# Strongly 1-bounded von Neumann algebras

Research paper by **Kenley Jung**

Indexed on: **31 Oct '05**Published on: **31 Oct '05**Published in: **Mathematics - Operator Algebras**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### Abstract

Suppose F is a finite set of selfadjoint elements in a tracial von Neumann
algebra M. For $\alpha >0$, F is $\alpha$-bounded if the free packing
$\alpha$-entropy of F is bounded from above. We say that M is strongly
1-bounded if M has a 1-bounded finite set of selfadjoint generators F such that
there exists an x in F with finite free entropy. It is shown that if M is
strongly 1-bounded, then any finite set of selfadjoint generators G for M is
1-bounded and the microstates free entropy dimension of G is less than or equal
to 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic
to an interpolated free group factor and the microstates free entropy dimension
is an invariant for these algebras. Examples of strongly 1-bounded von Neumann
algebras include (separable) II_1-factors which have property Gamma, have
Cartan subalgebras, are non-prime, or the group von Neumann algebras of
SL_n(Z), n >2. If M and N are strongly 1-bounded and their intersection is
diffuse, then the von Neumann algebra generated by M and N is strongly
1-bounded. In particular, a free product of two strongly 1-bounded von Neumann
algebras with amalgamation over a common, diffuse von Neumann subalgebra is
strongly 1-bounded. It is also shown that a II_1-factor generated by the
normalizer of a strongly 1-bounded von Neumann subalgebra is strongly
1-bounded.