Subfactor

In the theory of von Neumann algebras, a subfactor of a factor M {displaystyle M} is a subalgebra that is a factor and contains 1 {displaystyle 1} . The theory of subfactors led to the discovery of the Jones polynomial in knot theory. In the theory of von Neumann algebras, a subfactor of a factor M {displaystyle M} is a subalgebra that is a factor and contains 1 {displaystyle 1} . The theory of subfactors led to the discovery of the Jones polynomial in knot theory. Usually M {displaystyle M} is taken to be a factor of type I I 1 {displaystyle { m {II}}_{1}} , so that it has a finite trace.In this case every Hilbert space module H {displaystyle H} has a dimension dim M ⁡ ( H ) {displaystyle dim _{M}(H)} which is a non-negative real number or + ∞ {displaystyle +infty } . The index [ M : N ] {displaystyle } of a subfactor N {displaystyle N} is defined to be dim N ⁡ ( L 2 ( M ) ) {displaystyle dim _{N}(L^{2}(M))} . Here L 2 ( M ) {displaystyle L^{2}(M)} is the representation of N {displaystyle N} obtained from the GNS construction of the trace of M {displaystyle M} . This states that if N {displaystyle N} is a subfactor of M {displaystyle M} (both of type I I 1 {displaystyle { m {II}}_{1}} ) then the index [ M : N ] {displaystyle } is either of the form 4 c o s ( π / n ) 2 {displaystyle 4cos(pi /n)^{2}} for n = 3 , 4 , 5 , . . . {displaystyle n=3,4,5,...} , or is at least 4 {displaystyle 4} . All these values occur. The first few values of 4 cos ⁡ ( π / n ) 2 {displaystyle 4cos(pi /n)^{2}} are 1 , 2 , ( 3 + 5 ) / 2 = 2.618... , 3 , 3.247... , . . . {displaystyle 1,2,(3+{sqrt {5}})/2=2.618...,3,3.247...,...} Suppose that N {displaystyle N} is a subfactor of M {displaystyle M} , and that both are finite von Neumann algebras. The GNS construction produces a Hilbert space L 2 ( M ) {displaystyle L^{2}(M)} acted on by M {displaystyle M} with a cyclic vector Ω {displaystyle Omega } . Let e N {displaystyle e_{N}} be the projection onto the subspace N Ω {displaystyle NOmega } . Then M {displaystyle M} and e N {displaystyle e_{N}} generate a new von Neumann algebra ⟨ M , e N ⟩ {displaystyle langle M,e_{N} angle } acting on L 2 ( M ) {displaystyle L^{2}(M)} , containing M {displaystyle M} as a subfactor. The passage from the inclusion of N {displaystyle N} in M {displaystyle M} to the inclusion of M {displaystyle M} in ⟨ M , e N ⟩ {displaystyle langle M,e_{N} angle } is called the basic construction. If N {displaystyle N} and M {displaystyle M} are both factors of type I I 1 {displaystyle { m {II}}_{1}} and N {displaystyle N} has finite index in M {displaystyle M} then ⟨ M , e N ⟩ {displaystyle langle M,e_{N} angle } is also of type I I 1 {displaystyle { m {II}}_{1}} .Moreover the inclusions have the same index: [ M : N ] = [ ⟨ M , e N ⟩ : M ] , {displaystyle =,} and t r ⟨ M , e N ⟩ ( e N ) = [ M : N ] − 1 {displaystyle tr_{langle M,e_{N} angle }(e_{N})=^{-1}} . Suppose that N ⊂ M {displaystyle Nsubset M} is an inclusion of type I I 1 {displaystyle { m {II}}_{1}} factors of finite index. By iterating the basic construction we get a tower of inclusions where M − 1 = N {displaystyle M_{-1}=N} and M 0 = M {displaystyle M_{0}=M} , and each M n + 1 = ⟨ M n , e n + 1 ⟩ {displaystyle M_{n+1}=langle M_{n},e_{n+1} angle } is generated by the previous algebra and a projection. The union of all these algebras has a tracial state t r {displaystyle tr} whose restriction to each M n {displaystyle M_{n}} is the tracial state, and so the closure of the union is another type I I 1 {displaystyle { m {II}}_{1}} von Neumann algebra M ∞ {displaystyle M_{infty }} . The algebra M ∞ {displaystyle M_{infty }} contains a sequence of projections e 1 , e 2 , e 3 , . . . , {displaystyle e_{1},e_{2},e_{3},...,} which satisfy the Temperley–Lieb relations at parameter λ = [ M : N ] − 1 {displaystyle lambda =^{-1}} . Moreover, the algebra generated by the e n {displaystyle e_{n}} is a C ⋆ {displaystyle { m {C}}^{star }} -algebra in which the e n {displaystyle e_{n}} are self-adjoint, and such that t r ( x e n ) = λ t r ( x ) {displaystyle tr(xe_{n})=lambda tr(x)} when x {displaystyle x} is in the algebra generated by e 1 {displaystyle e_{1}} up to e n − 1 {displaystyle e_{n-1}} . Whenever these extra conditions are satisfied, the algebra is called a Temperly–Lieb–Jones algebra at parameter λ {displaystyle lambda } . It can be shown to be unique up to ⋆ {displaystyle star } -isomorphism. It exists only when λ {displaystyle lambda } takes on those special values 4 c o s ( π / n ) 2 {displaystyle 4cos(pi /n)^{2}} for n = 3 , 4 , 5 , . . . {displaystyle n=3,4,5,...} , or the values larger than 4 {displaystyle 4} .