arXiv subjects

Nico Spronk

Publications and source records attributed to Nico Spronk.

46 records · Page 3Linked to original sources

arXiv2007

Operator space structure on Feichtinger's Segal algebra

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger's remarkable Segal algebra S_0(G). In order to obtain functorial properties for non-abelain groups, in particular a tensor product formula, we endow S_0(G) with an operator space structure. With this structure S_0(G) is simultaneously an operator Segal algebra of the Fourier algebra A(G), and of the group algebra L^1(G). We show that this operator space structure is consistent with the major functorial properties: (i) S_0(G)\hat{\otimes}S_0(H)=S_0(G\times H) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map u\mapsto u|_H:S_0(G)\to S_0(H) is completely surjective, if H is a closed subgroup; and (iii) T_N:S_0(G)\to S_0(G/N) is completely surjective, where N is a normal subgroup and T_N u(sN)=\int_N u(sn)dn. We also show that S_0(G) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra.

math.FA↗

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

Operator Segal Algebras in Fourier Algebras

Let G be a locally compact group, A(G) its Fourier algebra and L1(G) the space of Haar integrable functions on G. We study the Segal algebra SA(G)=A(G)\cap L1(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of SA(G). We use it show that restriction operator u|->u|H:SA(G)->A(H), for some non-open closed subgroups H, is a surjective complete qutient map. We also show that if N is a non-compact closed subgroup, then the averaging operator tau_N:SA(G)->L1(G/N), tau_N u(sN)=\int_N u(sn)dn is a surjective complete quotient map. This puts an operator space perspective on the philosophy that SA(G) is ``locally A(G) while globally L1''. Also, using the operator space structure we can show that SA(G) is operator amenable exactly when when G is compact; and we can show that it is always operator weakly amenable. To obtain the latter fact, we use E. Samei's theory of hyper-Tauberian Banach algebras.

math.FA↗

arXiv2005

The spine of a Fourier-Stieltjes algebra

We define the spine A*(G) of the Fourier-Stieltjes algebra B(G) of a locally compact group G. A*(G) is graded over a certain semi-lattice, that of non-quotient locally precompact topologies on G. We compute the spine's spectrum G*, which admits a semi-group structure. We discuss homomorphisms from A*(G) to B(H) where H is another locally compact group; and we show that A*(G) contains the image of every completely bounded homomorphism from the Fourier algebra A(H) of any amenable group H. We also show that A*(G) contains all of the idempotents in B(G). Finally, we compute examples for vector groups, abelian lattices, minimally almost periodic groups and the ax+b-group; and we explore the complexity of A*(G) for the discrete rational numbers and free groups.

math.FA↗

arXiv2005

The algebra generated by idempotents in a Fourier-Stieltjes algebra

We study the closed algebra B_I(G) generated by the idempotents in the Fourier-Stieltjes algebra of a locally compact group G. We show that it is a regular Banach algebra with computable spectrum G^I, which we call the idempotent compactification of G. For any locally compact groups G and H, we show that B_I(G) is completely isometrically isomorphic to B_I(H) exactly when G/G_e= H/H_e, where G_e and H_e are the connected components of the identities. We compute some examples to illustrate out results.

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↗

arXiv2004

Measurable Schur Multipliers and Completely Bounded Multipliers of the Fourier Algebras

Let G be a locally compact group L^p(G) be the usual L^p-space for 1 =< p =< infty and A(G) be the Fourier algebra of G. Our goal is to study, in a new abstract context, the completely bounded multipliers of A(G), which we denote M_{cb}A(G). We show that M_{cb}A(G) can be characterised as the ``invariant part'' of the space of (completely) bounded normal L^{infty}(G)-bimodule maps on B(L^2(G)), the space of bounded operator on L^2(G). In doing this we develop a function theoretic description of the normal L^{infty}(X,mu)-bimodule maps on B(L^2(X,mu)), which we denote by V^{infty}(X,μ), and name the measurable Schur multipliers of (X,mu). Our approach leads to many new results, some of which generalise results hitherto known only for certain classes of groups. Those results which we develop here are a uniform approach to the functorial properties of M_{cb}A(G), and a concrete description of a standard predual of M_{cb}A(G).

math.FA↗

arXiv2004

Representations of Group Algebras in Spaces of Completely Bounded Maps

Let G be a locally compact group, M(G) denote its measure algebra and L^1(G) denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitary representation, and let CB^{sigma}(B(H)) be the space of normal completely bounded maps on B(H). We study the range of the map Gamma_pi:M(G)->CB^sigma(B(H)), Gamma_pi(mu)= int_G pi(s)\otimes pi(s)^*dmu(s) where we identify CB^sigma(B(H)) with the extended Haagerup tensor product B(H)\otimes^{eh}B(H)$. We use the fact that the C*-algebra generated by integrating pi to L^1(G) is unital exactly when pi is norm continuous to show that Gamma_pi(L^1(G))\subset B(H)\otimes^{eh}B(H) exactly when pi is norm continuous. For the case that G is abelian, we study Gamma_pi(M(G)) as a subset of the Varopoulos algebra. We also characterise positive definite elements of the Varopoulos algebra in terms of completely positive operators.

math.FA↗

arXiv2003

Operator biflatness of the Fourier algebra and approximate indicators for subgroups

We investigate if, for a locally compact group $G$, the Fourier algebra $A(G)$ is biflat in the sense of quantized Banach homology. A central role in our investigation is played by the notion of an approximate indicator of a closed subgroup of $G$: The Fourier algebra is operator biflat whenever the diagonal in $G \times G$ has an approximate indicator. Although we have been unable to settle the question of whether $A(G)$ is always operator biflat, we show that, for $G = SL(3,C)$, the diagonal in $G \times G$ fails to have an approximate indicator.

math.FA↗

arXiv2003

Operator amenability of Fourier-Stieltjes algebras

The notion of operator amenability was introduced by Z.-J. Ruan in 1995. He showed that a locally compact group G is amenable if and only if its Fourier algebra A(G) is operator amenable. In this paper, we investigate the operator amenability of the Fourier-Stieltjes algebra B(G) and of the reduced Fourier-Stieltjes algebra B_r(G). The natural conjecture is that any of these algebras is operator amenable if and only if G is compact. We partially prove this conjecture with mere operator amenability replaced by operator C-amenability for some constant C < 5. In the process, we obtain a new decomposition of B(G), which can be interpreted as the non-commutative counterpart of the decomposition of M(G) into the discrete and the continuous measures. We further introduce a variant of operator amenability - called operator Connes-amenability - which also takes the dual space structure on B(G) and B_r(G) into account. We show that B_r(G) is operator Connes-amenable if and only if G is amenable. Surprisingly, B(F_2) is operator Connes-amenable although F_2, the free group in two generators, fails to be amenable.

math.FA↗