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Volker Runde

Publications and source records attributed to Volker Runde.

At least 37 records · Page 2Linked to original sources
arXiv2008

Uniform continuity over locally compact quantum groups

We define, for a locally compact quantum group $G$ in the sense of Kustermans--Vaes, the space of $LUC(G)$ of left uniformly continuous elements in $L^\infty(G)$. This definition covers both the usual left uniformly continuous functions on a locally compact group and Granirer's uniformly continuous functionals on the Fourier algebra. We show that $LUC(G)$ is an operator system containing the $C^*$-algebra $C_0(G)$ and contained in its multiplier algebra $M(C_0(G))$. We use this to partially answer an open problem by Bedos--Tuset: if $G$ is co-amenable, then the existence of a left invariant mean on $M(C_0(G))$ is sufficient for $G$ to be amenable. Furthermore, we study the space $WAP(G)$ of weakly almost periodic elements of $L^\infty(G)$: it is a closed operator system in $L^\infty(G)$ containing $C_0(G)$ and--for co-amenable $G$--contained in $LUC(G)$. Finally, we show that--under certain conditions, which are always satisfied if $G$ is a group--the operator system $LUC(G)$ is a $C^*$-algebra.

math.OA↗
arXiv2008

Can B(l^p) ever be amenable?

It is known that ${\cal B}(\ell^p)$ is not amenable for $p =1,2,\infty$, but whether or not ${\cal B}(\ell^p)$ is amenable for $p \in (1,\infty) \setminus \{2 \}$ is an open problem. We show that, if ${\cal B}(\ell^p)$ is amenable for $p \in (1,\infty)$, then so are $\ell^\infty({\cal B}(\ell^p))$ and $\ell^\infty({\cal K}(\ell^p))$. Moreover, if $\ell^\infty({\cal K}(\ell^p))$ is amenable so is $\ell^\infty(\mathbb{I},{\cal K}(E))$ for any index set $\mathbb I$ and for any infinite-dimensional ${\cal L}^p$-space $E$; in particular, if $\ell^\infty({\cal K}(\ell^p))$ is amenable for $p \in (1,\infty)$, then so is $\ell^\infty({\cal K}(\ell^p \oplus \ell^2))$. We show that $\ell^\infty({\cal K}(\ell^p \oplus \ell^2))$ is not amenable for $p =1,\infty$, but also that our methods fail us if $p \in (1,\infty)$. Finally, for $p \in (1,2)$ and a free ultrafilter $\cal U$ over $\posints$, we exhibit a closed left ideal of $({\cal K}(\ell^p))_{\cal U}$ lacking a right approximate identity, but enjoying a certain, very weak complementation property.

math.FA↗
arXiv2006

Characterizations of compact and discrete quantum groups through second duals

A locally compact group $G$ is compact if and only if $L^1(G)$ is an ideal in $L^1(G)^{**}$, and the Fourier algebra $A(G)$ of $G$ is an ideal in $A(G)^{**}$ if and only if $G$ is discrete. On the other hand, $G$ is discrete if and only if $C_0(G)$ is an ideal in $C_0(G)^{**}$. We show that these assertions are special cases of results on locally compact quantum groups in the sense of J. Kustermans and S. Vaes. In particular, a von Neumann algebraic quantum group $(M,Γ)$ is compact if and only if $M_*$ is an ideal in $M^*$, and a (reduced) $C^*$-algebraic quantum group $(A,Γ)$ is discrete if and only if $A$ is an ideal in $A^{**}$.

math.OA↗
arXiv2006

Operator amenability of Fourier-Stieltjes algebras, II

We give an example of a non-compact, locally compact group $G$ such that its Fourier-Stieltjes algebra $B(G)$ is operator amenable. Furthermore, we characterize those $G$ for which $A^*(G)$ - the spine of $B(G)$ as introduced by M. Ilie and the second named author - is operator amenable and show that $A^*(G)$ is operator weakly amenable for each $G$.

math.FA↗
arXiv2006

Harmonic operators: the dual perspective

The study of harmonic functions on a locally compact group $G$ has recently been transferred to a ``non-commutative'' setting in two different directions: C.-H. Chu and A. T.-M. Lau replaced the algebra $L^\infty(G)$ by the group von Neumann algebra $VN(G)$ and the convolution action of a probability measure $μ$ on $L^\infty(G)$ by the canonical action of a positive definite function $σ$ on $\VN(G)$; on the other hand, W. Jaworski and the first-named author replaced $L^\infty(G)$ by $B(L^2(G))$ to which the convolution action by $μ$ can be extended in a natural way. We establish a link between both approaches. The action of $σ$ on $VN(G)$ can be extended to $B (L^2(G))$. We study the corresponding space $\tilde{H}_σ$ of ``$σ$-harmonic operators'', i.e., fixed points in $B(L^2(G))$ under the action of $σ$. We show, under mild conditions on either $σ$ or $G$, that $\tilde{H}_σ$ is in fact a von Neumann subalgebra of $B (L^2(G))$. Our investigation of $\tilde{H}_σ$ relies, in particular, on a notion of support for an arbitrary operator in $B(L^2(G))$ that extends Eymard's definition for elements of $VN(G)$. Finally, we present an approach to $\tilde{H}_σ$ via ideals in $T (L^2(G))$ - where $T(L^2(G))$ denotes the trace class operators on $L^2(G)$, but equipped with a product different from composition -, as it was pioneered for harmonic functions by G. A. Willis.

math.FA↗
arXiv2006

Cohen-Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca-Herz algebras

Let $G$ be a locally compact group, and let ${\cal R}(G)$ denote the ring of subsets of $G$ generated by the left cosets of open subsets of $G$. The Cohen--Host idempotent theorem asserts that a set lies in ${\cal R}(G)$ if and only if its indicator function is a coefficient function of a unitary representation of $G$ on some Hilbert space. We prove related results for representations of $G$ on certain Banach spaces. We apply our Cohen--Host type theorems to the study of the Figà-Talamanca--Herz algebras $A_p(G)$ with $p \in (1,\infty)$. For arbitrary $G$, we characterize those closed ideals of $A_p(G)$ that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those $G$ such that $A_p(G)$ is 1-amenable for some -- and, equivalently, for all -- $p \in (1,\infty)$: these are precisely the abelian groups.

math.FA↗
arXiv2005

Operator amenability of the Fourier algebra in the cb-multiplier norm

Let $G$ be a locally compact group, and let $A_\cb(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_\cb(G)$ is operator amenable. In particular, $A_\cb(F_2)$ is operator amenable even though $F_2$, the free group in two generators, is not an amenable group. Moreover, we show that, if $G$ is a discrete group such that $A_\cb(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the cb-multiplier norm.

math.FA↗
arXiv2005

The amenability constant of the Fourier algebra

For a locally compact group $G$, let $A(G)$ denote its Fourier algebra and $\hat{G}$ its dual object, i.e. the collection of equivalence classes of unitary represenations of $G$. We show that the amenability constant of $A(G)$ is less than or equal to $\sup \{°(π) : π\in \hat{G} \}$ and that it is equal to one if and only if $G$ is abelian.

math.FA↗
arXiv2004

Representations of locally compact groups on QSL_p-spaces and a p-analog of the Fourier-Stieltjes algebra

For a locally compact group $G$ and $p \in (1,\infty)$, we define $B_p(G)$ to be the space of all coefficient functions of isometric representations of $G$ on quotients of subspaces of $L_p$ spaces. For $p =2$, this is the usual Fourier--Stieltjes algebra. We show that $B_p(G)$ is a commutative Banach algebra that contractively (isometrically, if $G$ is amenable) contains the Figà-Talamanca--Herz algebra $A_p(G)$. If $2 \leq q \leq p$ or $p \leq q \leq 2$, we have a contractive inclusion $B_q(G) \subset B_p(G)$. We also show that $B_p(G)$ embeds contractively into the multiplier algebra of $A_p(G)$ and is a dual space. For amenable $G$, this multiplier algebra and $B_p(G)$ are isometrically isomorphic.

math.FA↗
arXiv2004

A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal

Let $G$ be a locally compact group, and let $WAP(G)$ denote the space of weakly almost periodic functions on $G$. We show that, if $G$ is a $[SIN]$-group, but not compact, then the dual Banach algebra $WAP(G)^\ast$ does not have a normal, virtual diagonal. Consequently, whenever $G$ is an amenable, non-compact $[SIN]$-group, $WAP(G)^\ast$ is an example of a Connes-amenable, dual Banach algebra without a normal,virtual diagonal.

math.FA↗
arXiv2004

Amenability and weak amenability of the Fourier algebra

Let $G$ be a locally compact group. We show that its Fourier algebra $A(G)$ is amenable if and only if $G$ has an abelian subgroup of finite index, and that its Fourier-Stieltjes algebra $B(G)$ is amenable if and only if $G$ has a compact, abelian subgroup of finite index. We then show that $A(G)$ is weakly amenable if the component of the identity of $G$ is abelian, and we prove some partial results towards the converse.

math.FA↗
arXiv2003

Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule

Let $A$ be a dual Banach algebra with predual $A_\ast$ and consider the following assertions: (A) $A$ is Connes-amenable; (B) $A$ has a normal, virtual diagonal; (C) $A_\ast$ is an injective $A$-bimodule. For general $A$, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for $A = M(G)$ where $G$ is an infinite, locally compact group. Furthermore, we present partial solutions towards a characterization of (A) and (B) for $B(G)$ in terms of $G$: For amenable, discrete $G$ as well as for certain compact $G$, they are equivalent to $G$ having an abelian subgroup of finite index. The question of whether or not (A) and (B) are always equivalent remains open. However, we introduce a modified definition of a normal, virtual diagonal and, using this modified definition, characterize the Connes-amenable, dual Banach algebras through the existence of an appropriate notion of virtual diagonal.

math.FA↗
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.

math.FA↗