A Banach algebra A Banach algebra A such that Rad(A) is a proper subset of the set {x ; r(x) = 0} of all quasi-nilpotent elements.

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1.Suppose that H be a Hilbert space with dim H ³ 2. Let x, y Î H-{0} and < x,y > = 0. The norm of rank one operator (x[`(Ä)]y)(z) = < z,y > x is ||x ||||y || ¹ 0. So x[`(Ä)] y ¹ 0. Also (x[`(Ä)] y)2(z) = (x[`(Ä)] y)( < z,y > x) = < z,y > < x,y > x = 0 so (x[`(Ä)] y)2 = 0. Hence it is quasi-nilpotent. But B(H) is semi-simple. Therefore x[`(Ä)] y Ï Rad(B(H)) = {0}.

2.Let A = M2(C) @ B(C2) . A is a C*-algebra so Rad(A) = {0}.
The element (
0
1
0
0
) has the spectrum {0} and so r((
0
1
0
0
)) = 0. Hence Rad(A) is not equal to {x ; r(x) = 0}.

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On 22 Feb 2001, 00:13.