arXiv subjects

Florin G. Radulescu

Publications and source records attributed to Florin G. Radulescu.

3 recordsLinked to original sources

arXiv2000

Irreducible subfactors derived from Popa's construction for non-tracial states

For an inclusion of the form $\Bbb C\subseteq M_n(\Bbb C)$, where $M_n(\Bbb C)$ is endowed with a state with diagonal weights $λ=(λ_1, ..., λ_n)$, we use Popa's construction, for non-tracial states, to obtain an irreducible inclusion of $II_1$ factors, $N^λ(Q)\subseteq M^λ(Q) $ of index $\sum \frac{1}{λ_i}$. $M^λ(Q)$ is identified with a subfactor inside the centralizer algebra of the canonical free product state on $Q\star M_N(\Bbb C)$. Its structure is described by ``infinite'' semicircular elements as in \cite{Ra3}. The irreducible subfactor inclusions obtained by this method are similar to the first irreducible subfactor inclusions, of index in $[4,\infty)$ constructed in \cite {Po1}, starting with the Jones' subfactors inclusion $R^s\subseteq R$, $s>4$. In the present paper, since the inclusion we start with has a simpler structure, it is easier to control the algebra structure of the subfactor inclusions.

math.OA↗

arXiv2000

Non-Commutative Markov Processes in Free Group Factors, Related to Berezin's Quantization and Automorphic Forms

In this paper we use the description of free group factors as the von Neumann algebras of Berezin's deformation of the upper half-plane, modulo PSL$(2,{\Bbb Z})$. The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive Hochschild 2-cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator $\cal L$ for a quantum dynamical semigroup that implements the cocycle on a strongly dense subalgebra. For $x$ in the dense subalgebra, ${\cal L}(x)$ is the (diffusion) operator $$ {\cal L}(x)=Λ(x)-(1/2)\{T,x\}, $$ where $Λ$ is the pointwise (Schur) multiplication operator with a symbol function related to the logarithm of the automorphic form $Δ$. The operator $T$ is positive and affiliated with the algebra ${\cal A}_t$ and $T$ corresponds to ${\cal L}(1)$, in a sense to be made precise in the paper. After a suitable normalization, corresponding to a principal-value type method, adapted for II$_1$ factors, $Λ$ becomes (completely) positive on a union of weakly dense subalgebras. Moreover the 2-cyclic cohomology cocycle associated to the deformation may be expressed in terms of $Λ$.

math.OA↗

arXiv1998

On some finite generation properties

In the algebra $\Cal A=\Cal L(PSL(2,\Bbb Z)\otimes B(H)=\Cal L(F_N)\otimes B(H)$, $N$ finite, there exists a bounded subnormal operator $Z$, such that $\Cal A$ is the weak closure of linear span of the set ${(Z^*)^n Z^m| n,m=0,1,2...}$.

math.OA↗