Norms, States and Numerical Ranges on Direct Sums

Gaur, A. K.; Kováfik, Zdislav V.

doi:10.1524/anly.1991.11.23.155

Cited by 7 publications

(9 citation statements)

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“…Proposition 4.3 in [3] states that among these extensions, the /i-norm ||Ae + a||i = |A|-(-||a|| is maximal and the operator norm \\Xe + a\\op = sup{||Ax + ax\\ : \\x\\ < 1} is minimal, provided that it does extend || • ||, i.e., that || • || is a regular (= operator) norm.…”

Section: Introductionmentioning

confidence: 99%

Norms on Unitizations of Banach Algebras

Gaur¹,

Kovarik²

1993

Proceedings of the American Mathematical Society

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Abstract.Equivalence of various norms on the unitization of a nonunital Banach algebra is established, with bounds (1 and 6exp(l)) uniform over the class of such algebras. A tighter bound, 3, is obtained in C*-algebras for elements with Hermitian nonunital parts.The algebra norm || • || on a nonunital Banach algebra A can be extended to an algebra norm on the unitization A+ in many ways. Proposition 4.3 in [3] states that among these extensions, the /i-norm ||Ae + a||i = |A|-(-||a|| is maximal and the operator norm \\Xe + a\\op = sup{||Ax + ax\\ : \\x\\ < 1} is minimal, provided that it does extend || • ||, i.e., that || • || is a regular (= operator) norm.In the latter case, A+ is complete under both || • ||, and || • ||op, so by the "two-norm lemma" [2, II.2.5] these two norms are equivalent; the pure existence nature of the lemma does not yield an explicit bound M in || • ||i < M\\ • ||op and such a bound seems to depend on the algebra A .The present theorem establishes uniform equivalence of the two unitization norms over the class of nonunital Banach algebras with regular norms.Theorem. For every nonunital Banach algebra A with unitization A+ and with regular norm, and for every X £ C and a £ A, we have \\Xe + a\\op < ||/te + a||i < (6exp 1)||A^ + a||op.If A is a C*-algebra, a £ A is hermitian, and X is complex then \\Xe + a\\x <3||Ae + a||op and the constant 3 is best (minimal) possible. Proof. In a general algebra A with a regular norm, we have an extension of the classical inequality for the numerical radius v(a) [1, Theorem 4.1]:v(a) < \\a\\ < (expl)v(a).

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Section: Introductionmentioning

confidence: 99%

Norms on Unitizations of Banach Algebras

Gaur¹,

Kovarik²

1993

Proceedings of the American Mathematical Society

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“…Then a + λe 1 3 a + λe op for all a ∈ A and λ ∈ C. Remark 7. Sometimes (see [1], [2]) the regular norms are defined as those satisfying a = sup ax : x ∈ X, x = 1 for all a. All results of this paper remain true for this definition without any change.…”

Section: Remarkmentioning

confidence: 94%

“…By [1], both of the norms above on A + are extensions of and the 1 -norm is the maximal and the operator norm the minimal among all regular extensions of the original norm on A. It was shown in [2] that…”

mentioning

confidence: 99%

Norms on unitizations of banach algebras revisited

Arhippainen

Müller

2006

Acta Math Hung

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Let A be an algebra without unit. If is a complete regular norm on A it is known that among the regular extensions of to the unitization of A there exists a minimal (operator extension) and maximal ( 1 -extension) which are known to be equivalent. We shall show that the best upper bound for the ratio of these two extensions is exactly 3. This improves the results represented by A. K. Gaur and Z. V. Kovářík and later by T. W. Palmer.Let A be an algebra and let be a submultiplicative norm on A. So if is complete, then A, is a Banach algebra. We shall say that is regular if a = sup ax , xa : x ∈ A, x = 1 for all a ∈ A. A norm is called weakly regular if there exists a constant M > 0 such thatfor all a ∈ A. If an algebra A has an identity element (denoted by e) for which e = 1, then is automatically regular. * The second author was partially supported by the grant No. 201/03/0041 of GAČR.

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“…op ) is a complex Banach algebra. The following question was asked [3]: If (A e , . op ) is a complex Banach algebra, is the norm .…”

Section: Regular Norm and The Operator Seminormmentioning

confidence: 99%