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**Sample text**

Applying the Gelfand representation again, this time to the C*-algebra generated by 1 and a, we obtain the inequality ||a|| < ||6||. T o prove Condition ( 4 ) we first observe that if c > 1, then c is invertible and c " < 1. This is given by the Gelfand representation applied to the C*-subalgebra generated by 1 and c. Now a < b 1 = a~ l aa~ f < a~ l ba- l => (a-'^ba- / )< 1, that is, a l b~ a / < 1. Hence, b- ^ ( a / ) " ^ / ) - = a " . • 1 l 2 x 2 1 x 2 1 2 1 2 1 1 2 1 l 2 l l 2 1 2 1 1 2 . 2 .

T h e n 2 2 c = f ((* + b + a)(t + b-a))+(t = t 2 + 2tb + b 2 + b- a)(t + b + a)) a 2 >t . 2 Consequently, c is both invertible and positive. Since 1 + ic~" f dc~ ^ = c~ l (c + id)c~ l is invertible, therefore c + id is invertible. It follows that t + b — a is left invertible, and therefore invertible, because it is hermitian. Consequently, — t £ a(b — a). Hence, a(b — a) C R + , so b — a is positive, that is, a < b. • 1 2 x 2 1 2 x 2 It is not true that 0 < a < b a < b in arbitrary C*-algebras.

If f,g G L (G), then there is an element / *g G L (G) such that l l l l =J (f*g)(*) f(*-y)g(y)My) for almost all x in G. T h e product / * g is the convolution o f / and g. Under the multiplication operation given by (f,g) f * g, L (G) is an abelian Banach algebra, called the group algebra o f G. It has a unit if and only if G is discrete. Let G be the dual group of G, that is, the set of continuous h o m o morphisms 7 from G to T . This is endowed with a suitable topology making it a locally c o m p a c t group.