An Invitation to C*-Algebras
This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basic ideas, as simply and concretely as we could. So whenever it is convenient (and it usually is), Hilbert spaces become separable and C*-algebras become GCR. This practice probably creates an impression that nothing of value is known about other C*-algebras. Of course that is not true. But insofar as representations are con cerned, we can point to the empirical fact that to this day no one has given a concrete parametric description of even the irreducible representations of any C*-algebra which is not GCR. Indeed, there is metamathematical evidence which strongly suggests that no one ever will (see the discussion at the end of Section 3. 4). Occasionally, when the idea behind the proof of a general theorem is exposed very clearly in a special case, we prove only the special case and relegate generalizations to the exercises. In effect, we have systematically eschewed the Bourbaki tradition. We have also tried to take into account the interests of a variety of readers. For example, the multiplicity theory for normal operators is contained in Sections 2. 1 and 2. 2. (it would be desirable but not necessary to include Section 1. 1 as well), whereas someone interested in Borel structures could read Chapter 3 separately. Chapter I could be used as a bare-bones introduction to C*-algebras. Sections 2.
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abelian von Neumann analytic Borel space analytic set assertion assume Banach algebra belongs Borel cross section Borel function Borel map Borel measure Borel set Borel structure Borel subset bounded C*-algebra C*-subalgebra CCR ideal central subrepresentation choose claim commutative C*-algebra compact operators complex composition series conclude contains a nonzero continuous function Corollary countably separated decomposition define denote direct sum disjoint equivalence class equivalence relation finite Borel measure follows Hilbert space homomorphism identity representation implies invariant irr(A irreducible representations isometric lemma let f measure classes metric multiplicity multiplicity-free representation Neumann algebra norm normal operators Note open sets orthogonal Polish space Proof properties prove quotient range space resp satisfying self-adjoint element separable C*-algebra separable Hilbert space sequence spectral spectrum standard Borel space strong closure subrepresentation subspace suffices to show theorem unique unitarily unitary equivalence von Neumann algebra weak closure