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For a tuple $A=(A_1, A_2, ..., A_n)$ of elements in a unital algebra ${mathcal B}$ over $mathbb{C}$, its {em projective spectrum} $P(A)$ or $p(A)$ is the collection of $zin mathbb{C}^n$, or respectively $zin mathbb{P}^{n-1}$ such that the multi-parameter pencil $A(z)=z_1A_1+z_2A_2+cdots +z_nA_n$ is not invertible in ${mathcal B}$. ${mathcal B}$-valued $1$-form $A^{-1}(z)dA(z)$ contains much topological information about $P^c(A):=mathbb{C}^nsetminus P(A)$. In commutative cases, invariant multi-linear functionals are effective tools to extract that information. This paper shows that in non-commutative cases, the cyclic cohomology of ${mathcal B}$ does a similar job. In fact, a Chen-Weil type map $kappa$ from the cyclic cohomology of ${mathcal B}$ to the de Rham cohomology $H^*_d(P^c(A), mathbb{C})$ is established. As an example, we prove a closed high-order form of the classical Jacobis formula.
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For a tuple $A=(A_0, A_1, ..., A_n)$ of elements in a unital Banach algebra ${mathcal B}$, its {em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]in pn$ such that $A(z)=z_0A_0+z_1A_1+... +z_nA_n$ is not invertible in ${mathcal B}$. The pre-image of $p(A)$ in ${cc}^{n+1}$ is denoted by $P(A)$. When ${mathcal B}$ is the $ktimes k$ matrix algebra $M_k(cc)$, the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When $A$ is commutative, $P(A)$ is a union of hyperplanes. When ${mathcal B}$ is reflexive or is a $C^*$-algebra, the {em projective resolvent set} $P^c(A):=cc^{n+1}setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type ${mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ on $P^c(A)$. As a consequence, we show that if ${mathcal B}$ is a $C^*$-algebra with a trace $phi$, then $phi(A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H^1_d(P^c(A), cc)$.
In this article we provide a classification of the projective transformations in $PSL(n+1,Bbb{C})$ considered as automorphisms of the complex projective space $Bbb{P}^n$. Our classification is an interplay between algebra and dynamics, which just as in the case of isometries of CAT(0)-spaces, can be given by means of tree three types, namely: elliptic, parabolic and loxodromic. We carefully describe the dynamic in each case, more precisely we determine the corresponding Kulkarnis limit set, the equicontinuity region, the discontinuity region and in some cases we provide families of maximal regions where the respective cyclic group acts properly discontinuously. Also we provide, in each case, some equivalents ways to classify the projective transformations.
We prove that iterating projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bav{c}ak.
Let $alpha$ be a composition of $n$ and $sigma$ a permutation in $mathfrak{S}_{ell(alpha)}$. This paper concerns the projective covers of $H_n(0)$-modules $mathcal{V}_alpha$, $X_alpha$ and $mathbf{S}^sigma_{alpha}$, which categorify the dual immaculate quasisymmetric function, the extended Schur function, and the quasisymmetric Schur function when $sigma$ is the identity, respectively. First, we show that the projective cover of $mathcal{V}_alpha$ is the projective indecomposable module $mathbf{P}_alpha$ due to Norton, and $X_alpha$ and the $phi$-twist of the canonical submodule $mathbf{S}^{sigma}_{beta,C}$ of $mathbf{S}^sigma_{beta}$ for $(beta,sigma)$s satisfying suitable conditions appear as $H_n(0)$-homomorphic images of $mathcal{V}_alpha$. Second, we introduce a combinatorial model for the $phi$-twist of $mathbf{S}^sigma_{alpha}$ and derive a series of surjections starting from $mathbf{P}_alpha$ to the $phi$-twist of $mathbf{S}^{mathrm{id}}_{alpha,C}$. Finally, we construct the projective cover of every indecomposable direct summand $mathbf{S}^sigma_{alpha, E}$ of $mathbf{S}^sigma_{alpha}$. As a byproduct, we give a characterization of triples $(sigma, alpha, E)$ such that the projective cover of $mathbf{S}^sigma_{alpha, E}$ is indecomposable.
We provide a projective description of the space $mathcal{E}^{{mathfrak{M}}}(Omega)$ of ultradifferentiable functions of Roumieu type, where $Omega$ is an arbitrary open set in $mathbb{R}^d$ and $mathfrak{M}$ is a weight matrix satisfying the analogue of Komatsus condition $(M.2)$. In particular, we obtain in a unified way projective descriptions of ultradifferentiable classes defined via a single weight sequence (Denjoy-Carleman approach) and via a weight function (Braun-Meise-Taylor approach) under considerably weaker assumptions than in earli
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