By John M. Bachar, William G. Bade, Philip C. Curtis, H. Garth Dales, Marc P. Thomas (eds.)
Read or Download Radical Banach Algebras and Automatic Continuity: Proceedings of a Conference Held at California State University, Long Beach, July 17–31, 1981 PDF
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Additional resources for Radical Banach Algebras and Automatic Continuity: Proceedings of a Conference Held at California State University, Long Beach, July 17–31, 1981
Example text
4. 5. "1/ Proof. If p is prime then I/p ~ $3, so X = (x~/P) p has roots of all prime orders. Let u be a nonzero element of element t, then _t 2 l~tle > O. t ~ Supp uv Now suppose that the support of r < i, u If the support of v ~ g4" Hence, u has a smallest i n f v ~ 4 flu-uvll _> has no smallest element. If r c S3 and then, using elementary arguments of number theory, we see that the decom- position r = -Pl +... + -Pk In this case, put r + r' c (0,1), Now write 9 > 0 ~4" for each n<_ ~. ,m k positive is unigue.
PROPOSITION 7-7. 7, but the only prime ideals of are ~7 and Proof. 3, there exists such that u c l~ Xn ~ , and u ~ 0. 7. All elements of Let ~l are nilpotent, so every prime ideal of ~7 contains ~. ~=a+x and ~ = b + y be two elements of ~7 such that ~ ~ ~l (where 2 a,b c ~1(]I, e -n ), x,y c ~l). Then ab + xa + yb £ ~ so that ab = O. Since 2 ~ l ( ~ e-n ) is an integral domain, one of the two elements a or b, say a, equals zero~ and ~ c ~l" So ~ is a prime ideal of be a prime 2 ideal of ~7" The projection p : a+x ~a from ~7 onto ~ l ( ~ e-n ) is a n 2 algebra homomorphism~ sothat p(J) is a prime ideal of ~l(l~, e -n ).
Such that function lira inf f3(t) = + ~. t-~0 + if We may f2(t) > fl(t) f3 such that and Put t > t 0, 0 if~(s)[d s _ f 3 ( t 0 ) t > 0, Then there is not a decreasing function there exists since if f4(t) > f3(t) t < tO . for every t > O. Also, Now set 1/t re(t) : f4(1/(s+l))ds . ~0 Then m g(t) f0(t) _> ~(t) (t > 0). lira inf fl(t)/[f0(t/n)]n = + ~ t-~0 + for every t > 0. and there exists a continuously differentiable f¼(t) < 0 g(t))= form a countable set and so there exists a con- f4(t) = f3(t0 ) Then 2 h(t)_>(~)(t such that f3(t) > Log f2(t) (t > 0).
