Microstates Free Entropy Dimension and the Non-Isomorphism of Free Group Factors in von Neumann Algebras

The isomorphism problem for free group factors, asking whether the von Neumann algebras L(Fn) and L(Fm) are isomorphic for distinct n, m N2, remains one of the most obstinate open questions in operator algebras. This monograph investigates the utility of Voiculescu's microstates free entropy dimension, denoted 0, as a potential invariant to distinguish these factors. We rigorously define the space of microstates, the free entropy , and the associated dimension 0, establishing the fundamental equality 0(X1, …, Xn) = n for a free generating tuple of L(Fn). We analyze the interpolated free group factors L(Ft) for t (1, ] introduced by Dykema and Radulescu, and discuss the dichotomy that either all L(Ft) are isomorphic or they are pairwise non-isomorphic. Furthermore, we examine the implications of Jung's strong 1-boundedness and the absence of Cartan subalgebras in L(Fn), providing a comprehensive survey of the obstructions to isomorphism imposed by free probabilistic invariants.