Student Operator Algebras Seminar

• An Introduction to C*-Algebras and the Classification Program by Karen Strung

  • • https://link.springer.com/book/10.1007/978-3-030-47465-2

• Von Neumann Algebras by Rolando de Santiago and Brent Nelson

  • • https://users.math.msu.edu/users/banelson/conferences/GOALS/notes/vNa_notes.pdf

• An introduction to K-theory for C*-algebras by Rørdam, Larsen, and Laustsen

  • • https://wiki.math.ntnu.no/_media/ma8107/2014h/rordam_m.pdf

• An Introduction to C*-Algebras and Noncommutative Geometry by Heath Emerson

  • • https://link.springer.com/book/10.1007/978-3-031-59850-0

The seminar is on Mondays at 4:30, and if you would like to give at talk, please email cano13 at purdue dot edu.

• September 10th, Lectures on KK-theory, Location: PSYC 3187 

• Abstract: A guide through Kasparov's K-theory, a critical tool in modern C*-algebra classification theory.

• Speaker: Marius Dadarlat, Purdue University

• January 27, Classification of Injective Factors (Part I), Location: SCHM 117

• Abstract: I will give an overview of the history of the classification program for von Neumann Algebras focusing on the work of Alain Connes. I discuss critical invariants such as property Gamma, the Araki-Woods classification, and the pioneering work of Murray and von Neumann.

• Speaker: Alejandro Cano, Purdue University

• February 3, Classification of Injective Factors (Part II), Location: SCHM 117

• Abstract: I will give an overview of the history of the classification program for von Neumann Algebras focusing on the work of Alain Connes. In this second part I give a rough sketch of the tools that Masamichi Takesaki and Connes used in order to give a full description of type III factors. 

  • • • Speaker: Alejandro Cano, Purdue University

• February 10, Classification of Approximately Finite-Dimensional C*-algebras Part I,

• Location: SCHM 117

• Abstract: I will introduce George Elliott's classification theorem of approximately finite dimensional C*-algebras. In this first talk I will cover preliminaries for this classification theorem such as the ordered abelian groups K_0(A), K_1(A), inductive limits of abelian groups, inductive limits of C*-algebras and set the stage to showcase his fundamental intertwining argument.

  • • • Speaker: Hao Wan, Purdue University

• February 17, Classification of Approximately Finite-Dimensional C*-algebras Part II,

• Location: SCHM 117

• Abstract: I will finish the inductive limits of C*-algebras, Abelian groups, and ordered Abelian groups. Then I will briefly cover Bratteli’s local characterization of AF-algebras and a type of diagram he used to represent AF-algebras. Time permitted, I will introduce two lemmas to set the stage for the proof of Elliotts’s classification theorem of AF-algebras. 

  • • • Speaker: Hao Wan, Purdue University

• February 24, Classification of Approximately Finite-Dimensional C*-algebras Part III,

• Location: SCHM 117

• Abstract: I will present Elliott's proof of the classification theorem of AF algebras.

  • • • Speaker: Hao Wan, Purdue University

• March 3, Measure equivalence and Orbit equivalence ,

• Location: SCHM 117

• Abstract: Two important equivalence relations on class of countable discrete groups that have been studied for a long time are orbit equivalence and measure equivalence. A celebrated result of Singer from 1950s relates orbit equivalence to the theory of crossed product von Neumann algebras, whereas measure equivalence was introduced by Gromov in the 90s as a measurable analogue of the notion of quasi-isometry of finitely generated groups. In this talk, I will introduce these equivalence relations, provide some basic examples, and survey some results in the theory together with, time permitting, a few applications to the theory of von Neumann algebras. 

  • • • Speaker: Ishan Ishan, University of Nebraska-Lincoln

• March 17, The K_1 Group of a C*-algebra

• Location: SCHM 117

• Abstract: We introduce the  K_1 group of a C*-algebra, focusing on its construction using unitary elements in matrix amplifications of the unitarization of  A , modulo homotopical equivalence with the identity. We will explore some examples regarding the computation of K_1(A) for key C*-algebras. Additionally, we will discuss the functorial interpretation of K_1, emphasizing exactness properties, homotopy invariance, and its role in the classification programme of C*-algebras by K-theoretical invariants.   

  • • • Speaker: Jose Manuel Barrientos Lopez, Purdue University

• March 31, An AF embedding

• Location: SCHM 117

• Abstract:  In this talk we will study examples of embeddings of a C*-algebra into an AF-algebra. This technique is a novel paradigm in the classification programme of C*-algebra  

  • • • Speaker: Jose Manuel Barrientos Lopez, Purdue University

• April 14, AF embeddability 

• Location: SCHM 117

• Abstract:  We will present a timeline of significant results concerning the conditions under which a C*-algebra can be realized as a sub-C*-algebra of an Approximately Finite Dimensional-algebra. This property culminates in a landmark theorem by C. Schafhauser, published in 2019, which provides conditions for resolving the problem in an abstract setting. Moreover, the technical tools developed in that work played a crucial role in the culminating classification theorem for unital C*-algebras.

  • • • Speaker: Jose Manuel Barrientos Lopez, Purdue University

• April 21, Classification of C*-algebras: The Finale 

• Location: SCHM 117

• Abstract:  The Classification Program of C*-algebras really kicked off in the 1990s with the spectacular resolution of the purely infinite case along with progress made in the stably finite case for AT algebras with real rank zero, which includes the irrational rotation algebras. The stably finite case, however, ended up being much more complicated than its the von Neumann counterpart, as evidenced by exotic counterexamples narrowing the scope of the program as well as the requirement of more sophisticated machinery in the form of tracial approximations, dimension theory and regularity conditions. The work of "many hands" culminated in 2015 with the proof of the celebrated classification theorem. We will try to weave together all these narrative strands developed over the course of the talks this semester in the hope of elucidating the bigger themes. 

  • • • Speaker: Chrisil Ouseph Purdue University

• April 28, Classification of C*-algebras: The Epilogue 

• Location: SCHM 117

• Abstract:  The talk will tie up all the loose ends of the classification program with a view towards future directions of research. We will start off with a brief overview of the main proof strategy for the so-called "abstract classification theorems," viz., first classifying approximately multiplicative maps and using the total K-theory invariant. We will then discuss the progress made on the Toms-Winter conjecture that predicts the equivalence of three regularity conditions that would help demarcate the algebras falling under the purview of classifiability. We will end with an introduction to the Cuntz semigroup, a powerful yet slightly unwieldy object that has emerged as the frontrunner for the base of finer invariants. 

  • • • Speaker: Chrisil Ouseph Purdue University

• August 29, Introductions and interests, location: MATH

  • Speakers: Alejandro Cano, Hao Wan, Chrisil Ouseph

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  September 9, Direct integral decomposition of von Neumann algebras, location: SCHM 314

• Abstract: In this survey talk, we will explore the direct integral of von Neumann algebras and its applications to addressing problems related to the rigidity of group von Neumann algebras with diffuse center.

  • Speaker: Adriana Fernández Quero, University of Iowa

• September 16, Group C*-Algebras, the Fourier transform and locally compact groups.

• Abstract: We define two types of C*-Algebras which we can associate to a large class of groups, and study the properties which each object can reflect from the other. In doing so, we state a number of theorems which tie the Fourier transform to a dual object of the group C*-Algebra we construct. The tools along the way will include representation theory, abstract integration theory, and functional analysis.

  • Speaker: Alejandro Cano, Purdue University

• September 23, Hilbert Hotels on Hilbert Spaces. location: SCHM 314

• Abstract: The talk will shine a light on the rich and underrated history of the interaction between set theory and operator algebras. We will warm up with a brisk jog through the necessary prerequisites in logic. A surprising survey of several results will follow, starting with an independence result in functional analysis. The remainder of the talk will be a showcase of some of the unexpected consequences in operator algebras of the Continuum Hypothesis and other independent axioms: a representation theorem for the double dual of C[0,1], weak expectations on von Neumann algebras, ultrapowers, and automorphisms on the Caulkin Algebra.

  • Speaker: Chrisil Ouseph, Purdue University

• September 30, Introduction to finite free probability. location: SCHM 314

• Abstract: Finite free additive and multiplicative convolutions are binary operations of polynomials that behave well with respect to the roots. These operations have gained interest in recent years due to its interpretation as expected characteristic polynomials of random matrix operations and their connection to free probability, geometry of polynomials, representation theory and combinatorics. We will study in detail the basic properties of these convolutions of polynomials, survey the results in the area, and mention some interesting open problems.

  • Speaker: Daniel Perales, Texas A&M

• October 21, The K-theoretic Atiyah-Singer Index Theorem. location: SCHM 314

• Abstract: Index theorems relate global analytic information of a manifold (the number of solutions to an elliptic PDE on the manifold) to a sum of topological invariants over the manifold. Index theorems relate topology and analysis with applications ranging from mathematical physics to operator algebras. I will state and unpack the Atiyah-Singer Index theorem in the language of K-theory, which provides the most "big picture" overview of the result.

  • Speaker: General Ozochiawaeze, Purdue University

• October 28, The C*-algebra version of the topological 2-torus. (Part 1) location: SCHM 314

• Abstract: Irrational rotation algebras, also known as noncommutative tori, are noncommutative C*-algebras that generalize the algebra of continuous functions on the two-dimensional torus. These algebras are defined as universal C*-algebras associated with a specific set of generators and relations, in terms of a rotation angle \theta\notin\mathbb{Q}. In particular, irrational rotation algebras have a unique tracial state and can be embedded into an AF-algebra, allowing us to explore their invariants via Elliott’s Theorem for AF-algebras. In this talk, we will discuss the construction of irrational rotation algebras and explore how the uniqueness of the tracial state facilitates the identification of complete invariants in K-theoretic terms. These invariants enable us to determine when two irrational rotation algebras are isomorphic based on their rotation parameters. If time permits, we will present details on the embedding technique used for the classification theorem. This talk will be mainly based on Chapter VI of the book C*-algebras by Example by Ken Davidson.

  • Speaker: Jose Manuel Barrientos Lopez, Purdue University

• November 4, The C*-algebra version of the topological 2-torus. (Part 2) location: SCHM 314

• Abstract: Irrational rotation algebras, also known as noncommutative tori, are noncommutative C*-algebras that generalize the algebra of continuous functions on the two-dimensional torus. These algebras are defined as universal C*-algebras associated with a specific set of generators and relations, in terms of a rotation angle \theta\notin\mathbb{Q}. In particular, irrational rotation algebras have a unique tracial state and can be embedded into an AF-algebra, allowing us to explore their invariants via Elliott’s Theorem for AF-algebras. In this talk, we will discuss the construction of irrational rotation algebras and explore how the uniqueness of the tracial state facilitates the identification of complete invariants in K-theoretic terms. These invariants enable us to determine when two irrational rotation algebras are isomorphic based on their rotation parameters. If time permits, we will present details on the embedding technique used for the classification theorem. This talk will be mainly based on Chapter VI of the book C*-algebras by Example by Ken Davidson.

  • Speaker: Jose Manuel Barrientos Lopez, Purdue University

• November 11 , Rigidity for von Neumann Algebras Arising from Groups. location: SCHM 314

• Abstract: I will survey some rigidity results for II$_1$ factors arising from countable groups, e.g. L(G) and L^{\infty}(X,\mu)\cross G. Time permitting, I will either go into more detail about the philosophy of deformation/rigidity or emphasize the geometric properties of groups and their actions necessary for rigidity results.

  • Speaker: Patrick Debonis, Purdue University