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If P is a closed prime ideal in a commutative unital complex Banach algebra, then the hull of P is connected (an easy consequence of Shilov's idempotent theorem) Is the same true for non-closed prime ideals?

ray
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  • Can I assume the algebra is semisimple? (Also, I assume you are referring to the hull-kernel topology) – Yemon Choi Feb 12 '17 at 17:21
  • I just edited the post; meant of course connected. One may restrict to the semi-simple case. – ray Feb 13 '17 at 07:57
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    @ Yemon Choi No, I refer to the Gelfand topology, the natural one which makes the maximal ideal space a compact Hausdorff space. – ray Mar 04 '17 at 15:28

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