arXiv subjects
Publications and source records attributed to Ken Dykema.
We prove a recursion formula for generating functions of certain renormalizations of *-moments of the DT(δ_0,1)-operator T, involving an operation \odot on formal power series and a transformation E that converts \odot to usual multiplication. This recursion formula is used to prove that all of these generating functions are rational functions, and to find a few of them explicitly.
Let Q be any II_1-factor. It is shown that any standard lattice G can be realized as the standard invariant of a free product of (several) rescalings of Q. In particular, if Q has fundamental group equal to the positive reals and if P is the free product of infinitely many copies of Q, then P has subfactors giving rise to all possible standard invariants. Similarly, given a II_1-subfactor N of M, it is shown there is a subfactor R of S having the same standard invariant as N in M but where S, respectively R, is the free product of M, respectively N, with rescalings of Q.
We introduce notation Q(1) * ... * Q(n) * L(F_r)$ for von Neumann algebra II_1 factors where $r$ is allowed to be negative. This notation is defined by rescalings of free products of II_1 factors, and is proved to be consistent with known results and natural operations. We also give two statements which we prove are equivalent to isomorphism of free group factors.
The constructions of free subproducts of von Neumann algebras and free scaled products are introduced, and results about them are proved, including rescaling results and results about free trade in free scaled products.
Given reduced amalgamated free products of C$^*$-algebras, $(A,phi)=*_i(A_i,phi_i)$ and $(D,psi)=*_i(D_i,psi_i)$, an embedding $A\to D$ is shown to exist assuming there are conditional expectation preserving embeddings $A_i\to D_i$. This result is extended to show the existance of the reduced amalgamated free product of certain classes of unital completely positive maps. Analogues of the above mentioned results are proved for von Neumann algebras.
A reduction formula for compressions of von Neumann algebras arising as free products is proved. This shows that the fundamental group is all of the positive reals for some such algebras. Additionally, by taking a sort of free product with an unbounded semicircular element, continuous one parameter groups of trace scaling automorphisms on II_infinity factors are constructed; this produces type III_1 factors with core M tensor B(H), where M can be a full II_1-factor without the Haagerup approximation property.
We show that Voiculescu's circular operator and, more generally, each circular free Poisson operator has a continuous family of invariant subspaces relative to the von Neumann algebra it generates. The proof relies on upper triangular random matrix models and consequent realizations of these operators as upper triangular matrices of free random variables.
It has recently been proved that the class of unital exact C*-algebras is closed under taking reduced amalgamated free products. In particular, this implied that the class of exact discrete groups is closed under taking amalgamated free products. Here a proof is presented of a special case: that the class of exact discrete groups is closed under taking free products (with amalgamation over the identity element). The proof of this special case is considerably simpler than in full generality.
Some completely positive maps on reduced amalgamated free products of C*-algebras are constructed; these allow a proof that the class of exact unital C*-algebras is closed under taking reduced amalgamated free products. Consequently, the class of exact discrete groups is closed under taking amalgamated free products.
Let H be a full Hilbert bimodule over a C*-algebra A. We show that the Cuntz-Pimsner C*-algebra associated to H is exact if and only if A is exact. Using this result, we give alternative proofs for exactness of reduced amalgamated free products of exact C*-algebras. In the case that A is a finite dimensional C*-algebra, we also show that the Brown-Voiculescu topological entropy of Bogljubov automorphisms of the Cuntz-Pimsner algebra associated to an A,A Hilbert bimodule is zero.
Certain reduced free products of C*-algebras, (A,phi)=(A_1,phi_1)*(A_2,ϕ_2), taken with respect to faithful states, at least one of which is not a trace, are shown to be purely infinite and simple. It is assumed that one of the A_i contain a partial isometry in the spectral subspace of phi_i corresponding to a positive number not equal to one. For example, if A_1 and A_2 are copies of the two-by-two complex matrices and if phi_1 and phi_2 are not unitarily conjugate, it is shown that A is simple and purely infinite.
In the reduced free product of C*-algebras (A,phi)=(A_1,phi_1)*(A_2,phi_2), A is shown to be purely infinite and simple under the hypothesis that A_1 is the crossed product of a C*-algebra by a discrete infinite group, phi_1 is well behaved with respect to this crossed product and A_2 is not one dimensional.
Let (A,phi) be the reduced free product of infinitely many pairs (A_i,phi_i) of C*-algebras with faithful states. Assume that the A_i are not too small, in a specific sense. It is shown that if phi is a trace then K_0(A) is determined entirely by K_0(phi). If, furthermore, the image of K_0(phi) is dense in the reals then A has real rank zero. On the other hand, if phi is not a trace then A is simple and purely infinite.
Certain classes of automorphisms of recued amalgamated free products of C*-algebras are shown to have Brown-Voiculescu topological entropy zero. Also, for automorphisms of exact C*-algebras, the Connes-Narnhofer-Thirring entropy is shown to be bounded above by the Brown-Voiculescu entropy. These facts are applied to generalize Stormer's result about entropy of automorphisms of the II_1-factor of a free group.
Using Voiculsecu's free entropy, it is shown that the free group factors L(F_n) lack property C of Popa and that they lack finite multiplicity abelian subalgebras. Property C is an assymptotic commutivity property.
It is proved that the reduced group C*-algebra C*_{red}(G) has stable rank one (i.e. its group of invertible elements is a dense subset) if G is a discrete group arising as a free product G_1*G_2 where |G_1|>=2 and |G_2|>=3. This follows from a more general result where it is proved that if (A,tau) is the reduced free product of a family (A_i,tau_i), i\in I, of unital C*-algebras A_i with normalized faithful traces tau_i, and if the family satisfies the Avitzour condition (i.e. the traces, tau_i, are not too lumpy in a specific sense), then A has stable rank one.
The q-commutation relations in the title are those that have recently received much attention, and that for -1<q<1 provide an interpolation between Bosonic and Fermionic statistics, passing through free statistics at q=0. We look at the C*-algebra R^q generated by the representation of these relations on the twisted Fock space of Bozejko-Speicher, and we find a canonical unitary U_q from the twisted Fock space to usual full Fock space, such that U_q R^q (U_q)^* contains the Cuntz algebra R^0, and such that we have equality for |q|<0.44.
Voiculescu's random matrix model for freeness is extended to the non-Gaussian case and also the case of constant block diagonal matrices. Thus we are able to investigate free products of free group factors with matrix algebras and with the hyperfinite II$_1$ factor, showing that $$ L(F_n) * R = L(F_(n+1)) $$ for $n \ge 1$, (where $L(F_1)=L(Z)$).