Fundamentals of the Theory of Operator Algebras. V2: Advanced Theory

abelian abelian projections abelian von Neumann automorphic representation automorphism group bounded linear C*-algebra QI C*-norm C*-subalgebra central carrier central projection closure commutant compact contains Corollary corresponding countably countably decomposable defined dense direct sum element everywhere Exercise factor of type faithful normal follows Hence Hilbert space Hint homomorphism isomorphism Lemma linear functional linear mapping linear span matrix units maximal abelian measurable minimal projections modular automorphism group Moreover multiplication Neumann algebra acting non-zero projection norm normal semi-finite notation null ideal null set orthogonal family p(QI partial isometry polar decomposition positive integer positive linear Proof properly infinite properties Proposition prove range real number representation of QI Section self-adjoint operator separable Hilbert space sequence Show subalgebra subfactor subprojection subspace Suppose that QI t(QI tensor product Theorem two-sided ideal ultraweak ultraweakly continuous unique unit ball unit vector unitarily equivalent unitary operator von Neumann algebra weak-operator continuous