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K-theory for real C * -algebras and applications
... Additive topological K-theory extends to a homology theory K − for C * -algebras, called operator K-theory [8,32]. For a unital C * -algebra A consider the groupoid pr A of projection matrices, i.e. self-adjoint idempotent matrices, and partial isometries witnessing the Murray-von Neumann relation between them. ...
... We briefly review the basic K-theory of C * -algebras we will need. Standard references to the theory are [8,32]. We begin by establishing notations and basic results for matrix operations. ...
We construct commutative algebra spectra that represent the operator $K$-theory of $C^*$-algebras, which are algebras over the commutative ring spectra that represent topological $K$-theory. The spectral multiplicative structure introduces a new graded commutative ring structure on the $K$-groups, generalizing the well-known graded ring structure of commutative $C^*$-algebras. This last structure reflects the multiplicative structure of topological $K$-theory via Gelfand duality, Swan's theorem and the fiber tensor product. We introduce $\mathscr L$-permutative categories, a generalization of bipermutative categories, which are permutative categories equipped with a multiplicative structures induced by coherent actions of the linear isometries operad. The main class of examples of interest are categories whose objects are projection matrices of the unitization of the stabilizations $\widetilde{\mathfrak{KA}}$ of a $C^*$-algebras $\mathfrak A$, and morphisms partial isometries witnessing the Murray-von Neumann relation. We then construct $E_\infty$-ring spaces out of them by adapting the usual method applied to bipermutative categories. The delooping functor of the recognition principle, the homotopical augmentation ideal and localization at the Bott element then give us our algebra spectra.
... For the convenience of the reader and to establish notation we briefly summarise the results in real Kasparov theory of use to us for the bulk-edge correspondence. The reader may consult [27,54] or [9, Appendix A] for more information. ...
... , where i * comes from restricting a Clifford action of C j+1,d to C j,d . Next, we use an elementary extension of the Atiyah-Bott-Shapiro isomorphism, [1] and [54, §2.3], to make the identification We remark that Index k reproduces the usual (real) C * -module Fredholm index as studied in [19,Chapter 4] if k = 0, see [54]. ...
We study the application of Kasparov theory to topological insulator systems and the bulk-edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real C*-algebras and KKO-theory must be used.
... Given a complex and ungraded C * -algebra A with real structure r A , we know from [22, §5] that there are isomorphisms KKR(C r,s , A) ∼ = KO r−s (A r A ), where this identification is shown via a (generalised) Clifford-module index, see [43,Section 2.2]. ...
We use the Cayley transform to provide an explicit isomorphism at the level of cycles from van Daele $K$-theory to $KK$-theory for graded $C^*$-algebras with a real structure. Isomorphisms between $KK$-theory and complex or real $K$-theory for ungraded $C^*$-algebras are a special case of this map. In all cases our map is compatible with the computational techniques required in physical and geometrical applications, in particular index pairings and Kasparov products. We provide applications to real $K$-theory and topological phases of matter.
... Up to a finite-dimensional adjustment (see [6,Appendix B]), the topological information of interest of this Kasparov module is contained in the kernel, Ker(X), which is a finitely generated and projective C * -submodule of E B with a graded leftaction of C n,k . If B is ungraded, an Atiyah-Bott-Shapiro like map then gives an isomorphism KKO(C n,k , B) → KO n−k (B) via Clifford modules, see [34,Section 2.2]. ...
Recent work by Prodan and the second author showed that weak invariants of topological insulators can be described using Kasparov's KK-theory. In this note, a complementary description using semifinite index theory is given. This provides an alternative proof of the index formulae for weak complex topological phases using the semifinite local index formula. Real invariants and the bulk-boundary correspondence are also briefly considered.
... Most of the theory is parallel to the theory of complex C * -algebras. For more details on real C * -algebras and their K-theory, including the role this plays in index theory, compare [35]. ...
These notes are based on lectures on index theory, topology, and operator algebras at the "School on High Dimensional Manifold Theory" at the ICTP in Trieste, and at the Seminari di Geometria 2002 in Bologna. We describe how techniques coming from the theory of operator algebras, in particular $C^*$-algebras, can be used to study manifolds. Operator algebras are extensively studied in their own right. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. The link between topology and analysis is provided by index theorems. Starting with the classical Atiyah-Singer index theorem, we will explain several index theorems in detail. Our point of view will be in particular, that an index lives in a canonical way in the K-theory of a certain $C^*$-algebra. The geometrical context will determine, which $C^*$-algebra to use. A central pillar of work in the theory of $C^*$-algebras is the Baum-Connes conjecture. Nevertheless, it has important direct applications to the topology of manifolds, it impliese.g. the Novikov conjecture. We will explain the Baum-Connes conjecture and put it into our context. Several people contributed to these notes by reading preliminary parts and suggesting improvements, in particular Marc Johnson, Roman Sauer, Marco Varisco und Guido Mislin. I am very indebted to all of them. This is an elaboration of the first chapter of the author's contribution to the proceedings of the above mentioned "School on High Dimensional Manifold Theory" 2001 at the ICTP in Trieste.
... There is a real version of this result which is 24-cyclic [66,67]. We will use the PV exact sequence extensively for the computation of the bulk-boundary homomorphism ∂. ...
We state and prove a general result establishing that T-duality simplifies the bulk-boundary correspondence, in the sense of converting it to a simple geometric restriction map. This settles in the affirmative several earlier conjectures of the authors, and provides a clear geometric picture of the correspondence. In particular, our result holds in arbitrary spatial dimension, in both the real and complex cases, and also in the presence of disorder, magnetic fields, and H-flux. These special cases are relevant both to String Theory and to the study of the quantum Hall effect and topological insulators with defects in Condensed Matter Physics.
... We first consider the following class of examples. Recall from [18] that a C * -algebra over R is a Banach * -algebra over R, * -isometrically isomorphic to a norm-closed subalgebra of linear operators on a real Hilbert space. We extend the C * -structure of A (as an R-algebra) to a C * -structure of A ⊗ C (as a C-algebra) by setting (a ⊗ z) * = (a * ) ⊗z. ...
We introduce a general framework to unify several variants of twisted
topological $K$-theory. We focus on the role of finite dimensional real simple
algebras with a product-preserving involution, showing that Grothendieck-Witt
groups provide interesting examples of twisted $K$-theory. These groups are
linked with the classification of algebraic vector bundles on real algebraic
varieties.
We examine the non-commutative index theory associated with the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.
Recent work by Prodan and the second author showed that weak invariants of topological insulators can be described using Kasparov’s KK-theory. In this note, a complementary description using semifinite index theory is given. This provides an alternative proof of the index formulae for weak complex topological phases using the semifinite local index formula. Real invariants and the bulk-boundary correspondence are also briefly considered.
A bounded operator T on a separable, complex Hilbert space is said to be odd symmetric if I*TtI = T where I is a real unitary satisfying I2 = -1 and Tt denotes the transpose of T . It is proved that such an operator can always be factorized as T = I*AtIA with some operator A. This generalizes a result of Hua and Siegel for matrices. As application it is proved that the set of odd symmetric Fredholm operators has two connected components labelled by a Z2- index given by the parity of the dimension of the kernel of T . This recovers a result of Atiyah and Singer. Two examples of Z2-valued index theorems are provided, one being a version of the Noether- Gohberg-Krein theorem with symmetries and the other an application to topological insulators.
On decrit et on applique un nouveau theoreme d'indice pour des operateurs elliptiques sur certaines varietes non compactes. On calcule un indice a valeur reelle pour des operateurs de type Dirac sur des varietes non compactes de geometrie bornee
The purpose of this paper is to give an exposition of the various triviality theorems, the equivariant version of a result due to L. Brown, and a simplification of the proof of Kasparov's triviality theorems.