arXiv subjects

Matthias Neufang

Publications and source records attributed to Matthias Neufang.

24 records · Page 2Linked to original sources

arXiv2008

Column and row operator spaces over QSL_p-spaces and their use in abstract harmonic analysis

The notions of column and row operator space were extended by A. Lambert from Hilbert spaces to general Banach spaces. In this paper, we use column and row spaces over quotients of subspaces of general $L_p$-spaces to equip several Banach algebras occurring naturally in abstract harmonic analysis with canonical, yet not obvious operator space structures that turn them into completely bounded Banach algebras. We use these operator space structures to gain new insights on those algebras.

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arXiv2006

Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres

We study multiplier algebras for a large class of Banach algebras which contains the group algebra $L_1(G)$, the Beurling algebras $L_1(G, ω)$, and the Fourier algebra $A(G)$ of a locally compact group $G$. This study yields numerous new results and unifies some existing theorems on $L_1(G)$ and $A(G)$ through an abstract Banach algebraic approach. Applications are obtained on representations of multipliers over locally compact quantum groups and on topological centre problems. In particular, five open problems in abstract harmonic analysis are solved.

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arXiv2003

Operator space structure and amenability for Figà-Talamanca-Herz algebras

Column and row operator spaces - which we denote by COL and ROW, respectively - over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group $G$ and $p,p' \in (1,\infty)$ with $\frac{1}{p} + \frac{1}{p'} = 1$, we use the operator space structure on $CB(COL(L^{p'}(G)))$ to equip the Figa-Talamanca-Herz algebra $A_p(G)$ with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for $p \leq q \leq 2$ or $2 \leq q \leq p$ and amenable $G$, the canonical inclusion $A_q(G) \subset A_p(G)$ is completely bounded (with cb-norm at most $K_G^2$, where $K_G$ is Grothendieck's constant). As an application, we show that $G$ is amenable if and only if $A_p(G)$ is operator amenable for all - and equivalently for one - $p \in (1,\infty)$; this extends a theorem by Z.-J. Ruan.

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arXiv2002

Amplification of Completely Bounded Operators and Tomiyama's Slice Maps

Let (M,N) be a pair of von Neumann algebras, or of dual operator spaces with at least one of them having property S_σ, and let T be an arbitrary completely bounded mapping on M. We present an explicit construction of an amplification of T to a completely bounded mapping on the w^*-spatial tensor product of M and N. Our approach makes essential use of the description of the predual of the latter given by Effros and Ruan in terms of the projective operator space tensor product and is further based on the concept of slice maps as introduced by Tomiyama. We discuss several properties of an amplification in connection with the investigations made by May-Neuhardt-Wittstock, where the special case M=B(H) and N=B(K) has been considered (for Hilbert spaces H and K). We will then mainly focus on various applications, such as a purely algebraic characterization of w^*-continuity using amplifications, as well as a generalization of the so-called Ge-Kadison Lemma (in connection with the uniqueness problem of amplifications). Our study will eventually enable us to show that the essential assertion of the main result obtained in the above-mentioned work of May-Neuhardt-Wittstock concerning completely bounded bimodule homomorphisms actually relies on a basic property of Tomiyama's slice maps.

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arXiv2002

A Unified Approach to the Topological Centre Problem for Certain Banach Algebras Arising in Abstract Harmonic Analysis

Let G be a locally compact group. Consider the Banach algebra L_1(G)^**, equipped with the first Arens multiplication, as well as the algebra LUC(G)^*, the dual of the space of bounded left uniformly continuous functions on G, whose product extends the convolution in the measure algebra M(G). We present (for the most interesting case of a non-compact group) completely different - in particular, direct - proofs and even obtain sharpened versions of the results, first proved by Lau-Losert and Lau, that the topological centres of the latter algebras precisely are L_1(G) and M(G), respectively. The special interest of our new approach lies in the fact that it shows a fairly general pattern of solving the topological centre problem for various kinds of Banach algebras; in particular, it avoids the use of any measure theoretical techniques. At the same time, deriving both results in perfect parallelity, our method reveals the nature of their close relation.

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arXiv2002

On the Topological Centre Problem for Weighted Convolution Algebras

Let G be a locally compact non-compact group. We show that under a very mild assumption on the weight function w, the weighted group algebra L_1(G,w) is strongly Arens irregular in the sense of Dales-Lamb-Lau. To this end, we first derive a general factorization theorem for bounded families in the L_\infty(G,1/w)^*-module L_\infty(G,1/w).

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