Determinant Theory in Banach Algebras

A determinant theory is developed for Banach algebras and a characterization of those traced unital Banach algebras admitting a determinant is given.

Invariant Determinants in Banach Algebras

The invariant determinants ha Banach algebras are discussed and a necessary and sufficient condition for those integral traced unital Barach algebra(A,τ) admitting a G-invari- ant determinant is obtained,where G is a group of trace preserving automorphisms of A.

Asymptotic Excisions of Metric Spaces and Ideals of Asymptotic Coarse Roe Algebras

We introduce in this note the notions of asymptotic excision of proper metric spaces and asymptotic equivalence relation for subspaces of metric spaces, which are relevant in characterizing spatial ideals of the asymptotic coarse Roe algebras. We show that the lattice of the asymptotic equivalence classes of the subspaces of a proper metric space is isomorphic to the lattice of the spatial ideals of the asymptotic Roe algebra. For asymptotic excisions of the metric space, we also establish a MayeroVietoris sequence in K-theory of the asymptotic coarse Roe algebras.

The Gap Labelling Integrated Density of States for a Quasi Crystal Universe Is Identical to the Observed 4.5 Percent Ordinary Energy Density of the Cosmos

Condense matter methods and mathematical models used in solving problems in solid state physics are transformed to high energy quantum cosmology in order to estimate the magnitude of the missing dark energy of the universe. Looking at the problem from this novel viewpoint was rewarded by a rather unexpected result, namely that the gap labelling method of integrated density of states for three dimensional icosahedral quasicrystals is identical to the previously measured and theoretically concluded ordinary energy density of the universe, namely a mere 4.5 percent of Einstein’s energy density, i.e. E(O) = mc2/22 where E is the energy, m is the mass and c is the speed of light. Consequently we conclude that the missing dark energy density must be E(D) = 1 - E(O) = mc2(21/22) in agreement with all known cosmological measurements and observations. This result could also be interpreted as a strong evidence for the self similarity of the geometry of spacetime, which is an expression of its basic fractal nature.

LENTICULAR NONCOMMUTATIVE TORI

All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as an inductive limit of circle algebras, where Lki (ni) are lens spaces. Let Lcd be a cd-homogeneous C*-algebra over whose cd-homogeneous C*-subalgebra restricted to the subspace Tr × T2 is realized as C(Tr) A1/d Mc(C), and of which no non-trivial matrix algebra can be factored out.The lenticular noncommutative torus Lpcd is defined by twisting in by a totally skew multiplier p on Tr+2 × Zm-2. It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p, and that Lpcd is not stablyisomorphic to if the cd-homogeneous C*-subalgebra of Lpcd restricted to some subspace LkiLki (ni) is realized as the crossed product by the obvious non-trivial action of Zki on a cd/ki-homogeneous C*-algebra over S2ni+1 for ki an integer greater than 1.

A Note on Orthogonal Multiplication

Hurewicz-Radon theorem [1] gives a sufficient and necessary condition about the existence of an orthogonal multiplication f:Rk×Rn→Rn. In this paper,we shall discuss two extensions of the theorem.that is,the sufficient and necessary conditions about the existence of an orthogonal multiplication f:Rk×Rn→Rn+1or f:Rk×Rn→Rn+2

The Coarse l^(p)-Novikov Conjecture and Banach Spaces with Property(H)

In this paper,for 1<p<∞,the authors show that the coarse l^(p)-Novikov conjecture holds for metric spaces with bounded geometry which are coarsely embeddable into a Banach space with Kasparov-Yu’s Property(H).

K-Theory and the Quantization Commutes with Reduction Problem

The authors examine the quantization commutes with reduction phenomenon for Hamiltonian actions of compact Lie groups on closed symplectic manifolds from the point of view of topological K-theory and K-homology. They develop the machinery of K-theory wrong-way maps in the context of orbifolds and use it to relate the quantization commutes with reduction phenomenon to Bott periodicity and the K-theory formulation of the Weyl character formula.

The φ-Group for n Submodules Systems as a Generalization of the K-Theory

In this paper,the authors construct a φ-group for n submodules,which generalizes the classical K-theory and gives more information than the classical ones.This theory is related to the classification theory for indecomposable systems of n subspaces.

K-theory of Continuous Deformations of C＊-algebras

We study K-theory of continuous deformations of C＊-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz algebras.

Equivariant eta forms and equivariant differential K-theory

In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation.A discrete inf-sup condition is proved and the optimal error estimates are also derived.Numerical experiments validate the theoretical analysis.

Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds

Comparing to the construction of stringy cohomology ring of equivariant sta-ble almost complex manifolds and its relation with the Chen-Ruan cohomology ring of the quotient almost complex orbifolds,the authors construct in this note a Chen-Ruan cohomology ring for a stable almost complex orbifold.The authors show that for a finite group G and a G-equivariant stable almost complex manifold X,the G-invariant part of the stringy cohomology ring of(X,G)is isomorphic to the Chen-Ruan cohomology ring of the global quotient stable almost complex orbifold[X/G].Similar result holds when G is a torus and the action is locally free.Moreover,for a compact presentable stable almost complex orbifold,they study the stringy orbifold K-theory and its relation with Chen-Ruan cohomology ring.

Quantum K-theory and q-difference Equations

This is a set of lecture notes for the first author’s lectures on the difference equations in2019 at the Institute of Advanced Study for Mathematics at Zhejiang University.We focus on explicit computations and examples.The convergence of local solutions is discussed.

A note on quantum odometers

We discuss various aspects of noncommutative geometry of smooth subalgebras of Bunce-Deddens-Toeplitz algebras.

Verlinde/Grassmannian Correspondence and Rank 2 δ-Wall-Crossing

Motivated by Witten’s work,we propose a K-theoretic Verlinde/Grassmannian correspondence which relates the GL Verlinde numbers to the K-theoretic quasimap invariants of the Grassmannian.We recover these two types of invariants by imposing different stability conditions on the gauged linear sigma model associated with the Grassmannian.We construct two families of stability conditions connecting the two theories and prove two wall-crossing results.We confirm the Verlinde/Grassmannian correspondence in the rank two case.

Some G-M-type Banach spaces and K-groups of operator algebras on them

By providing several new varieties of G-M-type Banachspaces according to decomposable and compoundable properties, this paper discusses the operator structures of thesespaces and the K-theory of the algebra of the operators on these G-M-type Banach spaces throughcalculation of the K-groups of the operator ideals contained in the class of Riesz operators.

CENERALIZED COHOMOLOGY OPERATIONS AND THE FINITENESS OF MULTIPLICATIVE ψ_k-MODULES

In this paper we prove that for a fixed l, the number of multiplicative ψ_k-module M=Z/p[y_1,…,y_1]/(Z/p[y_1,…,y_1])^(p+1)with degy_i=2b_i, i=1,…, l, is finite, and for a fixed l, the unstable polynomial A_p-al-gebras P(B)=Z/p[x_1,…,x_1] which are realizable are finite.

ISOPARAMETRIC HYPERSURFACES WITH FOUR DISTINCT PRINCIPAL CURVATURES

I. PREPARATION AND LEMMASWe start recalling theMain Theorem of Abresch. Given an isoparametric hypersurface in Sn+1 with g= 4, then the pair （m, m+ ）-w. r. g, we may assume that m ≤m+, satisfies one of the three conditions below:

Elements of Small Orders in K_2FⅡ

In ＂Elements of small orders in K2（F）＂ （Algebraic K-Theory, Lecture Notes in Math., 966, 1982, 1-6.）, the author investigates elements of the form {α, Фn（α）} in the Milnor group K2F of a field F, where Фn（x） is the n-th cyclotomic polynomial. In this paper, these elements are generalized. Applying the explicit formulas of Rosset and Tate for the transfer homomorphism for K2, the author proves some new results on elements of small orders in K2F.

The Whitehead Groups of Semigroups

Let S be a semigroup with identity and zero,（S）the category of finitely generated projective S-systems.In this paper,the Whitehead group K1S of S is defined to be K0Ω（S）, where Ω（S）is the Loop category of（S）,and the structure of K1S is given.