An Invitation to C*-Algebras by William Arveson

By William Arveson

This e-book provides an creation to C*-algebras and their representations on Hilbert areas. now we have attempted to provide merely what we think are the main uncomplicated principles, as easily and concretely as lets. So every time it truly is handy (and it always is), Hilbert areas turn into separable and C*-algebras develop into GCR. this custom most likely creates an impact that not anything of price is understood approximately different C*-algebras. in fact that isn't precise. yet insofar as representations are con­ cerned, we will element to the empirical undeniable fact that to at the present time not anyone has given a concrete parametric description of even the irreducible representations of any C*-algebra which isn't GCR. certainly, there's metamathematical proof which strongly means that not anyone ever will (see the dialogue on the finish of part three. 4). sometimes, whilst the assumption at the back of the facts of a common theorem is uncovered very in actual fact in a distinct case, we turn out merely the targeted case and relegate generalizations to the routines. In influence, we've systematically eschewed the Bourbaki culture. now we have additionally attempted take into consideration the pursuits of various readers. for instance, the multiplicity concept for regular operators is contained in Sections 2. 1 and a couple of. 2. (it will be fascinating yet now not essential to contain part 1. 1 as well), while somebody attracted to Borel buildings may learn bankruptcy three individually. bankruptcy i'll be used as a bare-bones creation to C*-algebras. Sections 2.

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0 Corollary. For every nonzero element z in A, there is an irreducible representation it of A on a Hilbert space A° and a unit vector in A° such that lin(z)il = lizli > O. Taking x = z*z in the preceding theorem, we obtain a pure state f of A such that f(z* z) = liz * zil = 11z112. 3). Then PROOF. lin(z)11 2 = (7r(z), 7r(z)) = (7r(z* z), ) = fir z) = 11z112,, and hence lirc(z)ll = lizli. 6, complete. it is irreducible, and the proof is 0 This corollary can be used to prove that every locally compact group has "sufficiently many" irreducible unitary representations.

This property can be generalized in the following way. A composition series in a C*-algebra A is a family of ideals {JOE ; 0 a ao } indexed by the ordinals a, 0 a oco , having the following properties: (i) for all a < ao , JOE is contained properly in JOE+1 ; (ii) J0 = 0, JOE° = A; (iii) if fi is a limit ordinal then 4 is the norm closure of UOE<13 JOE. Thus in the example we have a composition series of length 3, though in general of course a composition series can be infinite. 5. Every GCR algebra A has exactly one composition series a °c o} with the property that JOE+ i lJOE is the largest CCR ideal in {JŒ :0 AIJOE for every a, 0 a < Geo .

We first claim that f is bounded on the positive part 13 + of the unit ball of A (here, of course, 13 + denotes the set of self-adjoint elements x satisfying x 0 and 11x11 s 1). Notice that f(x) 0 for every x in B + , so if f is unbounded on B + then we may find a sequence xn e B + such that f(x n) > 1/n. Consider the infinite series En - 2 xn . B). Similarly, letting yk denote the k-th partial sum of the series, we have x yk and so f(x) f(yk). This implies that f(x) En - 2f(xn) En - 1 = + oo, a contradiction.

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