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Register a new userCorrigendum: Similarity degree of Fourier algebras
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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.
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We show that for a locally compact group $G$, amongst a class which contains amenable and small invariant neighbourhood groups, that its Fourier algebra $A(G)$ satisfies a completely bounded version Pisiers similarity property with similarity degree at most $2$. Specifically, any completely bounded homomorphism $pi: A(G)to B(H)$ admits an invertible $S$ in $B(H)$ for which $|S||S^{-1}|leq ||pi||_{cb}^2$ and $S^{-1}pi(cdot)S$ extends to a $*$-representation of the $C^*$-algebra $C_0(G)$. This significantly improves some results due to Brannan and Samei (J. Funct. Anal. 259, 2010) and Brannan, Daws and Samei (M{u}nster J. Math 6, 2013). 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 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 prove that if $rho: A(H) to B(G)$ is a homomorphism between the Fourier algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a locally compact group $G$ induced by a mixed piecewise affine map $alpha : G to H$, then $rho$ extends to a w*-w* continuous map between the corresponding $L^infty$ algebras if and only if $alpha$ is an open map. Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $alpha$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure of $H$, then $(alphatimesalpha)^{-1}$ preserves sets of compact operator synthesis, and conversely when $alpha$ is onto. We also prove similar preservation properties for operator Ditkin sets and operator M-sets, obtaining preservation properties for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov and Turowska.
This paper concerns the study of regular Fourier hypergroups through multipliers of their associated Fourier algebras. We establish hypergroup analogues of well-known characterizations of group amenability, introduce a notion of weak amenability for hypergroups, and show that every discrete commutative hypergroup is weakly amenable with constant 1. Using similar techniques, we provide a sufficient condition for amenability of hypergroup Fourier algebras, which, as an immediate application, answers one direction of a conjecture of Azimifard--Samei--Spronk [J. Funct. Anal. 256(5) 1544-1564, 2009] on the amenability of $ZL^1(G)$ for compact groups $G$. In the final section we consider Fourier algebras of hypergroups arising from compact quantum groups $mathbb{G}$, and in particular, establish a completely isometric isomorphism with the center of the quantum group algebra for compact $mathbb{G}$ of Kac type.
Let $G$ be a compact group. For $1leq pleqinfty$ 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 $p ot=2$ 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 weighte
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