### Joint spectra and multiplicative functionals

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Some inequalities are proved between the geometric joint spectral radius (cf. [3]) and the joint spectral radius as defined in [7] of finite commuting families of Banach algebra elements.

We present several notions of joint spectral radius of mutually commuting elements of a locally convex algebra and prove that all of them yield the same value in case the algebra is pseudo-complete. This generalizes a result proved by the author in 1993 for elements of a Banach algebra.

The aim of this paper is to characterize a class of subspectra for which the geometric spectral radius is the same and depends only upon a commuting $n$-tuple of elements of a complex Banach algebra. We prove also that all these subspectra have the same capacity.

A formula is given for the (joint) spectral radius of an n-tuple of mutually commuting Hilbert space operators analogous to that for one operator. This gives a positive answer to a conjecture raised by J. W. Bunce in [1].

Agrafeuil and Zarrabi in [1] characterized all closed ideals with at most countable hull in a unital Banach algebra embedded in the classical disc algebra and satisfying certain conditions ((H1), (H2), (H3)), and the analytic Ditkin condition. We modify Ditkin’s condition and show that analogous result is true for a wider class of algebras. This is an extension of the result obtained in [1].

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