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Tutorial on Von Neumann Algebras

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Tutorial on Von Neumann Algebras Tutorial on von Neumann algebras Adrian Ioana University of California, San Diego Model Theory and Operator Algebras UC Irvine September 20-22, 2017 1/174 the adjoint T ⇤ B(H) given by T ⇠,⌘ = ⇠,T ⇤⌘ ,forall⇠,⌘ H. 2 h i h i 2 the spectrum of T is σ(T )= λ C T λ I not invertible . { 2 | − } Fact: σ(T ) is a compact non-empty subset of C. Definition An operator T B(H)iscalled 2 1 self-adjoint if T = T ⇤. 2 positive if T ⇠,⇠ 0, for all ⇠ H. (positive self-adjoint) h i 2 ) 3 a projection if T = T = T 2 T is the orthogonal projection onto ⇤ , a closed subspace K H. ⇢ 4 a unitary if TT ⇤ = T ⇤T = I . The algebra of bounded linear operators H separable Hilbert space, e.g. `2(N), L2([0, 1], Leb). B(H) algebra of bounded linear operators T : H H, ! i.e. such that T =sup T ⇠ ⇠ =1 < . k k {k k|k k } 1 2/174 the spectrum of T is σ(T )= λ C T λ I not invertible . { 2 | − } Fact: σ(T ) is a compact non-empty subset of C. Definition An operator T B(H)iscalled 2 1 self-adjoint if T = T ⇤. 2 positive if T ⇠,⇠ 0, for all ⇠ H. (positive self-adjoint) h i 2 ) 3 a projection if T = T = T 2 T is the orthogonal projection onto ⇤ , a closed subspace K H. ⇢ 4 a unitary if TT ⇤ = T ⇤T = I . The algebra of bounded linear operators H separable Hilbert space, e.g. `2(N), L2([0, 1], Leb). B(H) algebra of bounded linear operators T : H H, ! i.e. such that T =sup T ⇠ ⇠ =1 < . k k {k k|k k } 1 the adjoint T ⇤ B(H) given by T ⇠,⌘ = ⇠,T ⇤⌘ ,forall⇠,⌘ H. 2 h i h i 2 3/174 Definition An operator T B(H)iscalled 2 1 self-adjoint if T = T ⇤. 2 positive if T ⇠,⇠ 0, for all ⇠ H. (positive self-adjoint) h i 2 ) 3 a projection if T = T = T 2 T is the orthogonal projection onto ⇤ , a closed subspace K H. ⇢ 4 a unitary if TT ⇤ = T ⇤T = I . The algebra of bounded linear operators H separable Hilbert space, e.g. `2(N), L2([0, 1], Leb). B(H) algebra of bounded linear operators T : H H, ! i.e. such that T =sup T ⇠ ⇠ =1 < . k k {k k|k k } 1 the adjoint T ⇤ B(H) given by T ⇠,⌘ = ⇠,T ⇤⌘ ,forall⇠,⌘ H. 2 h i h i 2 the spectrum of T is σ(T )= λ C T λ I not invertible . { 2 | − } Fact: σ(T ) is a compact non-empty subset of C. 4/174 3 a projection if T = T = T 2 T is the orthogonal projection onto ⇤ , a closed subspace K H. ⇢ 4 a unitary if TT ⇤ = T ⇤T = I . The algebra of bounded linear operators H separable Hilbert space, e.g. `2(N), L2([0, 1], Leb). B(H) algebra of bounded linear operators T : H H, ! i.e. such that T =sup T ⇠ ⇠ =1 < . k k {k k|k k } 1 the adjoint T ⇤ B(H) given by T ⇠,⌘ = ⇠,T ⇤⌘ ,forall⇠,⌘ H. 2 h i h i 2 the spectrum of T is σ(T )= λ C T λ I not invertible . { 2 | − } Fact: σ(T ) is a compact non-empty subset of C. Definition An operator T B(H)iscalled 2 1 self-adjoint if T = T ⇤. 2 positive if T ⇠,⇠ 0, for all ⇠ H. (positive self-adjoint) h i 2 ) 5/174 4 a unitary if TT ⇤ = T ⇤T = I . The algebra of bounded linear operators H separable Hilbert space, e.g. `2(N), L2([0, 1], Leb). B(H) algebra of bounded linear operators T : H H, ! i.e. such that T =sup T ⇠ ⇠ =1 < . k k {k k|k k } 1 the adjoint T ⇤ B(H) given by T ⇠,⌘ = ⇠,T ⇤⌘ ,forall⇠,⌘ H. 2 h i h i 2 the spectrum of T is σ(T )= λ C T λ I not invertible . { 2 | − } Fact: σ(T ) is a compact non-empty subset of C. Definition An operator T B(H)iscalled 2 1 self-adjoint if T = T ⇤. 2 positive if T ⇠,⇠ 0, for all ⇠ H. (positive self-adjoint) h i 2 ) 3 a projection if T = T = T 2 T is the orthogonal projection onto ⇤ , a closed subspace K H. ⇢ 6/174 The algebra of bounded linear operators H separable Hilbert space, e.g. `2(N), L2([0, 1], Leb). B(H) algebra of bounded linear operators T : H H, ! i.e. such that T =sup T ⇠ ⇠ =1 < . k k {k k|k k } 1 the adjoint T ⇤ B(H) given by T ⇠,⌘ = ⇠,T ⇤⌘ ,forall⇠,⌘ H. 2 h i h i 2 the spectrum of T is σ(T )= λ C T λ I not invertible . { 2 | − } Fact: σ(T ) is a compact non-empty subset of C. Definition An operator T B(H)iscalled 2 1 self-adjoint if T = T ⇤. 2 positive if T ⇠,⇠ 0, for all ⇠ H. (positive self-adjoint) h i 2 ) 3 a projection if T = T = T 2 T is the orthogonal projection onto ⇤ , a closed subspace K H. ⇢ 4 a unitary if TT ⇤ = T ⇤T = I . 7/174 strong operator topology (SOT): T T T ⇠ T ⇠ 0, i ! () k i − k! for all ⇠ H. 2 weak operator topology (WOT): T T T ⇠,⌘ T ⇠,⌘ , i ! () h i i!h i for all ⇠,⌘ H. 2 Exercise 1: Suppose that T T (WOT) and T , T are all projections i ! i (or all unitaries). Prove that T T (SOT). i ! Definition A subalgebra A B(H)iscalleda -algebra if T ⇤ A,forallT A. ⇢ ⇤ 2 2 A -subalgebra A B(H)iscalleda ⇤ ⇢ 1 C⇤-algebra if it is closed in the norm topology. 2 von Neumann algebra if it is WOT-closed. Terminology. A homomorphism ⇡ : A B between two -algebras is ! ⇤ called a -homomorphism if ⇡(T )=⇡(T ) ,forallT A. ⇤ ⇤ ⇤ 2 Abijective -homomorphism ⇡ : A B is called a -isomorphism. ⇤ ! ⇤ Topologies on B(H) norm topology: T T T T 0. i ! () k i − k! 8/174 Exercise 1: Suppose that T T (WOT) and T , T are all projections i ! i (or all unitaries). Prove that T T (SOT). i ! Definition A subalgebra A B(H)iscalleda -algebra if T ⇤ A,forallT A. ⇢ ⇤ 2 2 A -subalgebra A B(H)iscalleda ⇤ ⇢ 1 C⇤-algebra if it is closed in the norm topology. 2 von Neumann algebra if it is WOT-closed. Terminology. A homomorphism ⇡ : A B between two -algebras is ! ⇤ called a -homomorphism if ⇡(T )=⇡(T ) ,forallT A. ⇤ ⇤ ⇤ 2 Abijective -homomorphism ⇡ : A B is called a -isomorphism. ⇤ ! ⇤ Topologies on B(H) norm topology: T T T T 0. i ! () k i − k! strong operator topology (SOT): T T T ⇠ T ⇠ 0, i ! () k i − k! for all ⇠ H. 2 weak operator topology (WOT): T T T ⇠,⌘ T ⇠,⌘ , i ! () h i i!h i for all ⇠,⌘ H. 2 9/174 Definition A subalgebra A B(H)iscalleda -algebra if T ⇤ A,forallT A. ⇢ ⇤ 2 2 A -subalgebra A B(H)iscalleda ⇤ ⇢ 1 C⇤-algebra if it is closed in the norm topology. 2 von Neumann algebra if it is WOT-closed. Terminology. A homomorphism ⇡ : A B between two -algebras is ! ⇤ called a -homomorphism if ⇡(T )=⇡(T ) ,forallT A. ⇤ ⇤ ⇤ 2 Abijective -homomorphism ⇡ : A B is called a -isomorphism. ⇤ ! ⇤ Topologies on B(H) norm topology: T T T T 0. i ! () k i − k! strong operator topology (SOT): T T T ⇠ T ⇠ 0, i ! () k i − k! for all ⇠ H. 2 weak operator topology (WOT): T T T ⇠,⌘ T ⇠,⌘ , i ! () h i i!h i for all ⇠,⌘ H. 2 Exercise 1: Suppose that T T (WOT) and T , T are all projections i ! i (or all unitaries). Prove that T T (SOT). i ! 10 / 174 A -subalgebra A B(H)iscalleda ⇤ ⇢ 1 C⇤-algebra if it is closed in the norm topology. 2 von Neumann algebra if it is WOT-closed. Terminology. A homomorphism ⇡ : A B between two -algebras is ! ⇤ called a -homomorphism if ⇡(T )=⇡(T ) ,forallT A. ⇤ ⇤ ⇤ 2 Abijective -homomorphism ⇡ : A B is called a -isomorphism. ⇤ ! ⇤ Topologies on B(H) norm topology: T T T T 0. i ! () k i − k! strong operator topology (SOT): T T T ⇠ T ⇠ 0, i ! () k i − k! for all ⇠ H. 2 weak operator topology (WOT): T T T ⇠,⌘ T ⇠,⌘ , i ! () h i i!h i for all ⇠,⌘ H. 2 Exercise 1: Suppose that T T (WOT) and T , T are all projections i ! i (or all unitaries). Prove that T T (SOT). i ! Definition A subalgebra A B(H)iscalleda -algebra if T ⇤ A,forallT A. ⇢ ⇤ 2 2 11 / 174 2 von Neumann algebra if it is WOT-closed. Terminology. A homomorphism ⇡ : A B between two -algebras is ! ⇤ called a -homomorphism if ⇡(T )=⇡(T ) ,forallT A. ⇤ ⇤ ⇤ 2 Abijective -homomorphism ⇡ : A B is called a -isomorphism. ⇤ ! ⇤ Topologies on B(H) norm topology: T T T T 0. i ! () k i − k! strong operator topology (SOT): T T T ⇠ T ⇠ 0, i ! () k i − k! for all ⇠ H. 2 weak operator topology (WOT): T T T ⇠,⌘ T ⇠,⌘ , i ! () h i i!h i for all ⇠,⌘ H. 2 Exercise 1: Suppose that T T (WOT) and T , T are all projections i ! i (or all unitaries). Prove that T T (SOT). i ! Definition A subalgebra A B(H)iscalleda -algebra if T ⇤ A,forallT A. ⇢ ⇤ 2 2 A -subalgebra A B(H)iscalleda ⇤ ⇢ 1 C⇤-algebra if it is closed in the norm topology. 12 / 174 Terminology. A homomorphism ⇡ : A B between two -algebras is ! ⇤ called a -homomorphism if ⇡(T )=⇡(T ) ,forallT A.
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