##
**Introduction to subfactors.**
*(English)*
Zbl 0903.46062

London Mathematical Society Lecture Note Series. 234. Cambridge: Cambridge University Press. xii, 162 p. (1997).

Since the seminal work of V. Jones on the index of subfactors in the early 1980s, there have been considerable research activities in subfactors which are known to have significant connections with other fields such as low-dimensional topology and algebraic quantum field theory.

This masterful book is about subfactors of type \(\text{II}_1\) factors and is centred around the index theory. Many important devices are developed in relation to the index of subfactors. Let us first recall that a type \(\text{II}_1\) factor \(M\) is a von Neumann algebra with trivial centre and a unique normal tracial state \(tr\) which induces a bijection between the equivalent classes of its projections and the continuum \([0,1]\). The factor \(M\) is usually assumed to have separable predual. Let \(L^2(M)\) be the Hilbert space associated with the GNS-representation induced by \(tr\). Given an inclusion \(N\subset M\) of type \(\text{II}_1\) factors, the index \([M:N]\) is the ‘\(N\)-dimension’ of \(L^2(M)\) which is considered as a module over \(N\). There is a restriction of the values that \([M:N]\) can take. In fact, we have Jones’ remarkable result that \([M:N]\in \{4\cos^2{\pi\over n}:n= 3,4,\dots\}\cup[4, \infty]\). To investigate this, the first tool is the so-called basic construction. There is a \(tr\)-preserving conditional expectation \(E_N:M\to N\) which is implemented by a projection \(e_1\) in that we have \(e_1xe_1= E_N(x)e_1\) for \(x\in M\). This gives rise to a tower \(N\subset M\subset\langle M\cup\{e_1\}\rangle= M_1\), where \(M_1\) is the von Neumann algebra generated by \(M\) and \(e_1\), and is also a type \(\text{II}_1\) factor. Hence one can iterate this construction to produce a tower of type \(\text{II}_1\) factors \[ N\subset M\subset M_1\subset\cdots\subset M_n\subset\cdots, \] where \(M_{n+ 1}= \langle M_n, e_{n+1}\rangle\). The sequence \(\{e_n\}\) of projections plays an important role in the theory of subfactors. Its importance and the deep relationship between index of subfactors and norms of nonnegative integral matrices are discussed in Chapter 3.

The basic construction leads to the important concepts of principal graph and dual graph invariants of subfactor inclusions. They are explained in Chapter 4 as well as their use in reducing the above result on restriction of index values to the classification of nonnegative integral matrices of small norm.

Another important tool for computing the index is Pimsner and Popa’s commuting squares and their ‘minmax’ formula \[ [M: N]= (\sup\{\lambda: E_N(x)\geq \lambda x\;\forall x\geq 0\})^{-1} \] the latter can also be taken as an alternative definition of the index. The usefulness of the commuting squares for computing the index of hyperfinite subfactors together with Ocneanu’s compactness theorem for computing higher relative commutants are demonstrated in Chapter 5. A large class of examples provided by the so-called vertex and spin models are given in the last chapter. There are also useful bibliographical remarks at the end of the book.

This book contains a wealth of beautiful ideas and results in the theory of subfactors. It will remain an important reference for students and researchers for many years to come.

This masterful book is about subfactors of type \(\text{II}_1\) factors and is centred around the index theory. Many important devices are developed in relation to the index of subfactors. Let us first recall that a type \(\text{II}_1\) factor \(M\) is a von Neumann algebra with trivial centre and a unique normal tracial state \(tr\) which induces a bijection between the equivalent classes of its projections and the continuum \([0,1]\). The factor \(M\) is usually assumed to have separable predual. Let \(L^2(M)\) be the Hilbert space associated with the GNS-representation induced by \(tr\). Given an inclusion \(N\subset M\) of type \(\text{II}_1\) factors, the index \([M:N]\) is the ‘\(N\)-dimension’ of \(L^2(M)\) which is considered as a module over \(N\). There is a restriction of the values that \([M:N]\) can take. In fact, we have Jones’ remarkable result that \([M:N]\in \{4\cos^2{\pi\over n}:n= 3,4,\dots\}\cup[4, \infty]\). To investigate this, the first tool is the so-called basic construction. There is a \(tr\)-preserving conditional expectation \(E_N:M\to N\) which is implemented by a projection \(e_1\) in that we have \(e_1xe_1= E_N(x)e_1\) for \(x\in M\). This gives rise to a tower \(N\subset M\subset\langle M\cup\{e_1\}\rangle= M_1\), where \(M_1\) is the von Neumann algebra generated by \(M\) and \(e_1\), and is also a type \(\text{II}_1\) factor. Hence one can iterate this construction to produce a tower of type \(\text{II}_1\) factors \[ N\subset M\subset M_1\subset\cdots\subset M_n\subset\cdots, \] where \(M_{n+ 1}= \langle M_n, e_{n+1}\rangle\). The sequence \(\{e_n\}\) of projections plays an important role in the theory of subfactors. Its importance and the deep relationship between index of subfactors and norms of nonnegative integral matrices are discussed in Chapter 3.

The basic construction leads to the important concepts of principal graph and dual graph invariants of subfactor inclusions. They are explained in Chapter 4 as well as their use in reducing the above result on restriction of index values to the classification of nonnegative integral matrices of small norm.

Another important tool for computing the index is Pimsner and Popa’s commuting squares and their ‘minmax’ formula \[ [M: N]= (\sup\{\lambda: E_N(x)\geq \lambda x\;\forall x\geq 0\})^{-1} \] the latter can also be taken as an alternative definition of the index. The usefulness of the commuting squares for computing the index of hyperfinite subfactors together with Ocneanu’s compactness theorem for computing higher relative commutants are demonstrated in Chapter 5. A large class of examples provided by the so-called vertex and spin models are given in the last chapter. There are also useful bibliographical remarks at the end of the book.

This book contains a wealth of beautiful ideas and results in the theory of subfactors. It will remain an important reference for students and researchers for many years to come.

Reviewer: C.-h.Chu (London)

### MSC:

46L37 | Subfactors and their classification |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |