), || TIj = 1). Standard results for n(X) are given in [1, ?32]. THEOREM. Let X be a complex C*-algebra. Then n(X) is 1 or 2 according as X is commutative or not commutative. PROOF. If X is commutative, n(X) = 1 by [1, Theorem 32.8]. If X is not commutative, the numerical index of X as an algebra is 2 by [2, Theorem 3]. Since the left regular representation of X is isometric, it follows that n(X ) 2. We may assume (by embedding X in its Arens second dual) that X is unital. Let e > 0. Choose y E X with IIYII = 1, IlTyll > 1 E. By [3, Theorem 1] there exist positive real numbers a1, ..., an with IL aj = 1 and unitary elements Ul, . . ., uo of X such that LIy Ejn a1ju,![ I E. For some j with 1 1 2e. Choose a state g of X such that jg((Tu1)u7*)l > '(I 2e). Then (z '(1 2c). The proof is complete. REMARK. The final step in the proof is based on an idea of Crabb (see [1, Theorem 32.9]). Received by the editors July 30, 1976 and, in revised form, November 22, 1976. AMS (MOS) subject classifications (1970). Primary 46L05, 47A10. |
| Databáze: | OpenAIRE |
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