By Ola Bratteli
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Extra resources for Operator Algebras and Quantum Statistical Mechanics 1 : C*- and W*-Algebras. Symmetry Groups. Decomposition of States (Texts and Monographs in Physics)
The defines norm uniform are satisfying e the requirements topology. given by a The topology on neighborhoods of an metric ,&(A; e) = JB; B c- 91, JIB % which is element A - All < c- referred to % in this as the topology ej, complete with respect to the uniform topology then it is called a Banach algebra. A normed algebra with involution which is complete and has the property 11 A 11 11 A* 11 is called a Banach *-algebra. Our principal definition is the following: where E > 0. 1. C*-algebra Banach is a IIA*All for all A The e = *-algebra % with the property JIAI12 91.
A, contradiction. The C*-norm property follows 11 (a, A) 112 < = B and A = = = JIB B 91 is A)* 1111 (a, A) 11. Therefore (a, A) 11 < 11 (a, A)* 11. = 11i C*-Algebras But the reverse follows Inequality 11(a, A) 11 and the desired conclusion The completeness is 2 by replacing (a, A) with 11(a, A)*(a, A)11 :! 23 Hence A 11(a, A) 112 :! established. it follows straightforwardly from the completeness of C of and 91. 6. 5. We C1 + 91. pair (oc, A) and write W without is identity. defined use as the The C*-algebra algebra of pairs the notation aT + A for the = Note that if W is have a identity T identity.
1. - it c- an invertible and the spectrum r,(A) in C. The inverse (Al - 1. The resolvent set algebra with identity W is defined as the set of A u,(A) of A is defined where A c- r,(A), A) c- C such that Al is - A is the complement of called the resolvent of as A at A. in The spectrum of an element of a general algebra can be quite arbitrary but a Banach algebra, and in particular in a C*-algebra, the situation is quite simple, as we There one are of the example, A will see. various simplest c- C and techniques for analyzing resolvents and spectra, and by series expansion and analytic continuation.