Singular masas and measure-multiplicity invariant (Q2839885)

Let \(M\) be a separable II\(_1\) factor with faithful normal tracial state \(\tau\) and \(A\subset M\) be a maximal abelian selfadjoint subalgebra (masa). The author studies the measure multiplicity invariant of \(A\) in \(M\), denoted by \(mm_M(A)\). This invariant is the equivalence class of the quadruple \((X,\lambda_X, [\eta|_{\Delta(X)^c}], m|_{\Delta(X)^c})\), where {\parindent=6mm \begin{itemize} \item[(1)] \(X\) is a compact Hausdorff space such that \(C(X)\) is a norm separable and wot dense subalgebra of \(A\), \item [(2)] \(\lambda_X\) is the Borel probability measure obtained by restricting the trace \(\tau\) to \(C(X)\), \item [(3)] \(\eta|_{\Delta(X)^c}\) is the measure on \(X\times X\), concentrated on the complement \(\Delta(X)^c\) of the diagonal set \(\Delta(X)\) of \(X\times X\), \item [(4)] \(m|_{\Delta(X)^c}\) is the multiplicity function. NEWLINENEWLINE\end{itemize}} Equivalence of quadruples is implemented by the existence of a Borel isomorphism on \(X\), intertwining the data in a natural way.NEWLINENEWLINEThe author studies conditions implying the property that the left right measure \(\eta|_{\Delta(X)^c}\) is the class of a product measure, as well as consequences of this property. In the next sections, these ideas are used to compute the aforementioned invariant in several classes of examples.NEWLINENEWLINEIn Section 3, Tauer masas are considered. Let \(R\) be the hyperfinite II\(_1\) factor. A masa \(A\subset R\) is called a Tauer masa if there exists a sequence of type I subfactors \(N_n\) of \(R\), \(n\geq 1\), such that (1) \(N_n\subset N_{n+1}\), (2) \((\bigcup_{n\geq 1} N_n)''=R\), and (3) \(A\cap N_n\) is a masa in \(N_n\), for all \(n\geq 1\).NEWLINENEWLINEIn Section 4, properties of the left right measure of masas that possess non-trivial centralizing sequences of \(M\) are considered.NEWLINENEWLINESection 5 treats the same problem in masas in the free group factors \(L(\mathbb{F}_k)\). The author proves the following result (Theorem 5.5): There exist non-conjugate singular masas \(A\), \(B\) in \(L(\mathbb{F}_k)\) (\(2\leq k\leq \infty\)) with the same measure multiplicity invariant.