Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate involutions.
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$\begingroup$ How do you define "real-world"? $\endgroup$– Robert IsraelCommented May 26, 2020 at 16:58
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3$\begingroup$ I guess I'll take what I can get, with the bare minimum being "not specifically constructed for the purpose of providing an example", but it would be nice if there was some application. My best hope for a fully "natural" example is an algebra constructed from a group that only has a zero *-radical when the group has some approximation property. $\endgroup$– Cameron ZwarichCommented May 26, 2020 at 17:05
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2$\begingroup$ Hi Cam, is the Volterra algebra with conjugation of functions as the star-operation good enough? :) $\endgroup$– Yemon ChoiCommented May 26, 2020 at 23:26
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3$\begingroup$ In similar vein, I vaguely recall that the disc algebra with involution given by "take complex conjugates of all the Taylor coefficients" has bad properties as a star-algebra, this crops up somewhere in Palmer vol 2 but unfortunately I don't have a copy at hand $\endgroup$– Yemon ChoiCommented May 26, 2020 at 23:29
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$\begingroup$ Hey Yemon! The Volterra algebra is actually a pretty great example. I'm surprised it's not in Palmer volume 2 along with the others. The disc algebra example is 9.7.25 in Palmer, but it has a faithful *-representation. I'm curious whether there's a infinite-dimensional non-commutative example that isn't somehow based on a finite-dimensional or commutative example. Maybe a semigroup algebra of some kind? $\endgroup$– Cameron ZwarichCommented May 27, 2020 at 5:28
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