Dense subalgebras of purely infinite simple groupoid C*-algebras

A simple Steinberg algebra associated to an ample Hausdorff groupoid G is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid $C^*$-algebra $C^*_r(G)$ is simple and purely infinite. But the Steinberg algebra seems too small for the converse to hold. For this purpose we introduce an intermediate *-algebra B(G) constructed using corners $1_U C^*_r(G) 1_U$ for all compact open subsets U of the unit space of the groupoid. We then show that if G is minimal and effective, then B(G) is algebraically properly infinite if and only if $C^*_r(G)$ is purely infinite simple. We apply our results to the algebras of higher-rank graphs.

Is Learning in Developmental Math Associated With Community College Outcomes?

Objective: Remedial mathematics courses are widely considered a barrier to student success in community college, and there has been a significant amount of work recently to reform them. Yet, there is little research that explicitly examines whether increasing learning in remedial classes improves grades or completion rates. This study examines the relationship between procedural and conceptual learning in developmental math and measures of progress toward a degree, such as grades. Method: A mathematical skills assessment was given to all intermediate algebra students at a large, urban community college, and to students in the following college-level class at the beginning of the next term. Assessment scores were compared with student characteristics, grades in intermediate algebra, grades in college-level math, and whether the student earned a credential. Results: After controlling for grades in previous classes, procedural algebra skills were not associated with higher grades in college-level math. Conceptual mathematics proficiency was associated with higher grades in general education math but not in precalculus. In developmental classes, however, learning gains were primarily procedural, which were correlated with grades. In addition, students who took at least one term off of math had significantly lower procedural skills but not conceptual skills. Contributions: The findings challenge the assumption in community college research that increased student learning in remedial mathematics will improve student outcomes. The results suggest that the type of mathematics taught in developmental classes can have an effect on student outcomes. Instruction focused on procedural skills may not be preparing students for college mathematics.

The cup subalgebra has the absorbing amenability property

Consider an inclusion of diffuse von Neumann algebras [Formula: see text]. We say that [Formula: see text] has the absorbing amenability property (AAP) if for any diffuse subalgebra [Formula: see text] and any amenable intermediate algebra [Formula: see text] we have that [Formula: see text] is contained in [Formula: see text] We prove that the cup subalgebra associated to any subfactor planar algebra has the AAP.

Sharing Teaching Ideas: Correspondence from Mathematicians

When struggling with mathematics problems in today's classroom, students occasionally experience a flash of discovery that is inspired by the past. An example happened in an intermediate algebra class at the end of a lesson on completing the square. In an attempt to pique students' interest and to connect completing the square with other mathematics, one of the authors, Jennifer Horn, challenged the students to complete the square on the standard quadratic equation, ax2 + bx + c = 0. Obviously, she intended for them to “derive” the quadratic formula that they had used in previous lessons.