Let $mathfrak{A}$ be a Banach algebra. We say that a sequence ${D_n}_{n=0}^infty$ of continuous operators form $mathfrak{A}$ into $mathfrak{A}$ is a textit{local higher derivation} if to each $ainmathfrak{A}$ there corresponds a continuous higher derivation ${d_{a,n}}_{n=0}^infty$ such that $D_n(a)=d_{a,n}(a)$ for each non-negative integer $n$. We show that if $mathfrak{A}$ is a $C^*$-algebra t...

We show that any free product of finite-dimensional von Neumann algebras equipped with non-tracial states is isomorphic to a Araki–Woods factor its quasi-free state possibly direct sum algebra. This gives complete answer questions posed by Dykema in [5] and Shlyakhtenko [10], which had been partially answered Houdayer [7] Ueda [16]. also extend this suitable infinite-dimensional almost periodic...

Suppose X = {x1, . . . , xn} is an n-tuple of selfadjoint elements in a tracial von Neumann algebra M . If z = z∗ ∈ M and for some y = y∗ in the von Neumann algebra generated by X δ0(y, z) < δ0(y) + δ0(z), then χ(X ∪ {z}) = −∞ (here χ and δ0 denote the microstates free entropy and free entropy dimension, respectively). In particular, if z lies in the von Neumann algebra generated by X , then χ(...

Let A be a unital separable simple infinite dimensional C∗-algebra with tracial rank no more than one and with the tracial state space T (A) and let U(A) be the unitary group of A. Suppose that u ∈ U0(A), the connected component of U(A) containing the identity. We show that, for any ǫ > 0, there exists a selfadjoint element h ∈ As.a such that ‖u− exp(ih)‖ < ǫ. We also study the problem when u c...

We construct a simple, nuclear, stably projectionless C∗-algebra W which has trivial Ktheory and a unique tracial state, and we investigate the extent to which W might fit into the hierarchy of strongly self-absorbing C∗-algebras as an analogue of the Cuntz algebra O2. In this context, we show that every nondegenerate endomorphism of W is approximately inner and we construct a tracepreserving e...

This paper deals with mathematical issues relating to the computation of spectra of self adjoint operators on Hilbert spaces. We describe a general method for approximating the spectrum of an operator A using the eigenvalues of large finite dimensional truncations of A. The results of several papers are summarized which imply that the method is effective in most cases of interest. Special atten...

If X,Y, Z are finite sets of selfadjoint elements in a tracial von Neumann algebra and X generates a hyperfinite von Neumann algebra, then δ0(X ∪Y ∪Z) ≤ δ0(X ∪Y )+δ0(X ∪Z)−δ0(X). We draw several corollaries from this inequality. In [9] Voiculescu describes the role of entropy in free probability. He discusses several problems in the area, one of which is the free entropy dimension problem. Free...

We give complete descriptions of the tracial states on both universal and reduced crossed products a C*-dynamical system consisting unital C*-algebra discrete group. In particular, we also answer question when are in canonical bijection with invariant original C*-algebra. This generalizes unique trace property for groups. The analysis simplifies greatly various cases, example conjugacy classes ...

Let A and C be two unital simple C*-algebras with tracial rank zero. Suppose that C is amenable and satisfies the Universal Coefficient Theorem. Denote by KKe(C,A) ++ the set of those κ for which κ(K0(C)+ \ {0}) ⊂ K0(A)+ \ {0} and κ([1C ]) = [1A]. Suppose that κ ∈ KKe(C,A) . We show that there is a unital monomorphism φ : C → A such that [φ] = κ. Suppose that C is a unital AH-algebra and λ : T(...

Given a representation of a unimodular locally compact group, we discuss criteria for associated coherent state expansions in terms of the commuting algebra. It turns out that for those representations that admit such expansions there exists a unique finite trace on the commuting algebra such that the admissible vectors are precisely the tracial vectors for that trace. This observation is immed...