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We extend some classical results of Cowling and Meda to the noncommutative setting. Let $(T_t)_{t>0}$ be a symmetric contraction semigroup on a noncommutative space $L_p(\mathcal{M}),$ and let the functions $\phi$ and $\psi$ be regularly…

We develop a microspectral theory for quasinilpotent linear operators $Q$ (i.e., those with $\sigma(Q) = \{0}$) in a Banach space. When such $Q$ is not compact, normal, or nilpotent, the classical spectral theory gives little information,…

The class of separable C*-algebras which can be written as inductive limits of continuous-trace C*-algebras with spectrum homeomorphic to a disjoint union of trees and trees with a point removed is classified by the Cuntz semigroup.

For a linear operator $T$ in a Banach space let $\sigma_p(T)$ denote the point spectrum of $T$, $\sigma_{p[n]}(T)$ for finite $n > 0$ be the set of all $\lambda \in \sigma_p(T)$ such that $\dim \ker (T - \lambda) = n$ and let…

We are interested in the spectrum of the Hodge-de Rham operator on a cyclic covering $X$ over a compact manifold $M$ of dimension $n+1$. Let $\Sigma$ be a hypersurface in $M$ which does not disconnect $M$ and such that $M-\Sigma$ is a…

We extend the analysis of Balmer and Gallauer on the tt-geometry of the small derived category of permutation modules for a finite group over a field to the setting of a commutative Noetherian base. In this general context, we provide a…

Thoma's theorem states that a group algebra $C^*(\Gamma)$ is of type I if and only if $\Gamma$ is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually…

The famous 1960s Lumer-Phillips Theorem states that a closed and densely defined operator $A\colon D(A)\subseteq X\rightarrow X$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if $(A,D(A))$ is…

Since its introduction by Symons, the semigroup of maps with restricted range has been studied in the context of transformations on a set, or of linear maps on a vector space. Sets and vector spaces being particular examples of independence…

In recent work, for a triangulated category $\cT$, the author introduced a topological space $\tSpec(\cT)$ which we call the triangular spectrum of $\cT$ as a tensor-free analog of the Balmer spectrum for a tensor triangulated category. In…

For a class of $*$-algebras, where $*$-algebra $A_{\Gamma,\tau}$ is generated by projections associated with vertices of graph $\Gamma$ and depends on a parameter $\tau$ $(0 < \tau \leq 1)$, we study the sets $\Sigma_\Gamma$ of values of…

We prove that there is a universal constant $C>0$ with the following property. Suppose that $n\in \mathbb{N}$ and that $\mathsf{A}=(a_{ij})\in M_n(\mathbb{R})$ is a symmetric stochastic matrix. Denote the second-largest eigenvalue of…

Following a result of Hatori, Miura and Tagaki ([4]) we give here a spectral characterization of an isomorphism from a $C^\star$-algebra onto a Banach algebra. We then use this result to show that a $C^\star$-algebra $A$ is isomorphic to a…

Let $H_0$ and $H$ be self-adjoint operators in a Hilbert space. We consider the spectral projections of $H_0$ and $H$ corresponding to a semi-infinite interval of the real line. We discuss the index of this pair of spectral projections and…

Sufficient conditions for a semigroup measure algebra to have contractible Gelfand spectrum are given and it is shown that for a wide class of semigroups these conditions are also necessary.

Let $A$ and $B$ be unital semi-simple commutative Banach algebras. In this paper we study two-variable polynomials $p$ which satisfy the following property: a map $T$ from $A$ onto $B$ such that the equality \[ \sigma (p(Tf,Tg))=\sigma…

Let $C_b(X)$ be the C*-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space $X$. Moreover, let $A_0$ be some ideal and $A_1$ be some unital C*-subalgebra of $C_b(X)$. For $A_0$ and $A_1$ having…

We treat spectral problems by twisted groupoid methods. To Hausdorff locally compact groupoids endowed with a continuous $2$-cocycle one associates the reduced twisted groupoid $C^*$-algebra. Elements (or multipliers) of this algebra admit…

Let $A$ be a C*-algebra that is the norm closure $A = \overline{\sum_{\beta \in \alpha} I_\beta}$ of an arbitrary sum of C*-ideals $I_\beta \subseteq A$. We construct a homological spectral sequence that takes as input the K-theory of…

Motivated by a question of L. Robert, asking whether $\rm L(T(A)) = Lsc_{C}(T(A))$ for any separable C*-algebra A, we introduce and initiate the study of \emph{tracially reflexive C*-algebras}. We first prove that commutative C*-algebras…