By I. Kaplansky

ISBN-10: 0821816500

ISBN-13: 9780821816509

An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and uncomplicated effects Derivations of $C^*$-algebras Homogeneous $C^*$-algebras CCR-algebras $W^*$ and $AW^*$-algebras Miscellany Mappings conserving invertible components Nonassociativity Bibliography

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**Sample text**

To conclude this section, we note that there is the notion of least common multiple or lcm of integers a and b. This is deﬁned to be the positive integer m such that both a and b divide m (so m is a common multiple of a and b), and such that m divides every common multiple of a and b. It is denoted by lcm(a, b). The proof that such an integer m does exist, and is unique, is left as an exercise. More generally, given non-zero integers a1 , . . , an , we deﬁne their least common multiple, lcm(a1 , .

Suppose that [a]n = [c]n . 4 Congruence classes 41 Then: (i) [a + b]n = [c + b]n , and (ii) [ab]n = [cb]n . Proof (i) Since [a]n = [c]n , n divides c − a. So we can write c = a + kn for some integer k. Therefore [c + b]n = [a + kn + b]n = [a + b + kn]n = [a + b]n (by deﬁnition of congruence class) as required. (ii) With the above notation, we have that [cb]n = [(a + kn)b]n = [ab + nkb]n = [ab]n . Comment The proof itself is, we hope, easy to follow line by line. In the discussion before the result we tried to explain the purpose of the theorem and proof.

The following result is sometimes referred to as the Fundamental Theorem of Arithmetic. It says that, in some sense, the primes are the multiplicative building blocks from which every (positive) integer may be produced in a unique way. 28 Number theory Therefore positive integers, other than 1, which are not prime are referred to as composite. The distinction between prime and composite numbers, and the importance of this distinction, was recognised at least as early as the time of Philolaus (who died around 390 bc).

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