arXiv subjects
Publications and source records attributed to Nico Spronk.
We define the Schur multipliers of a separable von Neumann algebra M with Cartan masa A, generalising the classical Schur multipliers of $B(\ell^2)$. We characterise these as the normal A-bimodule maps on M. If M contains a direct summand isomorphic to the hyperfinite II_1 factor, then we show that the Schur multipliers arising from the extended Haagerup tensor product $A \otimes_{eh} A$ are strictly contained in the algebra of all Schur multipliers.
We consider the minimal boundedly-translation-invariant Segal algebra $S_0^p(G)$ in the Figà-Talamanca--Herz algebra $A_p(G)$ of a locally compact group $G$. In the case that $p=2$ and $G$ is abelian this is the classical Segal algebra of Feichtinger. Hence we call this the Feichtinger--Figà-Talamanca--Herz Segal algebra of $G$. Remarkably, this space is also a Segal algebra in $L^1(G)$ and is, in fact, the minimal such algebra which is closed under pointwise multiplication by $\apg$. Even for $p=2$, this result is new for non-abelian $G$. We place a $p$-operator space structure on $S_0^p(G)$, and demonstrate the naturality of this by showing that it satisfies all natural functiorial properties: projective tensor products, restriction to subgroups and averaging over normal subgroups. However, due to complications arising within the theory of $p$-operator spaces, we are forced to work with weakly completely bounded maps in many of our results.
We study the ranges of the maps of convolution $u\otimes v\mapsto u\ast v$ and a `twisted' convolution $u\otimes v\mapsto u\ast \check{v}$ ($\check{u}(s)=u(s^{-1})$) and on the Haagerup tensor product of a Fourier algebra of a compact group $A(G)$ with itself. We compare the results to result of factoring these maps through projective and operator projective tensor products. We notice that $(A(G),\ast)$ is an operator algebra and observe an unexpected set of spectral synthesis.
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.
Let $G$ be a finitely generated group with polynomial growth, and let $\om$ be a weight, i.e. a sub-multiplicative function on $G$ with positive values. We study when the weighted group algebra $\ell^1(G,\om)$ is isomorphic to an operator algebra. We show that $\ell^1(G,\om)$ is isomorphic to an operator algebra if $\om$ is a polynomial weight with large enough degree or an exponential weight of order $0<α<1$. We will demonstrate the order of growth of $G$ plays an important role in this question. Moreover, the algebraic centre of $\ell^1(G,\om)$ 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 $G$ is the $d$-dimensional integers $\Z^d$ and 3-dimensional discrete Heisenberg group $\mathbb{H}_3(\Z)$. The case of the free group with two generators will be considered as a counter example of groups with exponential growth.
A general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators. As a consequence a one-to-one correspondence between contractive idempotents and a certain class of ternary rings of operators is established.
For a locally compact group $G$, the first-named author considered the closed subspace $a_0(G)$ which is generated by the pure positive definite functions. In many cases $a_0(G)$ is itself an algebra. We illustrate using Heisenburg groups and the $2\times 2$ real special linear group, that this is not the case in general. We examine the structures of the algebras thereby created and examine properties related to amenability.
We study, for a locally compact group $G$, the compactifications $(π,G^π)$ associated with unitary representations $π$, which we call {\it $π$-Eberlein compactifications}. We also study the Gelfand spectra $Φ_{\mathcal{A}}(π)}$ of the uniformly closed algebras $\mathcal{A}(π)$ generated by matrix coefficients of such $π$. We note that $Φ_{\mathcal{A}(π)}\cup\{0\}$ is itself a semigroup and show that the Šilov boundary of $\mathcal{A}(π)$ is $G^π$. We study containment relations of various uniformly closed algebras generated by matrix coefficients, and give a new characterisation of amenability: the constant function 1 can be uniformly approximated by matrix coefficients of representations weakly contained in the left regular representation if and only if $G$ is amenable. We show that for the universal representation $ω$, the compactification $(ω,G^ω)$ has a certain universality property: it is universal amongst all compactifications of $G$ which may be embedded as contractions on a Hilbert space, a fact which was also recently proved by Megrelishvili. We illustrate our results with examples including various abelian and compact groups, and the $ax+b$-group. In particular, we witness algebras $\fA(π)$, for certain non-self-conjugate $π$, as being generalised algebras of analytic functions.
We show that $L^\infty(μ)$, in its capacity as multiplication operators on $L^p(μ)$, is minimal as a $p$-operator space for a decomposable measure $μ$. We conclude that $L^1(μ)$ has a certain maximal type $p$-operator space structure which facilitates computations with $L^1(μ)$ and the projective tensor product.
For a locally compact group $G$ and $p \in (1,\infty)$, we define and study the Beurling-Figa-Talamanca-Herz algebras $A_p(G,ω)$. For $p=2$ and abelian $G$, these are precisely the Beurling algebras on the dual group $\hat{G}$. For $p =2$ and compact $G$, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We prove that a locally compact group $G$ is amenable if and only if one - and, equivalently, every - Beurling-Figa-Talamanca-Herz algebra $A_p(G,ω)$ has a bounded approximate identity.
Some unfortunate errors from our paper math/0505591 are corrected.
For a compact group $G$ we define the Beurling-Fourier algebra $A_ω(G)$ on $G$ for weights $ω$ defined on the dual $\what G$ and taking positive values. The classical Fourier algebra corresponds to the case $ω$ is the constant weight 1. We study the Gelfand spectrum of the algebra realizing it as a subset of the complexification $G_{\mathbb C}$ defined by McKennon and Cartwright and McMullen. In many cases, such as for polynomial weights, the spectrum is simply $G$. We discuss the questions when the algebra $A_ω(G)$ is symmetric and regular. We also obtain various results concerning spectral synthesis for $A_ω(G)$.
Let G be a locally compact group, and let A(G) and B(G) denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, L^1(G) and M(G), in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of A(G) and B(G) and compare them to such properties for L^1(G) and M(G). For us, ``amenability properties'' refers to amenability, weak amenability, and biflatness, as well as some properties which are more suited to special settings, such as the hyper-Tauberian property for semisimple commutative Banach algebras. We wish to emphasize that the theory of operator spaces and completely bounded maps plays an indispensable role when studying A(G) and B(G). We also show some applications of amenability theory to problems of complemented ideals and homomorphisms.
We give for a compact group G, a full characterisation of when its Fourier algebra A(G) is weakly amenable: when the connected component of the identity G_e is abelian. This condition is also equivalent to the hyper-Tauberian property for A(G), and to having the anti-diagonal D^v={(s,s^{-1}):s is in G} being a set of spectral synthesis for A(GXG). We show the relationship between amenability and weak amenability of A(G), and (operator) amenability and (operator) weak amenability of A_D(G), an algebra defined by the authors in arXiv:0705.4277. We close by extending our results to some classes of non-compact, locally compact groups, including small invariant neighbourhood groups and maximally weakly almost periodic groups.
In this note we study two related questions. (1) For a compact group G, what are the ranges of the convolution maps on A(GXG) given for u,v in A(G) by $u X v |-> u*v' ($v'(s)=v(s^{-1})$) and $u X v |-> u*v$? (2) For a locally compact group G and a compact subgroup K, what are the amenability properties of the Fourier algebra of the coset space A(G/K)? The algebra A(G/K) was defined and studied by the first named author. In answering the first question, we obtain for compact groups which do not admit an abelian subgroup of finite index, some new subalgebras of A(G). Using those algebras we can find many instances in which A(G/K) fails the most rudimentary amenability property: operator weak amenability. However, using different techniques, we show that if the connected component of the identity of G is abelian, then A(G/K) always satisfies the stronger property that it is hyper-Tauberian, which is a concept developed by the second named author. We also establish a criterion which characterises operator amenability of A(G/K) for a class of groups which includes the maximally almost periodic groups.
Let G be a locally compact group, and ZL1(G) be the centre of its group algebra. We show that when $G$ is compact ZL1(G) is not amenable when G is either nonabelian and connected, or is a product of infinitely many finite nonabelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL1(G).
We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, L1(G), and the Fourier algebra, A(G), of a locally compact group, G.
For any finite unital commutative idempotent semigroup S, a unital semilattice, we show how to compute the amenability constant of its semigroup algebra l^1(S), which is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We show that there is no commutative semilattice with amenability constant between 5 and 9.