A. Yu. Pirkovskii
Given a metrizable K"othe algebra $λ(P)$, we compute the global dimension, the weak global dimension, the bidimension, and the weak bidimension of $λ(P)$ in terms of the K"othe set $P$.
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We give a survey of our recent results on homological properties of K"othe algebras, with an emphasis on biprojectivity, biflatness, and homological dimension. Some new results on the approximate contractibility of K"othe algebras are also presented.
The conditions on a Banach space, $E$, under which the algebra, $\mathcal{K}(E)$, of compact operators on $E$ is right flat or homologically unital are investigated. These homological properties are related to factorization in the algebra and it is shown that, for $\mathcal{K}(E)$, they are closely associated with the approximation property for $E$. The class of spaces, $E$, such that $\mathcal{K}(E)$ is known to be right flat and homologically unital is extended to include spaces which do not have the bounded compact approximation property.
In this paper, we study the notion of $ϕ$-biflatness, $ϕ$-biprojectivity, approximate biprojectivity and Johnson pseudo-contractibility for a new class of Banach algebras. Using this class of Banach algebras we give some examples which are approximately biprojective. Also some Banach algebras are given among matrix algebras which are never Johnson pseudo-contractible.
Algebras of analytic functionals and homological epimorphisms
本文证明连通复李群上的解析泛函代数在无需可解性假设下满足同调满同态性质，并推广到更一般的完备化同态，包括关于Banach PI-代数的包络。
Let $G$ and $H$ be locally compact groups. $BM(G, H)$ denoted the Banach algebra of bounded bilinear forms on $C_{0}(G)\times C_{0}(H)$.In this paper, the homological properties of Bimeasure algebras are investigated. It is found and approved that the Bimeasure algebras $BM(G, H)$ is amenable if and only if $G$ and $H$ are discrete. The correlation between the weak amenability of $BM(G, H)$ and $M(G\times H)$ is assessed. It is found and approved that the biprojectivity of the bimeasure algebra $BM(G, H)$ is equivalent to the finiteness of $G$ and $H$. Furthermore, we show that the bimeasure g…
Let T be a homomorphism from a Banach algebra B to a Banach algebra A.The Cartesian product space A * B with T-Lau multiplication and l^1-norm becomes a new Banach algebra A *_T B. We investigate the notions such as approximate amenability, pseudo amenability, phi-pseudo amenability,phi-biflatness and phi-biprojectivity for Banach algebra A *_T B. We also present an example to show that approximate amenability of A and B is not stable for A *_T B. Finally we characterize the double centralizer algebra of A *_T B and present an application of this characterization.
We show that there exists a Banach space $E$ with the following properties: the Banach algebra $\mathscr{B}(E)$ of bounded, linear operators on $E$ has a singular extension which splits algebraically, but it does not split strongly, and the homological bidimension of $\mathscr{B}(E)$ is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of Helemskii. The Banach space $E$ that we use was originally introduced by Read (J. London Math. Soc. 1989).
Our aim is to give an explicit description of the Arens-Michael envelope for the universal enveloping algebra of a finite-dimensional nilpotent complex Lie algebra. It turns out that the Arens-Michael envelope belongs to a class of completions introduced by R.~Goodman in 70s. To find a precise form of this algebra we preliminary characterize the set of holomorphic functions of exponential type on a simply connected nilpotent complex Lie group. This approach leads to unexpected connections to Riemannian geometry and the theory of order and type for entire functions. As a corollary, it is shown …
We give several characterizations of relative homological epimorphisms in the setting of locally convex topological algebras, thereby correcting a gap in our earlier paper [Trans. Moscow Math. Soc. 2008, 27-104].
In this paper, we countinue our work in \cite{11}. We show that $L^{1}(G,w)$ is $ϕ_{0}$-biprojective if and only if $G$ is compact, where $ϕ_{0}$ is the augmentation character. We introduce the notions of character Johnson amenability and character Johnson contractibility for Banach algebras. We show that $\ell^{1}(S)$ is pseudo-amenable if and only if $\ell^{1}(S)$ is character Johnson-amenable, provided that $S$ is a uniformly locally finite band semigroup. We give some conditions whether $ϕ$-biprojectivity ($ϕ$-biflatness) of $\ell^{1}(S)$ implies the finiteness (amenability) of $S$, respec…
In this paper, we investigate homological properties of Banach algebras. We show that retractions Banach algebras preserve biprojectivity, contractibility and biflatness. We also prove that contractibility of second dual of a Banach algebra implies contractibility of the Banach algebra. For a Banach algebra $A$ with $Δ(A)\neq\emptyset$, let $\frak{F}(X, A)$ be one of the Banach algebras $C_b(X, A)$, $C_0(X, A)$, $\hbox{Lip}_α(X, A)$ or $\hbox{lip}_α(X, A)$. In the following, we study homological properties of Banach algebra $\frak{F}(X, A)$, especially contractibility of it. We prove that cont…
Homological dimensions of algebras of analytic functionals and their completions
本文证明连通复李群上解析泛函代数及其某些完备化的主要同调维数等于该群线性化典范分解中单连通可解因子的维数，线性复约化因子不影响这些代数的同调性质。
We study general Hilbert modules over the disc algebra and exhibit necessary spectral conditions for the vanishing of certain associated extension groups. In particular, this sheds some light on the problem of identifying the projective Hilbert modules. Part of our work also addresses the classical derivation problem.
Given work contains the full text of the proof of the following assertion: For the topological algebra $C^{\infty}(\mathcal{M})$ of smooth functions on a smooth $m$-dimensional real manifold $\mathcal{M}$ the small global dimension $(\mathop{\mathrm{ds}} C^\infty (\mathcal{M}))$, the global homological dimension $(\mathop{\mathrm{dg}} C^\infty (\mathcal{M}))$ and the bidimension $(\mathop{\mathrm{db}} C^\infty(\mathcal{M}))$ are equal to $m$ (all dimensions are understood in the sense of the homology of topological (locally convex) algebras).
Grandis's non-abelian homological algebra generalizes standard homological algebra in abelian categories to \textit{homological categories}, which are a broader class of categories including for example the category of lattices and Galois connections. Here, we prove that if $\mathsf{C}$ is any category with an ideal of null morphisms with respect to which (co)kernels exist, then the arrow category of $\mathsf{C}$ is a homological category. This broadens the applicability of Grandis's framework substantially. In particular, one can form the homology of chain complexes in $\mathsf{C}$ by taking …
Homological algebra for nonarchimedean completions of group algebras
本文证明了群代数到其 dagger 完备化的典范态射是等上同调态射，从而将完备化代数的同调性质归约为较简单的群代数，并应用于 Hochschild 与循环同调计算。
We survey some results on homological dimensions of the algebraic, complex analytic, and smooth quantum tori. Our main theorem states, in particular, that the smooth and the complex analytic quantum n-tori have global dimension n. This contrasts with the result of McConnell and Pettit (1988) who proved that, in the generic case, the algebraic quantum n-torus has global dimension 1. In this connection we also formulate some general theorems on homological dimensions of nuclear Fréchet algebras.
The Arens-Michael envelope of a solvable Lie algebra is a homological epimorphism
本文证明了有限维复李代数的universal enveloping algebra的Arens-Michael包当且仅当该李代数为可解时是同调上模，填补了Pirkovskii之前证明的必要条件。
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