A Geometric Invariant for Metabelian Pro-p Groups

Abstract. Let A and B be complete normed algebras with a unit such that B is simple and power-associative, and let be a dense-range homomorphism. We prove that if θ is irreducible (that is, , for every closed proper ideal I of A), then θ is continuous. The continuity of non-irreducible homomorphisms is also obtained provided that the set of «spectrally rare »elements in the range algebra is not dense in B. These results extend the classical Rickart's dense-range homomorphism theorem to the non-associative setting.

Using quantum differential operators, we construct a super representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) -submodule isomorphic to the regular representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) . In this way, we obtain a presentation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) by a basis together with explicit multiplication formulas of the basis elements by generators.

The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the k \mathfrak {k} -finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

Norming the Algebra of Formal Power Series in n Indeterminates

A representation of solution of stochastic differential equations

Potentials for some tensor algebras

On the generalized spectra of topological algebras

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

Recent research on Dunford-Pettis operators associated with the group algebra L 1 (G) and the Fourier algebra A(G) of a locally compact group has motivated the study of the structure of the Dunford-Pettis space for a Banach module ϵ over a Banach algebra . In this paper, we study conditions under which (ϵ) coincides with the entire Banach right -module ϵ, and investigate its connections with some classical geometric properties. We analyze the structure of Dunford-Pettis elements for various classical Banach spaces and study their behavior under continuous homomorphisms and quotient maps. Moreover, we explore the geometric properties of certain ideals in commutative Banach algebras.

The Poincar-Birkhoff-Witt Theorem deals with the structure and universal property of the universal enveloping algebra U (L) of a Lie algebra L , e.g., over R or C .K. H. Hofmann and L. Kramer (HK) [On weakly complete group algebras of Compact Groups, J. Lie Theory 30 (2020) 407-426] recently introduced the weakly complete universal enveloping algebra U(g) of a profinite-dimensional topological Lie algebra g.Here it is shown that the classical universal enveloping algebra U (|g|) of the abstract Lie algebra underlying g is a dense subalgebra of U(g) , algebraically generated by g U(g) .It is further shown that, inspite of U being a left adjoint functor, it nevertheless preserves projective limits in the form U(lim i g/i) = lim i U(g/i) , for profinite-dimensional Lie algebras g represented as projective limits of their finite-dimensional quotients.The required theory is presented in an appendix which is of independent interest.In a natural way, a weakly complete enveloping algebra U(g) is a weakly complete symmetric Hopf algebra with a Lie subalgebra P(U(g)) of primitive elements containing g (indeed properly if g = {0}), and with a nontrivial multiplicative pro-Lie group G(U(g)) of grouplike units, having P(U(g)) as its Lie algebra -in contrast with the classical Poincar-Birhoff-Witt environment of U (L) , thus providing a new aspect of Lie's Third Fundamental Theorem: Indeed a canonical pro-Lie subgroup * (g) of G(U(g)) is identified whose Lie algebra is naturally isomorphic to g.The structure of U(g) is described in detail for dim g = 1 .The primitive and grouplike components and their mutual relationship are evaluated precisely.In (HK), cited above, and in the work of R. Dahmen and K. H. Hofmann [The pro-Lie group aspect of weakly complete algebras and weakly complete group Hopf algebras, J. Lie Theory 29 (2019) 413-455] the real weakly complete group Hopf algebra R[G] of a compact group G was described.In particular, the set P(R[G])) of primitive elements of R[G] was identified as the Lie algebra g of G .It is now shown that for any compact group G with Lie algebra g there is a natural morphism of weakly complete symmetric Hopf algebras g : U(g) R[G] , implementing the identity on g and inducing a morphism of pro-Lie groups * (G) G(R[G]) = G : yet another aspect of Sophus Lie's Third Fundamental Theorem !

In this paper, we prove the following theorem: Let [Formula: see text] be an algebraic closure of a finite field of characteristic [Formula: see text]. Let [Formula: see text] be a continuous homomorphism from the absolute Galois group of [Formula: see text] to [Formula: see text]) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of [Formula: see text] is contained in the intersection of the stabilizers of the line spanned by [Formula: see text] and the plane spanned by [Formula: see text], where [Formula: see text] denotes the standard basis. Such [Formula: see text] will not satisfy a certain strict parity condition. Under the conditions that the Serre conductor of [Formula: see text] is squarefree, that the predicted weight [Formula: see text] lies in the lowest alcove, and that [Formula: see text], we prove that [Formula: see text] is attached to a Hecke eigenclass in [Formula: see text], where [Formula: see text] is a subgroup of finite index in [Formula: see text] and [Formula: see text] is an [Formula: see text]-module. The particular [Formula: see text] and [Formula: see text] are as predicted by the main conjecture of the 2002 paper of the authors and David Pollack, minus the requirement for strict parity.

Weak amenability of a weighted group algebra, or a Beurling algebra, is a long-standing open problem. The commutative case has been extensively investigated and fully characterized. We study the non-commutative case. Given a weight function $\omega$ on a locally compact group $G$, we characterize derivations from $L^1(G,\omega)$ into its dual in terms of certain functions. Then we show that for a locally compact IN group $G$, if there is a non-zero continuous group homomorphism $\varphi$: $G\to \mathbb{C}$ such that $\varphi(x)/\omega(x)\omega(x^{-1})$ is bounded on $G$, then $L^1(G,\omega)$ is not weakly amenable. Some useful criteria that rule out weak amenability of $L^1(G,\omega)$ are established. Using them we show that for many polynomial type weights the weighted Heisenberg group algebra is not weakly amenable, neither is the weighted $\boldsymbol{ax+b}$ group algebra. We further study weighted quotient group algebra $L^1(G/H,\hat\omega)$, where $\hat\omega$ is the canonical weight on $G/H$ induced by $\omega$. We reveal that the kernel of the canonical homomorphism from $L^1(G,\omega)$ to $L^1(G/H,\hat\omega)$ is complemented. This allows us to obtain some sufficient conditions under which $L^1(G/H,\hat\omega)$ inherits weak amenability of $L^1(G,\omega)$. We study further weak amenability of Beurling algebras of subgroups. In general, weak amenability of a Beurling algebra does not pass to the Beurling algebra of a subgroup. However, in some circumstances this inheritance can happen. We also give an example to show that weak amenability of both $L^1(H,\omega|_H)$ and $L^1(G/H,\hat\omega)$ does not ensure weak amenability of $L^1(G,\omega)$.

A treatment of induced transformations of measures and measurable functions is presented. Given a diagram $\varphi :G \to H$ in the category of locally compact groups and continuous proper surjective group homomorphisms, functors are produced which on objects are given by $G \to {L^2}(G),{L^1}(G)$, $M(G),W(G)$, denoting, resp., the ${L^2}$-space, ${L^1}$-algebra, measure algebra, and von Neu mann algebra generated by left regular representation of ${L^1}$ on ${L^2}$. All functors but but the second are shown to preserve projective limits; by example, the second is shown not to do so. The category of Hilbert spaces and linear transformations of norm $\leqslant 1$ is shown to have projective limits; some propositions on such limits are given. Also given is a type and factor characterization of projective limits in the category of ${W^ \ast }$-algebras and surjective normal $\ast$-algebra homomorphisms.

Geometric representations of the formal affine Hecke algebra