By Dickson L.E.

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**Extra resources for History of the theory of numbers: diophantine analysis**

**Example text**

13). Assume that x is not invertible in F . We want to show that x is not invertible in E . 10 a :=v b, x is a topological divisor of 0 in F . Then x is afortiori a topological divisor of 0 in E . 10 b =~ a, x is not invertible in E . 3, F(x, 1) is a Gelfand C*-algebra. Since x is not invertible in F(x, 1), it is not invertible in E by the first case. Case 3 The General Case Assume that x is invertible in E . 14). 6). 6), which is a contradiction. Hence x is not invertible in E . 14 a). I ( 0 ) Let E be a (unital) C*-algebra.

By continuity, this extention is an involutive algebra homomorphism. 20. 23 I Let E be a finite-dimensional symmetric, involutive, semi-simple, complex Gelfand algebra and u the Gelfand transform on E . 14, u is injective. 21. 4). Given x E E , define 5"E >E , y~ ) xyx*. Then ~ = 5" E ~ for every x E U, the map U ~, x: ;5 is a continuous group homomorphism, and EC= {x E E ] ~ = identity map} = {x E U I ~ = identity map}. Take x E U. Then 5 is linear, bijective, 51 = xlx* = 1, ~y* = xy*x* = ( ~ j ) * = (~y)*, 22 ,~.

In particular, a C*-subalgebr~ of a Gelfand C*-algebra is a Gelfand C*-algebra. 2 Let E be a commutative real C*-algebra such that Re E = E . 1, /~ is symmetric. 29, for every x E E . Hence E is a Gelfand C*-subalgebra of E . 3 ( 0 ) L~t ~ b~ ~ C*-~gg~b~a ~ m A a ~omm=t~t~v~ subset of E . If IK = qJ (IK - JR), then we assume that A U A* is commutative (that A C Re E ). Then the C*-subalgebra of E generated by A is a Gelfand C*-algebra. Moreover, A is contained in a maximal Gelfand C*-subalgebra of E.