In CQ -type and -type direct sums of Banach algebras, we study the behavior of regular (also called operator) norms and of state preserving operators, and how numerical ranges are built from those of the components. We also discuss the importance of regular norms in unitization of algebras. Finally, numerical ranges in ε Θ (Χ) are described.
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).
ABSTRACT. In this paper, the notion of spatial numerical range of elements of Banach algebras without identity is studied. Specifically, the relationship between spatial numerical ranges, numerical ranges and spectra is investigated. Among other results, it is shown that the closure of the spatial numerical range of an element of a Banach algebra without Identity but wlth regular norm Is exactly its numerical range as an element of the unitized algebra.Futhermore, the closure of the spatial numerical range of a hermltlan element coincides with the convex hull of Its spectrum. In particular, spatial numerical ranges of the elements of the Banach algebra C (X) are o described.
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