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By Waclaw Sierpinski, I. N. Sneddon, M. Stark

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**Extra resources for A Selection of Problems in the Theory of Numbers **

**Example text**

This factorization is obviously an involutive algebra homomorphism. 20), -1 it may be extended continuously to the completion of E~ q (0). 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}.

In particular, a Gelfand C*-algebra E is unital iff a ( E ) is compact. Let E be a Gelfand C*-algebra. 16 b), r(x) = I1~11 for every x C E . 13, the Gelfand transform of E is an isometry of C*-algebras. m Remark. H. Stone proved (1940) the above result for Gelfand C*-subalgebras of the C*-algebra of operators on a given Hilbert space. 6 Let E be a (unital) C*-algebra and F a (unital) Gelfand C*-subalgebra of E . Then the following are equivalent: a) F is separable. b) There is an x E S n E such that F = E(x) (F = E(x, 1)).

Step 1 i ~ a(x) Assume that i E a ( x ) . 1). We deduce that 2~ + 1 <__ I1~= I, which is a contradiction. 1 The General Theory 17 Take a, 3 C lR with a + i3 C a ( x ) . Assume that ~ # 0. 4 b)) which contradicts Step 1. Hence Z = 0 and a(x) C JR. 16 ( 0 ) Let x be a self-normal element of a C*-algebra. e b) r ( x ) = Ilxll. c) (x quasinilpotent) r ( a ( x ) = {0}) v:~ (x = 0). 2 b). 15). 4, r(x) = lim Ilxnll ! 21 d), (x = 0 ) ~ (x q u a s i n i l p o t e n t ) ==~ (a(x)= {0}) (~(x) = O) ~ (llxll = 0) ~ (x = 0).

### A Selection of Problems in the Theory of Numbers by Waclaw Sierpinski, I. N. Sneddon, M. Stark

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