K-theoretic Invariants Research Articles

On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoids

We study semigroup $C^*$-algebras of semigroups associated with number fields and initial data arising naturally from class field theory. These semigroup $C^*$-algebras turn out to have an interesting $C^*$-algebraic structure, giving access to many new examplesof classifiable $C^*$-algebras and exhibi ting phenomena which have not appeared before. Moreover, using K-theoretic invariants, we investigate how much information about the initial number-theoretic data is encoded in our semigroup $C^*$-algebras.

Virtual Riemann–Roch theorems for almost perfect obstruction theories

This is the third in a series of works devoted to constructing virtual structure sheaves and K-theoretic invariants in moduli theory. The central objects of study are almost perfect obstruction theories, introduced by Y.-H. Kiem and the author as the appropriate notion in order to define invariants in K-theory for many moduli stacks of interest, including generalized K-theoretic Donaldson–Thomas invariants. In this paper, we prove virtual Riemann–Roch theorems in the setting of almost perfect obstruction theory in both the non-equivariant and equivariant cases, including cosection localized versions. These generalize and remove technical assumptions from the virtual Riemann–Roch theorems of Fantechi–Göttsche and Ravi–Sreedhar. The main technical ingredients are a treatment of the equivariant K-theory and equivariant Gysin map of sheaf stacks and a formula for the virtual Todd class.

Classifying right-angled Hecke C*-algebras via K-theoretic invariants

Exploiting the graph product structure and results concerning amalgamated free products of C⁎-algebras we provide an explicit computation of the K-theoretic invariants of right-angled Hecke C⁎-algebras, including concrete algebraic representants of a basis in K-theory. On the way, we show that these Hecke algebras are KK-equivalent with their undeformed counterparts and satisfy the UCT. Our results are applied to study the isomorphism problem for Hecke C⁎-algebras, highlighting the limits of K-theoretic classification, both for varying Coxeter type as well as for fixed Coxeter type.

Characteristic classes of Borel orbits of square-zero upper-triangular matrices

Anna Melnikov provided a parametrization of Borel orbits in the affine variety of square-zero n×n matrices by the set of involutions in the symmetric group. A related combinatorics leads to a construction a Bott-Samelson type resolution of the orbit closures. This allows to compute cohomological and K-theoretic invariants of the orbits: fundamental classes, Chern-Schwartz-MacPherson classes and motivic Chern classes in torus-equivariant theories. The formulas are given in terms of Demazure-Lusztig operations. The case of square-zero upper-triangular matrices is rich enough to include information about cohomological and K-theoretic classes of the double Borel orbits in Hom(Ck,Cm) for k+m=n. We recall the relation with double Schubert polynomials and show analogous interpretation of Rimányi-Tarasov-Varchenko trigonometric weight function.

The virtual [formula omitted]-theory of Quot schemes of surfaces

We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all surfaces and dimension 1 quotients for surfaces X with pg>0. We also show that the generating series of virtual cobordism classes can be irrational.Given a K-theory class on X of rank r, we associate natural series of virtual Segre and Verlinde numbers. We show that the Segre and Verlinde series match in the following cases: •[(i)] Quot schemes of dimension 0 quotients,•[(ii)] Hilbert schemes of points and curves over surfaces with pg>0,•[(iii)] Quot schemes of minimal elliptic surfaces for quotients supported on fiber classes. Moreover, for punctual quotients of the trivial sheaf of rank N, we prove a new symmetry of the Segre/Verlinde series exchanging r and N. The Segre/Verlinde statements have analogues for punctual Quot schemes over curves.

Wilson loop algebras and quantum K-theory for Grassmannians

We study the algebra of Wilson line operators in three-dimensional mathcal{N} = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

Localizing virtual structure sheaves for almost perfect obstruction theories

Abstract Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

Ladders and simplicity of derived module categories

Recollements of derived module categories are investigated, using a new technique, ladders of recollements, which are maximal mutation sequences. The position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category. Ladders also turn out to control derived simplicity on all levels. An algebra is derived simple if its derived category cannot be deconstructed, that is, if it is not the middle term of a non-trivial recollement whose outer terms are again derived categories of algebras. Derived simplicity on each level is characterised in terms of heights of ladders.These results are complemented by providing new classes of examples of derived simple rings, in particular indecomposable commutative rings, as well as by a finite dimensional counterexample to the Jordan–Hölder problem for derived module categories. Moreover, recollements are used to compute homological and K-theoretic invariants.

On K-theoretic invariants of semigroup C*-algebras attached to number fields, Part II

This paper continues the study of K-theoretic invariants for semigroup C*-algebras attached to ax+b-semigroups over rings of algebraic integers in number fields. We show that from the semigroup C*-algebra together with its canonical commutative subalgebra, it is possible to reconstruct the zeta function of the underlying number field as well as its ideal class group (as a group). In addition, we give an alternative interpretation of this result in terms of dynamical systems.

The ranges of 𝐾-theoretic invariants for nonsimple graph algebras

There are many classes of nonsimple graph C ∗ C^* -algebras that are classified by the six-term exact sequence in K K -theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C ∗ C^* -algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of K K -groups by splicing together smaller graphs whose C ∗ C^* -algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C ∗ C^* -algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C ∗ C^* -algebras under investigation have more than one ideal and where there are currently no relevant classification theories available.

On K-theoretic invariants of semigroup C*-algebras attached to number fields

We show that semigroup C*-algebras attached to ax+b-semigroups over rings of integers determine number fields up to arithmetic equivalence, under the assumption that the number fields have the same number of roots of unity. For finite Galois extensions, this means that the semigroup C*-algebras are isomorphic if and only if the number fields are isomorphic.

K-theory of Furstenberg Transformation Group C*-algebras

AbstractThis paper studies the K-theoretic invariants of the crossed product C*-algebras associated with an important family of homeomorphisms of the tori Tn called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given n, the K-groups of those crossed products whose corresponding n × n integer matrices are unipotent of maximal degree always have the same rank an. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these K-groups is false. Using the representation theory of the simple Lie algebra sl(2;C), we show that, remarkably, an has a combinatorial significance. For example, every a2n+1 is just the number of ways that 0 can be represented as a sum of integers between–n and n (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple explicit formula for the asymptotic behavior of the sequence {an} is given. Finally, we describe the order structure of the K0-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

C*-Algebras, Dynamics, and Classification

Classification is a central theme in mathematics, and a particularly rich one in the theory of operator algebras. Indeed, one of the first major results in the theory is Murray and von Neumann’s type classification of factors (weakly closed self-adjoint algebras of operators on Hilbert space with trivial center), and one of its modern touchstones is the mid-1970s Connes-Haagerup classification of amenable factors with separable predual. Several significant themes in the classification theory of norm-separable C‡-algebras have emerged since the work of Connes-Haagerup, and these were the focus of our workshop. They include Elliott’s program to classify separable nuclear C‡-algebras via K-theoretic invariants, the role of C‡-algebras in the classification of orbit equivalence relations of discrete countable group actions, and the more recent contact between descriptive set theorists and operator algebraists which seeks to quantify the Borel complexity of the isomorphism relation for various natural classes of algebras.

Classification of a class of crossed product [formula omitted]-algebras associated with residually finite groups

A residually finite group acts on a profinite completion by left translation. We consider the corresponding crossed product C ⁎ -algebra for discrete countable groups that are central extensions of finitely generated abelian groups by finitely generated abelian groups (these are automatically residually finite). We prove that all such crossed products are classifiable by K-theoretic invariants using techniques from the classification theory for nuclear C ⁎ -algebras.

Non-simple purely infinite rings

In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-) unital $K$-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over $K$ with many non-simple Leavitt path algebras is purely infinite.

Simple real rank zero algebras with locally Hausdorff spectrum

Let A be a unital, simple, separable C*-algebra with real rank zero, stable rank one, and weakly unperforated ordered K 0 group. Suppose, also, that A can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then A is tracially approximately finite dimensional (i.e., A has tracial rank zero). Hence, A is an AH-algebra with bounded dimension growth and is determined by K-theoretic invariants. The above result also gives the first proof for the locally AH case.

Reviewer Acknowledgment

Exploiting the graph product structure and results concerning amalgamated free products of C⁎-algebras we provide an explicit computation of the K-theoretic invariants of right-angled Hecke C⁎-algebras, including concrete algebraic representants of a basis in K-theory. On the way, we show that these Hecke algebras are KK-equivalent with their undeformed counterparts and satisfy the UCT. Our results are applied to study the isomorphism problem for Hecke C⁎-algebras, highlighting the limits of K-theoretic classification, both for varying Coxeter type as well as for fixed Coxeter type.

K -theoretic invariants for Floer homology

This paper defines two K-theoretic invariants, Wh 1 and Wh 2 ,f or individual and one-parameter families of Floer chain complexes. The chain complexes are generated by intersection points of two Lagrangian submanifolds of a symplectic manifold, and the boundary maps are determined by holomorphic curves connecting pairs of intersection points. The paper proves that Wh 1 and Wh 2 do not depend on the choice of almost complex structures and are invariant under Hamiltonian deformations. The proof of this invariance uses properties of holomorphic curves, parametric gluing theorems, and a stabilization process.

When are two commutative C*-algebras stably homotopy equivalent?

Let X and Y be finite connected CW complexes with base points, and let K denote the C*algebra of compact operators on a separable infinite dimensional complex Hilbert space. The purpose of this paper is to study the question of when C0(X)⊗K is homotopy equivalent to C0(Y )⊗K; here C0(X) is, as usual, the C*-algebra of continuous complex-valued functions which vanish at the base point. Recall that two C-algebras A and B are said to be homotopy equivalent, written A ' B, if there are ∗-homomorphisms φ : A → B and ψ : B → A for which ψ ◦ φ and φ ◦ ψ may be deformed by a path of endomorphisms to the identity maps idA : A → A and idB : B → B, respectively. Two C*-algebras A and B are called ‘stably’ homotopy equivalent if A⊗K ' B⊗K (this should not be confused with the notion of stable homotopy of spaces used in topology), so another way of stating the problem we consider is: when are C0(X) and C0(Y ) ‘stably’ homotopy equivalent? To begin with we should remark that the analogous question for homotopy equivalence has a simple answer: C0(X) and C0(Y ) are homotopy equivalent if and only if X and Y are based homotopy equivalent. The situation for ‘stable’ homotopy equivalence is more complicated and is closely related to K-theory: for example, if two C*-algebras are ‘stably’ homotopy equivalent then they have the same K-theoretic invariants. The main purpose of this paper is to show that the converse is not true: we give an example of two spaces X and Y which cannot be distinguished by

K-theoretic invariants for C∗-algebras associated to transformations and induced flows

We discuss a technique of studying the K-theory of a unital C ∗-algebra associated to a homomorphism on a compact metric space ( Y Z ) by examining the non-unital C ∗-algebra associated to the induced topological flow (Ind Z R ( Y), R ). The Thom isomorphism of Connes and the Schwartzman asymptotic cycle are used to calculate the range of the trace corresponding to an invariant measure on the K 0 group of C ∗( X, R ) for a continuous flow on a compact metric space ( X, R ). Under certain conditions projections in C ∗( X, R ) with trace r corresponding to cross sections to the flow can be constructed for every positive real number r in this range, again by combining techniques of Connes and Schwartzman. Applications to the calculation of the tracial range of K 0( C ∗( X, R )) are discussed. In particular, this invariant is calculated for minimal affine actions of Z on n-tori which have quasi-discrete spectrum, and for minimal actions of Z on compact abelian groups with topologically discrete spectrum. In both cases this tracial invariant is shown to be the preimage of the eigenvalues for ( Y, Z ) under the natural projection ø: R → R / Z = S 1.