For a locally compact group G we introduce and study the reduced Beurling-Fourier-Stieltjes algebra, a weighted analogue of the reduced Fourier-Stieltjes algebra, together with the algebra of completely bounded multipliers of the associated weighted Fourier algebra. We show, in particular, that these two algebras coincide when G is amenable. For a general locally compact group G, we identify them as subspaces of the reduced Fourier-Stieltjes algebra and of the space of functions that locally belong to the Fourier algebra, respectively. Furthermore, we establish sufficient conditions on the group and the weight under which the algebra of completely bounded multipliers of the weighted Fourier algebra embeds into its unweighted counterpart.

Let G be a countable group and μ a probability measure on G. We build a new framework to compute asymptotic quantities associated with the μ-random walk on G, using methods from harmonic analysis on groups and Banach space theory, most notably complex interpolation. It is shown that under mild conditions, the Lyapunov exponent of the μ-random walk with respect to a weight ω on G can be computed in terms of the asymptotic behavior of the spectral radius of μ in an ascending class of weighted group algebras, and we prove that for natural choices of ω and μ, the Lyapunov exponent vanishes. Also, we show that the Avez entropy of the μ-random walk can be realized as the Lyapunov exponent of μ with respect to a suitable weight. We apply our results to stationary dynamical systems consisting of an action of a group with the property of rapid decay on a probability space. We prove that whenever the associated Koopman representation is weakly contained in the left-regular representation of the group, then the Avez entropy coincides with the Furstenberg entropy of the stationary space. This gives a characterization of (Zimmer) amenability for actions of rapid decay groups on stationary spaces. Next, by considering the spectral radius in the algebras of p-pseudofunctions on G, we introduce a new asymptotic quantity, which we call convolution entropy. We show that for groups with the property of rapid decay, the convolution entropy coincides with the Avez entropy.

We construct several new classes of bifunctors $(A,B)\mapsto A\otimes_{\alpha} B$, where $A\otimes_\alpha B$ is a cross norm completion of $A\odot B$ for each pair of C*-algebras $A$ and $B$. For the first class of bifunctors considered $(A,B)\mapsto A\otimes_p B$ ($1\leq p\leq\infty$), $A\otimes_p B$ is a Banach algebra cross-norm completion of $A\odot B$ constructed in a fashion similar to $p$-pseudofunctions of a locally compact group. We also consider $\otimes_{p,q}$ for H\"older conjugate $p,q\in [1,\infty]$ -- a Banach $*$-algebra analogue of the tensor product $\otimes_p$. By taking enveloping C*-algebras of $A\otimes_{p,q} B$, we arrive at a third bifunctor $(A,B)\mapsto A\otimes_{\mathrm C^*_{p,q}} B$ where the resulting algebra $A\otimes_{\mathrm C^*_{p,q}} B$ is a C*-algebra. For groups belonging to a large class of non-amenable discrete groups possessing both the rapid decay and Haagerup property, we show that the tensor products $\mathrm C^*_{\mathrm r}(G_1)\otimes_{\mathrm C^*_{p,q}}\mathrm C^*_{\mathrm r}(G_2)$ coincide with a Brown-Guentner type C*-completion of $\mathrm \ell^1(G_1\times G_2)$ and conclude that if $2\leq p'

We consider a new class of potentially exotic group C*algebras C*(PF*p(G)) for a locally compact group G, and its connection with the class of potentially exotic group C*-algebras C*Lp(G) introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show C*Lp(G) = C*(PF*p(G)) for p.(2, 8), and the C*-algebras C*Lp(G) are pairwise distinct for p.(2, 8) when Gbelongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of C*Lp(G) and C*(PF*p(G)). This greatly generalizes earlier results of Okayasu (see [30]) and the second author (see [40]) on the pairwise distinctness of C*Lp(G) for 2 < p <8 when Gis either a noncommutative free group or the group SL(2, R), respectively. As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group G. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem, which presents sufficient condition implying the weak containment of a cyclic unitary representation of Gin the left regular representation of G( see [14]). Also we give a near solution to a 1978 conjecture of Cowling stated in [10]. This conjecture of Cowling states if Gis a Kunze-Stein group and pis a unitary representation of Gwith cyclic vector.such that the map G s . (pi(s)xi xi) belongs to Lp(G) for some 2 < p <8, then Ap. Lp(G). We show Bp. Lp+ (G) for every > 0(recall Ap. Bp). (c) 2023 Elsevier Inc. All rights reserved.

Motivated by the recent result in Samei and Wiersma (2020, Advances in Mathematics 359, 106897) that quasi-Hermitian groups are amenable, we consider a generalization of this property on discrete groups associated to certain Roe-type algebras; we call it uniformly quasi-Hermitian. We show that the class of uniformly quasi-Hermitian groups is contained in the class of supramenable groups and includes all subexponential groups. We also show that they are invariant under quasi-isometry.

A locally compact group G is Hermitian if the spectrum SpL1(G)(f)⊆R for every f∈L1(G) satisfying f=f⁎, and quasi-Hermitian if SpL1(G)(f)⊆R for every f∈Cc(G) satisfying f=f⁎. We show that every quasi-Hermitian locally compact group is amenable. This, in particular, confirms the long-standing conjecture that every Hermitian locally compact group is amenable, a problem that has remained open since the 1960s. Our approach involves introducing the theory of “spectral interpolation of triple Banach ⁎-algebras” and applying it to a family PFp⁎(G) (1≤p≤∞) of Banach ⁎-algebras related to convolution operators that lie between L1(G) and Cr⁎(G), the reduced group C⁎-algebra of G. We show that if G is quasi-Hermitian, then PFp⁎(G) and Cr⁎(G) have the same spectral radius on Hermitian elements in Cc(G) for p∈(1,∞), and then deduce that G must be amenable. We also give an alternative proof to Jenkin's result in [23] that a discrete group containing a free sub-semigroup on two generators is not quasi-Hermitian. This, in particular, provides a dichotomy on discrete elementary amenable groups: either they are non quasi-Hermitian or they have subexponential growth. Finally, for a non-amenable group G with either rapid decay or Kunze-Stein property, we prove the stronger statement that PFp⁎(G) is not “quasi-Hermitian relative to Cc(G)” unless p=2.

Let G be a locally compact group, let Ω:G×G→C⁎ be a 2-cocycle, and let ⊛ denote the twisted convolution associated to Ω. Given a complementary pair of Young functions (Φ,Ψ), we prove that if Φ is continuous and strictly increasing, LΨ(G)⊆L2(G) and(0.1)|Ω(s,t)|≤u(s)+v(t)(s,t∈G) for some u,v∈SΨ(G), then (LΦ(G),⊛) with the maximal operator space structure is completely isomorphic to an operator algebra. We also present various classes of 2-cocycles for which one could obtain such algebras generalizing in part the results of [15]. We apply our methods to compactly generated groups of polynomial growth and demonstrate that our results could be applied to variety of cases.

Let G be a discrete group, let $$p\ge 1$$, and let $$\omega $$ be a weight on G. Using the approach from Gröchenig and Klotz (J Lond Math Soc (2) 88:49–64, 2013), we provide sufficient conditions on a weight $$\omega $$ for $$\ell ^p(G,\omega )$$ to be a Banach algebra admitting a norm-controlled inversion in $$C^*_r(G)$$. We show that our results can be applied to various cases including locally finite groups as well as finitely generated groups of polynomial or intermediate growth and a natural class of weights on them. These weights are of the form of polynomial or certain subexponential functions. We also consider the non-discrete case and study the existence of norm-controlled inversion in $$B(L^2(G))$$ for some related convolution algebras.

We address two errors made in our paper [7] . The most significant error is in Theorem 1.1. We repair this error, and show that the main result, Theorem 2.5 of [7] , is true. The second error is in one of our examples, Remark 2.4 (iv), and we partially resolve it.

Let G be a locally compact group, let Omega:GxG -> C be a 2-cocycle, and let (Phi,Psi) be a complementary pair of strictly increasing continuous Young functions. We continue our investigation in [14] of the algebraic properties of the Orlicz space L Phi(G) with respect to the twisted convolution ? coming from Omega. We show that the twisted Orlicz algebra (L Phi(G),?) posses a bounded approximate identity if and only if it is unital if and only if G is discrete. On the other hand, under suitable condition on Omega, (L Phi(G),?) becomes an Arens regular, dual Banach algebra. We also look into certain cohomological properties of (L Phi(G),?), namely amenability and Connes-amenability, and show that they rarely happen. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.

Let G be a locally compact abelian group, \(\omega :G\rightarrow (0,\infty )\) be a weight, and (\(\Phi ,\Psi \)) be a complementary pair of strictly increasing continuous Young functions. We show that for the weighted Orlicz algebra \(L^\Phi _\omega (G)\), the weak amenability is obtained under conditions similar to the ones considered in Zhang (Proc Amer Math Soc 142:1649–1661, 2014) for weighted group algebras. Our methods can be applied to various families of weighted Orlicz algebras, including weighted \(L^p\)-spaces.

We address two errors made in our paper arXiv:1511.03423. The most significant error is in Theorem 1.1. We repair this error, and show that the main result, Theorem 2.5 of arXiv:1511.03423, is true. The second error is in one of our examples, Remark 2.4 (iv), and we partially resolve it.

A locally compact group $G$ is called Hermitian if the spectrum $text{Sp}_{L^1(G)}(f)subseteqmathbb R$ for every $fin L^1(G)$ satisfying $f=f^*$, and called if $text{Sp}_{L^1(G)}(f)subseteqmathbb R$ for every $fin C_c(G)$ satisfying $f=f^*$. We show that every locally compact group is amenable. This, in particular, confirms the long-standing conjecture that every Hermitian locally compact group is amenable, a problem that has remained open since the 1960s. Our approach involves introducing the theory of interpolation of triple Banach and applying it to a family ${rm PF}_p^*(G)$ ($1leq pleq infty$) of Banach $*$-algebras related to convolution operators that lie between $L^1(G)$ and $C^*_r(G)$, the reduced group C$^*$-algebra of $G$. We show that if $G$ is quasi-Hermitian, then ${rm PF}_p^*(G)$ and $C^*_r(G)$ have the same spectral radius on Hermitian elements in for $pin (1,infty)$, and then deduce that $G$ must be amenable. We also give an alternative proof to Jenkinsu0027 result that a discrete group containing a free sub-semigroup on two generators is not quasi-Hermitian. This, in particular, provides a dichotomy on discrete elementary amenable groups: either they are non or they have subexponential growth. Finally, for a non-amenable group $G$ with either rapid decay or Kunze-Stein property, we prove the stronger statement that ${rm PF}_p^*(G)$ is not quasi-Hermitian relative to $C_c(G)$ unless $p=2$.

A locally compact group $G$ is called Hermitian if the spectrum ${\rm Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in L^1(G)$ satisfying $f=f^*$, and called if ${\rm Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in C_c(G)$ satisfying $f=f^*$. We show that every locally compact group is amenable. This, in particular, confirms the long-standing conjecture that every Hermitian locally compact group is amenable, a problem that has remained open since the 1960s. Our approach involves introducing the theory of interpolation of triple Banach and applying it to a family ${\rm PF}_p^*(G)$ ($1\leq p\leq \infty$) of Banach $*$-algebras related to convolution operators that lie between $L^1(G)$ and $C^*_r(G)$, the reduced group C$^*$-algebra of $G$. We show that if $G$ is quasi-Hermitian, then ${\rm PF}_p^*(G)$ and $C^*_r(G)$ have the same spectral radius on Hermitian elements in for $p\in (1,\infty)$, and then deduce that $G$ must be amenable. When $G$ is a non-amenable group with either rapid decay or Kunze-Stein property, we prove the stronger statement that ${\rm PF}_p^*(G)$ is not quasi-Hermitian relative to $C_c(G)$ unless $p=2$.

Let $G$ be a compact group. For $1\leq p\leq\infty$ we introduce a class of Banach function algebras $\mathrm{A}^p(G)$ on $G$ which are the Fourier algebras in the case $p=1$, and for $p=2$ are certain algebras discovered in \cite{forrestss1}. In the case $p\not=2$ we find that $\mathrm{A}^p(G)\cong \mathrm{A}^p(H)$ if and only if $G$ and $H$ are isomorphic compact groups. These algebras admit natural operator space structures, and also weighted versions, which we call $p$-Beurling-Fourier algebras. We study various amenability and operator amenability properties, Arens regularity and representability as operator algebras. For a connected Lie $G$ and $p>1$, our techniques of estimation of when certain $p$-Beurling-Fourier algebras are operator algebras rely more on the fine structure of $G$, than in the case $p=1$. We also study restrictions to subgroups. In the case that $G=SU(2)$, restrict to a torus and obtain some exotic algebras of Laurent series. We study amenability properties of these new algebras, as well.

Weighted group algebras have been studied extensively in Abstract Harmonic Analysis where complete characterizations have been found for some important properties of weighted group algebras, namely amenability and Arens regularity. One of the generalizations of weighted group algebras is weighted hypergroup algebras. Defining weighted hypergroups, analogous to weighted groups, we study Arens regularity and isomorphism to operator algebras for them. We also examine our results on three classes of discrete weighted hypergroups constructed by conjugacy classes of FC groups, the dual space of compact groups, and hypergroup structure defined by orthogonal polynomials. We observe some unexpected examples regarding Arens regularity and operator isomorphisms of weighted hypergroup algebras.

We show that for a connected Lie group G, its Fourier algebra A(G) is weakly amenable only if G is abelian. Our main new idea is to show that weak amenability of A(G) implies that the anti-diagonal, ΔˇG={(g,g−1):g∈G}, is a set of local synthesis for A(G×G). We then show that this cannot happen if G is non-abelian. We conclude for a locally compact group G, that A(G) can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group G, A(G) is weakly amenable if and only if its connected component of the identity Ge is abelian.

We show that for a locally compact group G, amongst a class which contains amenable and small invariant neighbourhood groups, its Fourier algebra A(G) satisfies a completely bounded version Pisier's similarity property with similarity degree at most 2. Specifically, any completely bounded homomorphism π:A(G)→B(H) admits an invertible S in B(H) for which ‖S‖‖S−1‖≤‖π‖cb2 and S−1π(⋅)S extends to a ⁎-representation of the C*-algebra C0(G). This significantly improves some results due to Brannan and Samei (2010) [5] and Brannan, Daws and Samei (2013) [4]. We also note that A(G) has completely bounded similarity degree 1 if and only if it is completely isomorphic to an operator algebra if and only if G is finite.

Let $G$ be a compact connected Lie group. The question of when a weighted Fourier algebra on $G$ is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on $G$ with the order of growth strictly bigger than the half of the dimension of the group. The case of SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.

Abstract Let G be a finitely generated group with polynomial growth, and let ω be a weight, i.e. a sub-multiplicative function on G with positive values. We study when the weighted group algebra ℓ1 (G, ω) is isomorphic to an operator algebra. We show that ℓ1 (G, ω) is isomorphic to an operator algebra if ω is a polynomial weight with large enough degree or an exponential weight of order 0 < α < 1. We demonstrate that the order of growth of G plays an important role in this problem. Moreover, the algebraic centre of ℓ1 (G, ω) is isomorphic to a Q-algebra, and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when G consists of the d-dimensional integers ℤd or the three-dimensional discrete Heisenberg group ℍ3(ℤ). The case of the free group with two generators is considered as a counter-example of groups with exponential growth.