We prove two results on some special generators of finite simple groups and use them to prove that every non-abelian finite simple group S admits a non-congruence presentation (as conjectured by Chen, Lubotzky, and Tiep (2024)), and that if S has a non-trivial Schur multiplier, then it admits a smooth cover (as conjectured by Chen, Fan, Li, and Zhu (2024)).

We study in this paper analytic Schur multipliers on C + 2 \mathbb {C}_{+}^{2} and D 2 \mathbb {D}^{2} , i.e. Schur multipliers on R 2 \mathbb {R}^{2} and T 2 \mathbb {T}^{2} that are boundary-value functions of functions analytic in C + 2 \mathbb {C}_{+}^{2} and D 2 \mathbb {D}^{2} . Such Schur multipliers are important when studying properties of functions of maximal dissipative operators and contractions under perturbation. We show that if the boundary-value function of a Schur multiplier has certain regularity properties, then it can be represented as an element of the Haagerup tensor product of spaces of analytic functions with similar regularity properties.

For every n-dimensional nilpotent Lie algebra L, let ℳ(L)be the Schur multiplier of L. Then we define t(L) by t(L) = 1/2(n2 – n) – dimℳ(L). We classify nilpotent Lie algebras L satisfying t(L) = 13.

Abstract We call a matrix blocky if, up to row and column permutations, it can be obtained from an identity matrix by repeatedly applying one of the following operations: duplicating a row, duplicating a column, or adding a zero row or column. Blocky matrices are precisely the boolean matrices that are contractive when considered as Schur multipliers. It is conjectured that any boolean matrix with Schur multiplier norm at most $\gamma $ is expressible as a signed sum $$ \begin{align*}&A = \sum_{i=1}^{L} \pm B_{i}\end{align*} $$ for some blocky matrices $B_{i}$, where $L$ depends only on $\gamma $. This conjecture is an analogue of Green and Sanders’ quantitative version of Cohen’s idempotent theorem. In this paper, we prove bounds on $L$ that are polylogarithmic in the dimension of $A$. Concretely, if $A$ is an $n\times n$ matrix, we show that one may take $L = 2^{O(\gamma ^{7})} \log (n)^{2}$.

Abstract We introduce the notion of isoclinism between central extensions in the category of algebras with bracket. We provide several equivalent conditions under which algebras with bracket are isoclinic. We also study the connection between isoclinism and the Schur multiplier of algebras with bracket. It is shown that for finite-dimensional central extensions of algebras with bracket with same dimension, the notion of isoclinism and isomorphism are equivalent. Furthermore, we indicate that all stem covers of an algebra with bracket are isoclinic.

The Schur multiplier is a way of viewing the second cohomology group of an algebraic structure as the kernel of a maximal stem extension. We introduce an analogue of the Schur multiplier, along with the related notion of covers, in the context of triassociative algebras. This class of algebras generalizes associative algebras and is characterized by three multiplications and 11 identities. The paper highlights an extension-theoretic intersection of multipliers, covers and unicentral triassociative algebras, and we develop criteria for when the center of the cover maps onto the center of the algebra. Along the way, we obtain the uniqueness of the cover, two different characterizations of the multiplier, and several exact cohomological sequences.

Abstract Let $ {\mathcal M}$ be a von Neumann algebra equipped with a normal semifinite faithful (nsf) trace. We say that an operator $ T:{\mathcal M} \to{\mathcal M} $ is absolutely dilatable if there exist another von Neumann algebra $ M $ with an nsf trace, a unital normal trace preserving $\ast $-homomorphism $ J: {\mathcal M} \to M $, and a trace preserving $\ast $-automorphism $ U: M \to M $ such that $T^{k} = {\mathbb E}_{J} U^{k} J \ \textrm{for all}\ k \geq 0,$ where $ {\mathbb E}_{J}: M \to{\mathcal M} $ is the conditional expectation associated with $ J $. For a discrete amenable group $ G $ and a function $ u:G\to \mathbb{C} $ inducing a unital completely positive Fourier multiplier $ M_{u}: VN(G) \to VN(G) $, we establish the following transference theorem: the operator $ M_{u} $ admits an absolute dilation if and only if its associated Herz–Schur multiplier does. From this result, we deduce a characterization of Fourier multipliers with an absolute dilation in this setting. Building on the transference result, we construct the first known example of a unital completely positive Fourier multiplier that does not admit an absolute dilation. This example arises in the symmetric group $ {\mathcal{S}}_{3} $, the smallest group where such a phenomenon occurs. Moreover, we show that for every abelian group $ G $, every Fourier multiplier always admits an absolute dilation.

Let L be an ( m | n ) -dimensional nilpotent Lie superalgebra where m + n ≥ 4 and n ≥ 1 . This paper classifies such nilpotent Lie superalgebras L with a derived subsuperalgebra L 2 of dimension m + n − 2 such that γ ( L ) = m + 2 n − 2 ‐ dim M ( L ) , where γ ( L ) ∈ { 0 , 1 , 2 } and M ( L ) denotes the Schur multiplier of L. Furthermore, we show that all these superalgebras are capable.

A Marcinkiewicz multiplier theory for Schur multipliers

On the size of the Schur multiplier of finite groups

Abstract We establish regularity conditions for ‐boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural Hörmander–Mikhlin (HM) criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard, and de la Salle for . Our approach is partly based on a sharp local HM theorem for arbitrary Lie groups, which follows in turn from recent estimates by the authors on singular non‐Toeplitz Schur multipliers. We generalize the latter to arbitrary locally compact groups and refine the cocycle‐based approach to Fourier multipliers in group algebras by Junge, Mei, and Parcet. A few related open problems are also discussed.

We give a new proof of the boundedness of bilinear Schur multipliers of second order divided difference functions, as obtained earlier by Potapov, Skripka and Sukochev in their proof of Koplienko’s conjecture on the existence of higher order spectral shift functions. Our proof is based on recent methods involving bilinear transference and the Hörmander–Mikhlin–Schur multiplier theorem. Our approach provides a significant sharpening of the known asymptotic bounds of bilinear Schur multipliers of second order divided difference functions. Furthermore, we give a new lower bound of these bilinear Schur multipliers, giving again a fundamental improvement on the best known bounds obtained by Coine, Le Merdy, Potapov, Sukochev and Tomskova. More precisely, we prove that for f∈C2(R) and 1<p,p1,p2<∞ with 1p=1p1+1p2 we have ‖Mf[2]:Sp1×Sp2→Sp‖≲‖f′′‖∞D(p,p1,p2),where the constant D(p,p1,p2) is specified in Theorem 7.1 and D(p,2p,2p)≈p4p∗ with p∗ the Hölder conjugate of p. We further show that for f(λ)=λ|λ|,λ∈R, for every 1<p<∞ we have p2p∗≲‖Mf[2]:S2p×S2p→Sp‖.Here f[2] is the second order divided difference function of f with Mf[2] the associated Schur multiplier. In particular it follows that our estimate D(p, 2p, 2p) is optimal for p↘1.

The Schur Lie -multiplier of Leibniz algebras is the Schur multiplier of Leibniz algebras defined relative to the Liezation functor. In this paper, we study upper bounds for the dimension of the Schur Lie -multiplier of Lie -filiform Leibniz n-algebras and the Schur Lie -multiplier of its Lie -central factor. The upper bound obtained is associated to both the sequences of central binomial coefficients and the sum of the numbers located in the rhombus part of Pascal’s triangle. Also, the pattern of counting the number of Lie -brackets of a particular Leibniz n-algebra leads us to a new property of Pascal’s triangle. Moreover, we discuss some results which improve the existing upper bound for m-dimensional Lie -nilpotent Leibniz n-algebras with d-dimensional Lie -commutator. In particular, it is shown that if q is an m-dimensional Lie -nilpotent Leibniz 2-algebra with one-dimensional Lie -commutator, then dim M Lie ( q ) ≤ 1 2 m ( m − 1 ) − 1.

ABSTRACT Presenting a finite group by a free product of finite cyclic groups the Hopf formula for the Schur multiplier also affords a covering group, and the cover has a minimal exponent, provided that the presentation preserves the orders of the generators. This condition corresponds to a covering projection between Δ-complexes, and so a presentation by a Fuchsian group generated by elliptic elements corresponds to a covering projection between compact oriented surfaces.

Abstract A Schur multiplier is a linear map on matrices which acts on its entries by multiplication with some function, called the symbol. We consider idempotent Schur multipliers, whose symbols are indicator functions of smooth Euclidean domains. Given $1&lt;p\neq 2&lt;\infty $ , we provide a local characterization (under some mild transversality condition) for the boundedness on Schatten p-classes of Schur idempotents in terms of a lax notion of boundary flatness. We prove in particular that all Schur idempotents are modeled on a single fundamental example: the triangular projection. As an application, we fully characterize the local $L_p$ -boundedness of smooth Fourier idempotents on connected Lie groups. They are all modeled on one of three fundamental examples: the classical Hilbert transform and two new examples of Hilbert transforms that we call affine and projective. Our results in this paper are vast noncommutative generalizations of Fefferman’s celebrated ball multiplier theorem. They confirm the intuition that Schur multipliers share profound similarities with Euclidean Fourier multipliers – even in the lack of a Fourier transform connection – and complete, for Lie groups, a longstanding search of Fourier $L_p$ -idempotents.

Abstract In Caspers et al. (Can. J. Math. 75[6] [2022], 1–18), transference results between multilinear Fourier and Schur multipliers on noncommutative $L_p$ -spaces were shown for unimodular groups. We propose a suitable extension of the definition of multilinear Fourier multipliers for non-unimodular groups and show that the aforementioned transference results also hold in this more general setting.

Let $1&lt;p\neq 2&lt;\infty $ and let $S^p_n$ be the associated Schatten von Neumann class over $n\times n$ matrices. We prove new characterizations of unital positive Schur multipliers $S^p_n\to S^p_n$ which can be dilated into an invertible complete isometry acting on a non-commutative $L^p$-space. Then we investigate the infinite dimensional case.

On the finiteness of the non-abelian tensor product of groups

In this paper we introduce the concept of matrix-valued q-rational functions. In comparison to the classical case, we give different characterizations with principal emphasis on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna–Pick kernels in the context of q-deformed reproducing kernel Hilbert spaces and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna–Pick kernels.

Two-sided bounds for the tracial seminorm of multilinear Schur multipliers