Let M be a tracial von Neumann algebra and A be a weakly dense unital C-subalgebra of M . We say that a set X is a W -generating set for M if the von Neumann algebra generated by X is M and that X is a C-generating set for A if the unital C-algebra generated by X is A. For any finite W generating set X for M we show that δ0(X) ≤ sup{δ0(Y ) : Y is a finite C-generating set for A}. It follows tha...

We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C∗-algebras. The concept is particularly relevant for Elliott’s program to classify nuclear C∗algebras by K-theory data. We study some of its properties and show that a simple unital C∗-algebra, which has locally finite decomposition rank, real rank zero and which absorbs the...

Consider the B-valued probability space (A, E,B), where A is a tracial von Neumann algebra. We extend the theory of operator valued free probability to the algebra of affiliated operators Ã. For a random variable X ∈ Ã we study the Cauchy transform GX and show that the von Neumann algebra (B ∪ {X}) ′′ can be recovered from this function. In the case where B is finite dimensional, we show that, ...

It is shown that there exists a simple C*-algebra A such that the map k (Un(A)) ! k (U n+1 (A)) between homotopy groups which is induced by the usual embedding Un(A) U n+1 (A), where Un(A) is the group of unitary n n-matrices over A, is not surjective for any k 2 N 0 when n = sr(A) ? 1 and bijective for all k 2 N 0 when n sr(A). Two consequences are that K * (A) does not have the strict orderin...

Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra text] defined in Hartglass–Penneys’ article on canonical text]-algebras associated to planar algebra. Under mild conditions, is non-nuclear simple text]-algebra unique tracial state. There polynomial subalgebra together Dirac number operator such that spectral triple. We prove Haagerup-type bound of...

A polynomial f in non-commuting variables is trace-positive if the trace of f(A) is positive for all tuples A of symmetric matrices of the same size. The investigation of trace-positive polynomials and of the question of when they can be written as a sum of hermitian squares and commutators of polynomials are motivated by their connection to two famous conjectures: The BMV conjecture from stati...

We prove that every separable tracial von Neumann algebra embeds into a II1 factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. also establish an analogous result for the extension over non-atomic probability space of countable p.m.p. equivalence relation. In addition, we obtain two new results concerning structure infinitely generic factors. T...

We provide a generalization of Lieb’s triple matrix extension the Golden–Thompson inequality from algebras to setting traces on finite von Neumann algebras. More precisely, assume that ℳ is algebra equipped with tracial state τ. If 1≤p,q≤∞ 1/p+1/q=1, it shown whenever a, b, and c are self-adjoint τ-measurable operators satisfying a∈ℳ, e b ∈L p (ℳ,τ), q then following holds:

For an inclusion of the form C ⊆ Mn(C), where Mn(C) is endowed with a state with diagonal weights λ = (λ1, ..., λn), we use Popa’s construction, for non-tracial states, to obtain an irreducible inclusion of II1 factors, N(Q) ⊆ M(Q) of index ∑ 1 λi . M(Q) is identified with a subfactor inside the centralizer algebra of the canonical free product state on Q ⋆MN (C). Its structure is described by ...

Various subsets of the tracial state space of a unital C∗-algebra are studied. The largest of these subsets has a natural interpretation as the space of invariant means. II1-factor representations of a class of C∗-algebras considered by Sorin Popa are also studied. These algebras are shown to have an unexpected variety of II1-factor representations. In addition to developing some general theory...