### A canonical trace associated with certain spectral triples.

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For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that ${\omega}_{m}\left(t\right)/\omega \u2099\left(t\right)\to \infty $ as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists m ∈ ℕ such that...

Let φ be a Jordan automorphism of an algebra . The situation when an element a ∈ satisfies $1/2(\phi \left(a\right)+{\phi}^{-1}\left(a\right))=a$ is considered. The result which we obtain implies the Kleinecke-Shirokov theorem and Jacobson’s lemma.

Let $\mathscr{H}$ be a separable infinite dimensional complex Hilbert space, and let $\mathcal{L}\left(\mathscr{H}\right)$ denote the algebra of all bounded linear operators on $\mathscr{H}$ into itself. Let $A=({A}_{1},{A}_{2},\cdots ,{A}_{n})$, $B=({B}_{1},{B}_{2},\cdots ,{B}_{n})$ be $n$-tuples of operators in $\mathcal{L}\left(\mathscr{H}\right)$; we define the elementary operators ${\Delta}_{A,B}\phantom{\rule{0.222222em}{0ex}}\mathcal{L}\left(\mathscr{H}\right)\mapsto \mathcal{L}\left(\mathscr{H}\right)$ by ${\Delta}_{A,B}\left(X\right)={\sum}_{i=1}^{n}{A}_{i}X{B}_{i}-X.$ In this paper, we characterize the class of pairs of operators $A,B\in \mathcal{L}\left(\mathscr{H}\right)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A,B\in \mathcal{L}\left(\mathscr{H}\right)$ such that ${\sum}_{i=1}^{n}{B}_{i}T{A}_{i}=T$ implies ${\sum}_{i=1}^{n}{A}_{i}^{*}T{B}_{i}^{*}=T$ for all $T\in {\mathcal{C}}_{1}\left(\mathscr{H}\right)$ (trace class operators). The main result is the equivalence between this property and the fact that...

Let 𝓐 be a Banach algebra without nonzero finite dimensional ideals. Then every compact semiderivation on 𝓐 is a quasinilpotent operator mapping 𝓐 into its radical.

Denote by $C$ the commutator $AB-BA$ of two bounded operators $A$ and $B$ acting on a locally convex topological vector space. If $AC-CA=0$, we show that $C$ is a quasinilpotent operator and we prove that if $AC-CA$ is a compact operator, then $C$ is a Riesz operator.

Let L(H) denote the algebra of bounded linear operators on a complex separable and infinite dimensional Hilbert space H. For A, B ∈ L(H), the generalized derivation δA,B associated with (A, B), is defined by δA,B(X) = AX - XB for X ∈ L(H). In this note we give some sufficient conditions for A and B under which the intersection between the closure of the range of δA,B respect to the given topology and the kernel of δA*,B* vanishes.

Let $L\left(H\right)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L\left(H\right)$, we define the elementary operator ${\Delta}_{A}:L\left(H\right)\u27f6L\left(H\right)$ by ${\Delta}_{A}\left(X\right)=AXA-X$. In this paper we study the class of operators $A\in L\left(H\right)$ which have the following property: $ATA=T$ implies $A{T}^{*}A={T}^{*}$ for all trace class operators $T\in {C}_{1}\left(H\right)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of ${\Delta}_{A}$ is closed under taking...