We study the relationship between the dynamics of the action α of a discrete group G on a von Neumann algebra M, and structural properties of the associated crossed product inclusion L(G)⊆M⋊αG, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion L(G)⊆M⋊αG such as the Haagerup Approximation Property, and is essential to capturing “almost periodic” behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorem for general, tracial W⁎-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras.

We show that if M is a full factor and N ⊂ M is a co-amenable subfactor with expectation, then N is also full. This answers a question of Popa from 1986. We also generalize a theorem of Tomatsu by showing that if M is a full factor and σ :G ↷ M is an outer action of a compact group G , then σ is automatically minimal and M^G is a full factor which has w-spectral gap in M . Finally, in the appendix, we give a proof of the fact that several natural notions of co-amenability for an inclusion N⊂ M of von Neumann algebras are equivalent, thus closing the cycle of implications given in Anantharaman-Delaroche’s paper in 1995.

This note contains a short account of various mixing properties of coarse bimodules and its relations with associated completely positive maps.

This paper is a continuation of the authors' previous work on noncommutative joinings, and contains a study of relative independence of W ^* -dynamical systems. We prove that, given any separable locally compact group G , an ergodic W ^{*} -dynamical G -system \mathfrak{M} with compact subsystem \mathfrak{N} is disjoint relative to \mathfrak{N} from its maximal compact subsystem \mathfrak{M}_{K} if and only if \mathfrak{N}\cong\mathfrak{M}_{K} . This generalizes recent work of Duvenhage, which established the result for G abelian.

A state-preserving automorphism of a von Neumann algebra induces a canonical unitary operator on the GNS Hilbert space of the state which fixes the vacuum. This unitary commutes with both the modular operator of the state and its modular conjugation. We prove an extension of this result for state-preserving unital completely positive maps.

Let A is an element of M-n(C). We prove that if W-c(A), the correlation numerical range introduced in [2], is a subset of [0,infinity), then A = P + D where P is positive semidefinite and D is a diagonal matrix such that Tr(D) = 0. This answers a question of D. Hadwin and D. Han. Additionally, we explore a few properties of W-c(A) and W-uc(A), another numerical range introduced in [2] that is closely related to Connes's Embedding Conjecture.

Siena College is a liberal arts college with a strong School of Science. The college is in a region of upstate NY designated as Tech Valley for the recent growth in nanotech and other high tech research and industries. In recent years, over $3M in funding has been obtained from NSF for programs and research in STEM education at Siena College. The NSF Scholarships in Science, Technology, Engineering, and Mathematics (S-STEM) grant, obtained in 2009, was used to create a program we call Tech Valley Scholars (TVS). Per the NSF S-STEM guidelines, the TVS program's goal is to increase the number and quality of students graduating and entering the STEM workforce. In particular, we award scholarships based on unmet financial need and high level of academic promise. In addition to scholarships, a onecredit career preparation seminar, cohort activities, and extra mentoring are provided. The purpose of this paper is to present the results of statistical analyses on the persistence and performance of TVS students. Analyses investigated the retention rates and GPA for the TVS cohort compared to matched and unmatched control groups.RationaleThe U.S. must increase the number of majors in STEM fields and strengthen the science and technology workforce in order to lead the global economy. The February 2012 report by the President's Council of Advisors on Science and Technology (PCAST) set a goal of producing one million additional college graduates with STEM degrees over the next decade (Holdren & Lander, 2012). However, the PCAST report states that fewer than 40% of students who enter college intending to major in a STEM field actually complete a STEM degree. Low completion rates among STEM majors may be related to the difficulty colleges and universities have in recruiting and retaining sufficient numbers of STEM students. Specific reasons for the low completion rate among STEM majors include uninspiring introductory courses, an unwelcoming atmosphere in STEM departments, and lack of support/mentoring systems (Augustine, 2007; Holdren & Lander, 2012). These factors are complicated in the case of STEM majors, such as computer science, that are not readily taught in high school programs (Bowling, Bullen, Doyle, & Filaseta, 2013; Dahlberg, Barnes, Rorrer, Powell, & Cairco, 2008), and have even greater impact on at-risk students (Barlow & Villarejo, 2004; Herrera & Hurtado, 2011; IHEP, 2007; Landry, 2003). On the positive side, recent investigations suggest there are strong benefits of cohort programs, community building, and undergraduate research for recruiting and retaining STEM students (ACS, 2008; Angrist, Lang, & Oreopoulos, 2009; APS, 2014; CUR, 2007, 2014; Hathaway, Nagda, & Gregerman, 2002; Hodge, Pasquesi, Hirsh, & LaPore, 2007; Hunter, Laursen, & Seymour, 2007; Nagda, Gregerman, Jonides, von Hippel, & Lerner, 1998; Rauckhorst, Czaia, & Baxter Magolda, 2001; Russell, Hancock, & McCullough, 2007; Seymour, Hunter, Laursen, & Deantoni, 2004; Whalen & Shelley, 2010).Details about the Siena TVS S-STEM ProgramSince its inception in 2009, the Siena College Tech Valley Scholars (TVS) program will have impacted over 38 undergraduate Tech Valley Scholars in Biochemistry, Chemistry, Computer Science, Mathematics, and Physics, with a STEM graduation rate of greater than 90%. TVS students take part in summer research at Siena and at top NSF Research Experiences for Undergraduate (REU) programs. TVS students have an excellent record of going on to strong graduate schools and professional programs. TVS not only promotes a strong cohort atmosphere but also an important peer mentoring aspect (Gafney, 2005; Gottesman & Hoskins, 2013). Early exposure to faculty and upperclassmen from all disciplines has shown promise in broadening students' scientific curiosity (Barlow & Villarejo, 2004; Bauer & Bennett, 2003; Campbell & Skoog, 2004; Lopatto, 2004). …

By a result of the second author, the Connes embedding conjecture (CEC) is false if and only if there exists a self-adjoint noncommutative polynomial p(t 1 , t 2 ) in the universal unital C * -algebra Ꮽ = t 1 , t 2 : t j = t * j , 0 < t j ≤ 1 for 1 ≤ j ≤ 2 and positive, invertible contractions x 1 , x 2 in a finite von Neumann algebra ᏹ with trace τ such that τ ( p(x 1 , x 2 )) < 0 and Tr k ( p(A 1 , A 2 )) ≥ 0 for every positive integer k and all positive definite contractions A 1 , A 2 in M k ‫.)ރ(‬We prove that if the real parts of all coefficients but the constant coefficient of a self-adjoint polynomial p ∈ Ꮽ have the same sign, then such a p cannot disprove CEC if the degree of p is less than 6, and that if at least two of these signs differ, the degree of p is 2, the coefficient of one of the t 2 i is nonnegative and the real part of the coefficient of t 1 t 2 is zero then such a p disproves CEC only if either the coefficient of the corresponding linear term t i is nonnegative or both of the coefficients of t 1 and t 2 are negative.denote the set of tuples (A 1 , . . ., A N ) of those k × k self-adjoint matrices over ‫ރ‬ of operator norm at most R satisfying τ (x i 1 x i 2 . . .x i p ) -1 k Tr(A i 1 A i 2 , . . .A i p ) < ε,

In this paper, we investigate a notion of spectrum sigma(f) for Banach algebra-valued holomorphic functions on C-n. We prove that the resolvent sigma(c)(f) is a disjoint union of domains of holomorphy when B is a C*-algebra or is reflexive as a Banach space. Further, we study the topology of the resolvent via consideration of the B-valued Maurer-Cartan type 1-form f(z)(-1) df(z). As an example, we explicitly compute the spectrum of a linear function associated with the tuple of standard unitary generators in a free group factor von Neumann algebra.

We prove that various non-residually nite, non-residually solvable groups of the form ha; bjrrw = ri are so c. This paper concerns the so c property discussed in the survey [Pest08]. Particularly, we address Question 4.10 in this paper: the problem of Nate Brown asking whether or not every one relator group is so c. In [Ban10], the rst author proves that the example in [Baum69] of a non-residually nite non-residually solvable one relator group is a so c group. The purpose of this paper is to exhibit more such examples in the following large class of non-residually solvable one-relator groups introduced in [BaMiTro07]. Let F2 = ha; bj i denote the free group on two generators. Let r; w 2 F2 be two elements that do not commute. In [BaMiTro07], the authors show that the group r;w = ha; bjr w = ri = ha; bjr = [r; (r )]i has the same nite quotients as the group

In this short paper, we prove that the group < a, b vertical bar a = [a, a(b)]> is hyperlinear. Unlike the nonresidually finite Baumslag-Solitar groups, this group is not residually solvable.

A …nite von Neumann algebra M with a faithful normal trace has Haagerup’s approximation property if there exists a pointwise deformation of the identity in 2-norm by subtracial compact completely positive maps. In this paper we prove that the subtraciality condition can be removed. This enables us to provide a description of Haagerup’s approximation property in terms of correspondences. We also show that if N M is an amenable inclusion of …nite von Neumann algebras and N has Haagerup’s approximation property, then M also has Haagerup’s approximation property.

In this article, we introduce an isomorphism invariant for type II1 factors using the Connes–Følner condition. We compute bounds of this number for free group factors.

A finite von Neumann algebra $\mathcal{M}$ with a faithful normal trace $% \tau $ has Haagerup's approximation property (relative to a von Neumann subalgebra $\mathcal{N}$) if there exists a net $(\phi_{\alpha})_{\alpha\in \Lambda}$ of normal completely positive ($\mathcal{N}$-bimodular) maps from $\mathcal{M}$ to $\mathcal{M}$ that satisfy the subtracial condition $% \tau \circ \phi_{\alpha}\leq \tau $, the extension operators $% T_{\phi_{\alpha}}$ are bounded compact operators (in $ $), and pointwise approximate the identity in the trace-norm, i.e., $\lim_{\alpha}||\phi_{\alpha}(x)-x||_{2}=0$ for all $% x\in \mathcal{M}$. We prove that the subtraciality condition can be removed, and provide a description of Haagerup's approximation property in terms of Connes's theory of correspondences. We show that if $\mathcal{N}\subseteq \mathcal{M}$ is an amenable inclusion of finite von Neumann algebras and $% \mathcal{N}$ has Haagerup's approximation property, then $\mathcal{M}$ also has Haagerup's approximation property. This work answers two questions of Sorin Popa.

We introduce a notion of transitive family of subspaces relative to a type II1 factor, and hence a notion of transitive family of projections in such a factor. We show that whenever M is a factor of type II1 and M is generated by two self-adjoint elements, then M circle times M-2(C) contains a transitive family of 5 projections. Finally, we exhibit a free transitive family of 12 projections that generate a factor of type II1.