Projective Spectrum and Cyclic Cohomolgy

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.

Projective spectrum in Banach algebras

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 $H^1_d(P^c(A), cc)$.

$N_{p}$-type quotient modules on the torus

Structure of the quotient modules in $hh$ is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the so-call $N_{p}$-type quotient modules, namely, quotient modules of the form $hhominus [z-p]$, where $p (w)$ is a function in the classical Hardy space $H^2(G)$ and $[z-p]$ is the submodule generated by $z-p (w)$. This type of quotient modules serve as good examples in many studies. A notable feature of the $N_{p}$-type quotient module is its close connections with some classical single variable operator theories.