The spectral extension property and extension of multiplicative linear functionals

Author:
Michael J. Meyer

Journal:
Proc. Amer. Math. Soc. **112** (1991), 855-861

MSC:
Primary 46J05; Secondary 46H05

DOI:
https://doi.org/10.1090/S0002-9939-1991-1052578-X

MathSciNet review:
1052578

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Abstract: Let $\mathcal {A}$ be a commutative Banach algebra. Denote the spectral radius of an element $a$ in $\mathcal {A}$ by ${\rho _\mathcal {A}}(a)$. An extension of $\mathcal {A}$ is a Banach algebra $\mathcal {B}$ such that $\mathcal {A}$ is algebraically, but not necessarily continuously, embedded in $\mathcal {B}$. We view $\mathcal {A}$ as a subalgebra of $\mathcal {B}$. If $\mathcal {B}$ is an extension of $\mathcal {A}$ then $S{p_\mathcal {B}}(a) \cup \{ 0\} \subseteq S{p_\mathcal {A}}(a) \cup \{ 0\}$ and thus ${\rho _\mathcal {B}}(a) \leq {\rho _\mathcal {A}}(a),\forall a \in \mathcal {A}$. Let us say that $\mathcal {A}$ has the *spectral extension property* if ${\rho _\mathcal {B}}(a) = {\rho _\mathcal {A}}(a)$ for all $a \in \mathcal {A}$ and all extensions $\mathcal {B}$ of $\mathcal {A}$, that $\mathcal {A}$ has the *strong spectral extension property if* $S{p_\mathcal {B}}(a) \cup \{ 0\} = S{p_\mathcal {A}}(a) \cup \{ 0\}$ for all $a \in \mathcal {A}$ and all extensions $\mathcal {B}$ of $\mathcal {A}$, and that $\mathcal {A}$ has the *multiplicative Hahn-Banach property* if every multiplicative linear functional $\chi$ on $\mathcal {A}$ has a multiplicative linear extension to every commutative extension $\mathcal {B}$ of $\mathcal {A}$. We give characterizations of these properties for semisimple commutative Banach algebras.

- Bruce A. Barnes,
*Interpolation of spectrum of bounded operators on Lebesgue spaces*, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp. 359–378. MR**1065835**, DOI https://doi.org/10.1216/rmjm/1181073112 - Frank F. Bonsall and John Duncan,
*Complete normed algebras*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. MR**0423029** - I. M. Gel′fand, D. A. Raĭkov, and G. E. Šilov,
*Commutative normed rings*, Uspehi Matem. Nauk (N. S.)**1**(1946), no. 2(12), 48–146 (Russian). MR**0027130** - Irving Kaplansky,
*Normed algebras*, Duke Math. J.**16**(1949), 399–418. MR**31193** - M. H. A. Newman,
*Elements of the topology of plane sets of points*, Cambridge, At the University Press, 1951. 2nd ed. MR**0044820** - Theodore W. Palmer,
*Spectral algebras*, Rocky Mountain J. Math.**22**(1992), no. 1, 293–328. MR**1159960**, DOI https://doi.org/10.1216/rmjm/1181072812 - Charles E. Rickart,
*General theory of Banach algebras*, The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0115101** - C. E. Rickart,
*On spectral permanence for certain Banach algebras*, Proc. Amer. Math. Soc.**4**(1953), 191–196. MR**52700**, DOI https://doi.org/10.1090/S0002-9939-1953-0052700-4 - G. Šilov,
*On the extension of maximal ideals*, C. R. (Doklady) Acad. Sci. URSS (N.S.)**29**(1940), 83–84. MR**0004073**

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© Copyright 1991
American Mathematical Society