Involutions and stable subalgebras

On Coarse Spectral Geometry in Even Dimension

Let $σ$ be the involution of the Roe algebra $\Roe{\RR}$ which is induced from the reflection $\RR\to\RR; x\mapsto -x$. A graded Fredholm module over a separable $C^*$-algebra $A$ gives rise to a homomorphism $\tildeρ:A\to\Roe{\RR}^σ$ to the fixed-point subalgebra. We use this observation to give an even-dimensional analogue of a result of Roe. Namely, we show that the $K$-theory of this symmetric Roe algebra is $K_0(\Roe{\RR}^σ)\cong\ZZ$, $K_1(\Roe{\RR})=0$, and that the induced map $\tildeρ_*:K_0(A) \to \ZZ$ on $K$-theory gives the index pairing of $K$-homology with $K$-theory.

On finitely summable Fredholm modules from Smale spaces

We prove that all K-homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K-d…

Relatively spectral morphisms and applications to K-theory

Spectral morphisms between Banach algebras are useful for comparing their K-theory and their "noncommutative dimensions" as expressed by various notions of stable ranks. In practice, one often encounters situations where the spectral information is only known over a dense subalgebra. We investigate such relatively spectral morphisms. We prove a relative version of the Density Theorem regarding isomorphism in K-theory. We also solve Swan's problem for the connected stable rank, in fact for an entire hierarchy of higher connected stable ranks that we introduce.

Enriched set-valued P-partitions and shifted stable Grothendieck polynomials

We introduce an enriched analogue of Lam and Pylyavskyy's theory of set-valued $P$-partitions. An an application, we construct a $K$-theoretic version of Stembridge's Hopf algebra of peak quasisymmetric functions. We show that the symmetric part of this algebra is generated by Ikeda and Naruse's shifted stable Grothendieck polynomials. We give the first proof that the natural skew analogues of these power series are also symmetric. A central tool in our constructions is a "$K$-theoretic" Hopf algebra of labeled posets, which may be of independent interest. Our results also lead to some new exp…

关于高度2稳定茎中的周期族

On periodic families in the stable stems of height two

该研究在球面2-素数稳定同伦群中发现大量无限周期族，验证了Hopkins–Mahowald的相关预测，明确其挠性与检测性质，并证实特定维数下怪异球面的存在性。