A survey of the model theory of tracial von Neumann algebras Isaac Goldbring University of Illinois at Chicago Harvard-MIT Logic Seminar November 11, 2013 Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 1 / 34
Introduction to von Neumann algebras Introduction to von Neumann algebras 1 Isomorphic ultrapowers 2 Connes Embedding Problem 3 4 Model companions Computability theory 5 6 Future directions Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 2 / 34
Introduction to von Neumann algebras Von Neumann algebras Throughout, H is a complex Hilbert space and B ( H ) is the ∗ -algebra of bounded operators on H . For X ⊆ B ( H ) , we set X ′ := { T ∈ B ( H ) : TS = ST for all S ∈ X } . Observe that: X ′ is a unital subalgebra of B ( H ) that is closed under ∗ if X is. X ⊆ X ′′ . Definition A von Neumann algebra is a ∗ -subalgebra M of B ( H ) such that M = M ′′ . Equivalently, M is a ∗ -subalgebra of B ( H ) that is closed in either the weak operator topology or the strong operator topology . Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 3 / 34
Introduction to von Neumann algebras Examples of vNas Example B ( H ) is a von Neumann algebra. Example Suppose that ( X , µ ) is a finite measure space. Then L ∞ ( X , µ ) acts on the Hilbert space L 2 ( X , µ ) by left multiplication, yielding an embedding L ∞ ( X , µ ) ֒ → B ( L 2 ( X , µ )) , the image of which is a von Neumann algebra. (Actually, all abelian von Neumann algebras are isomorphic to some L ∞ ( X , µ ) , whence von Neumann algebra theory is sometimes dubbed “noncommutative measure theory.”) Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 4 / 34
Introduction to von Neumann algebras Group von Neumann algebras Example Suppose that G is a locally compact group and α : G → B ( H ) is a unitary group representation. Then the group von Neumann algebra of α is α ( G ) ′′ . (Understanding α ( G ) ′′ is tantamount to understanding the invariant subspaces of α .) In the important special case that α : G → B ( L 2 ( G )) (where G is equipped with its haar measure) is given by left translations α ( g )( f )( x ) := f ( g − 1 x ) , we call α ( G ) ′′ the group von Neumann algebra of G and denote it by L ( G ) . Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 5 / 34
Introduction to von Neumann algebras Group von Neumann algebras Example Suppose that G is a locally compact group and α : G → B ( H ) is a unitary group representation. Then the group von Neumann algebra of α is α ( G ) ′′ . (Understanding α ( G ) ′′ is tantamount to understanding the invariant subspaces of α .) In the important special case that α : G → B ( L 2 ( G )) (where G is equipped with its haar measure) is given by left translations α ( g )( f )( x ) := f ( g − 1 x ) , we call α ( G ) ′′ the group von Neumann algebra of G and denote it by L ( G ) . Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 5 / 34
Introduction to von Neumann algebras R Example Let M 2 denote the set of 2 × 2 matrices with entries from C . We consider the canonical embeddings M 2 ֒ → M 2 ⊗ M 2 ֒ → M 2 ⊗ M 2 ⊗ M 2 ֒ → · · · and set M := � ∞ � n M 2 . n = 1 The normalized traces on � n M 2 form a cohesive family of traces, yielding a trace tr : M → C . We can define an inner product on M by � A , B � := tr ( B ∗ A ) . Set H to be the completion of M with respect to this inner product. M acts on H by left multiplication, whence we can view M as a ∗ -subalgebra of B ( H ) . We set R to be the von Neumann algebra generated by M . R is called the hyperfinite II 1 factor . Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 6 / 34
Introduction to von Neumann algebras Tracial von Neumann algebras Suppose that A is a von Neumann algebra. A tracial state (or just trace ) on A is a linear functional τ : A → C satisfying: τ ( 1 ) = 1; τ ( x ∗ x ) ≥ 0 for all x ∈ A ; τ ( xy ) = τ ( yx ) for all x , y ∈ A . A tracial von Neumann algebra is a pair ( A , τ ) , where A is a von Neumann algebra and τ is a trace on A . In the case that τ is also faithful , meaning that τ ( x ∗ x ) = 0 ⇒ x = 0, the function � x , y � τ := τ ( y ∗ x ) is an inner product on A , yielding the so-called 2-norm � · � 2 on A . The associated metric is complete on any bounded subset of A . ( A , τ ) is called separable if the metric associated to the 2-norm is separable. Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 7 / 34
Introduction to von Neumann algebras II 1 Factors A von Neumann algebra A is said to be a factor if A ∩ A ′ = C · 1. Fact � ⊕ If A is a von Neumann algebra, then A ∼ X A x (a direct integral ) = where each A x is a factor. A factor is said to be of type II 1 if it is infinite-dimensional and admits a trace. Fact A II 1 factor admits a unique weakly continuous trace, which is automatically faithful. Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 8 / 34
Introduction to von Neumann algebras II 1 Factors A von Neumann algebra A is said to be a factor if A ∩ A ′ = C · 1. Fact � ⊕ If A is a von Neumann algebra, then A ∼ X A x (a direct integral ) = where each A x is a factor. A factor is said to be of type II 1 if it is infinite-dimensional and admits a trace. Fact A II 1 factor admits a unique weakly continuous trace, which is automatically faithful. Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 8 / 34
Introduction to von Neumann algebras Examples-revisited B ( H ) is a factor. If dim ( H ) < ∞ , then B ( H ) admits a trace, but is not a II 1 factor. If dim ( H ) = ∞ , then B ( H ) admits no trace. Thus, B ( H ) is never a II 1 factor. L ∞ ( X , µ ) admits a trace f �→ � X f d µ but is not a factor. If G is a countable group that is ICC , namely all conjugacy classes (other than { 1 } ) are infinite, then L ( G ) is a II 1 factor; the trace is given by T �→ � T δ e , δ e � . In particular, if n ≥ 2, then L ( F n ) is a II 1 factor. R is a II 1 factor; the trace tr : � � n M 2 → C extends uniquely to n the completion. Moreover, R embeds into any II 1 factor. Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 9 / 34
Introduction to von Neumann algebras Continuous model theory There is a natural language of continuous logic in which to discuss tracial von Neumann algebras. Theorem (Farah-Hart-Sherman) The class of tracial von Neumann algebras is universally axiomatizable. The class of II 1 factors is ∀∃ -axiomatizable. Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 10 / 34
Isomorphic ultrapowers Introduction to von Neumann algebras 1 Isomorphic ultrapowers 2 Connes Embedding Problem 3 4 Model companions Computability theory 5 6 Future directions Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 11 / 34
Isomorphic ultrapowers Ultrapowers of von Neumann algebras Suppose that ( A , τ ) is a tracial von Neumann algebra and U is a nonprincipal ultrafilter on N . We set ℓ ∞ ( A ) := { ( a n ) ∈ A N : � a n � is bounded } . Unfortunately, if we quotient this out by the ideal { ( a n ) ∈ A N : lim U � a n � = 0 } , the resulting quotient is usually never a von Neumann algebra. Rather, we have to quotient out by the smaller ideal { ( a n ) ∈ A N : lim U � a n � 2 = 0 } , yielding the tracial ultrapower A U of A . (This is the continuous logic ultrapower, so the result is once again a von Neumann algebra.) Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 12 / 34
Isomorphic ultrapowers Property (Γ) For a while, Murray and von Neumann could not figure out whether R and L ( F 2 ) were isomorphic or not. They finally figured out a property that distinguished them. Say that a II 1 factor M has property (Γ) if, for any finite F ⊆ M and any ǫ > 0, there is a trace 0 unitary u such that � ux − xu � 2 < ǫ for all x ∈ F . R has (Γ) (easy) while L ( F 2 ) does not (M-vN), so R �∼ = L ( F 2 ) . Equivalently, M has property (Γ) if and only if M ′ ∩ M U � = C ( M has nontrivial relative commutant ). Note that (Γ) is axiomatizable by the sentences � � yy ∗ − 1 � 2 + | tr ( y ) | + � � σ n := sup inf y � [ x i , y ] � 2 . � x Therefore, R and L ( F 2 ) are not elementarily equivalent! Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 13 / 34
Isomorphic ultrapowers Property (Γ) For a while, Murray and von Neumann could not figure out whether R and L ( F 2 ) were isomorphic or not. They finally figured out a property that distinguished them. Say that a II 1 factor M has property (Γ) if, for any finite F ⊆ M and any ǫ > 0, there is a trace 0 unitary u such that � ux − xu � 2 < ǫ for all x ∈ F . R has (Γ) (easy) while L ( F 2 ) does not (M-vN), so R �∼ = L ( F 2 ) . Equivalently, M has property (Γ) if and only if M ′ ∩ M U � = C ( M has nontrivial relative commutant ). Note that (Γ) is axiomatizable by the sentences � � yy ∗ − 1 � 2 + | tr ( y ) | + � � σ n := sup inf y � [ x i , y ] � 2 . � x Therefore, R and L ( F 2 ) are not elementarily equivalent! Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 13 / 34
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