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For rank 1 flat symmetric spaces, continuous orbital measures admit absolutely continuous convolution squares, except for Cartan type AI. Hence $L^1$-$L^2$ dichotomy for these spaces holds true in parallel to the compact and non-compact rank 1 symmetric spaces. We also study $L^1$-$L^2$ dichotomy for flat symmetric spaces of ranks $p=2,3$ of type AIII, i.e.\ associated with $SU(p,q)/S(U(p)\times U(q))$ where $q\geq p$. For continuous orbital measures given by regular points $L^1$-$L^2$ dichotomy holds. We study such measures given by certain singular points when $p=2$, and show that $L^1$-$L^2$ dichotomy fails. This is the first time such results are observed for any type of symmetric spaces of rank 2., Comment: 21 pages

We conduct a systematic study of traces on locally compact groups, in particular traces on their universal and reduced C*-algebras. We introduce the trace kernel, and examine its relation to the von Neumann kernel and to small-invariant neighbourhood (SIN) quotients. In doing so, we introduce the class of residually-$SIN$ groups, which contains both $SIN$ and maximally almost periodic groups. We examine in detail the trace kernel for connected groups. We study traces on reduced C*-algebras, giving a simple proof for compactly generated groups that existence of such a trace is equivalent to having an open normal amenable subgroup, and we display non-discrete groups admitting unique trace. We finish by examining amenable traces and the factorization property. We show for property (T) groups that amenable trace kernels coincide with von Neumann kernels. We show for totally disconnected groups that amenable trace separation implies the factorization property. We use amenable traces to give a simple proof that amenability of the group is equivalent to simultaneous nuclearity and possessing a trace of its reduced C*-algebra. As a final application of the results obtained in the paper, we address the embeddability of group C*-algebras into simple AF algebras. As a consequence, if a locally compact group is amenable and tracially separated (trace kernel is trivial), then its reduced C*-algebra is quasi-diagonal.

We study fixed points of contractive convolution operators associated to contractive quantum measures on locally compact quantum groups. We characterise the existence of non-zero fixed points respectively on $L^\infty(\mathbb{G})$ and on $C_0(\mathbb{G})$, and exploit these results to obtain for example the structure of the fixed points on the non-commutative $L_p$-spaces. Some consequences for the fixed points of classical convolution operators and Herz-Schur multipliers are also indicated., Comment: 26 pages, v2 contains very minor changes. The paper will appear in the Indiana University Mathematics Journal

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., Comment: 6 pages

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely $SU(n)$, the Heisenberg group $\mathbb{H}$, the reduced Heisenberg group $\mathbb{H}_r$, the Euclidean motion group $E(2)$ and its simply connected cover $\widetilde{E}(2)$. We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate "polynomially growing" weights does not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras., Comment: A few minor updates, to appear in Adv. Math.; 88 pages

We consider the Fourier-Stietljes algebra B(G) of a locally compact group G. We show that operator amenablility of B(G) implies that a certain semitolpological compactification of G admits only finitely many idempotents. In the case that G is connected, we show that operator amenability of B(G) entails that $G$ is compact., Comment: 16 pages

Let (G,tau_G) be a topological group. We establish relationships between weakly almost periodic topologies on G coarser than tau_G, central idempotents in the weakly almost periodic compactification G^W, and certain ideals in the algebra of weakly almost periodic functions W(G). We gain decompositions of weakly almost periodic representations, generalizing many from the literature. We look at the role of pre-locally compact topologies, unitarizable topologies, and extend or decompositions to Fourier-Stieltjes algebras B(G)., Comment: 31 pages, V2 has some corrected mistakes and typos, and a new example added to Section 6.7

Let $G$ be a locally compact group. It is not always the case that its reduced C*-algebra $C^*_r(G)$ admits a tracial state. We exhibit closely related necessary and sufficient conditions for the existence of such. We gain a complete answer when $G$ compactly generated. In particular for $G$ almost connected, or more generally when $C^*_r(G)$ is nuclear, the existence of a trace is equivalent to amenability. We exhibit two examples of classes of totally disconnected groups for which $C^*_r(G)$ does not admit a tracial state., Comment: The proof of Remark 2.6 (ii) has been corrected, and a typo has been fixed

We outline a simple proof of Hulanicki's theorem, that a locally compact group is amenable if and only if the left regular representation weakly contains all unitary representations. This combines some elements of the literature which have not appeared together, before., Comment: 4 pages

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 Pisier's 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., Comment: 14 pages; Final version to appear in JFA

We consider the issue of describing all self-adjoint idempotents (projections) in $L^1(G)$ when $G$ is a unimodular locally compact group. The approach is to take advantage of known facts concerning subspaces of the Fourier-Stieltjes and Fourier algebras of $G$ and the topology of the dual space of $G$. We obtain an explicit description of any projection in $L^1(G)$ which happens to also lie in the coefficient space of a finite direct sum of irreducible representations. This leads to a complete description of all projections in $L^1(G)$ for $G$ belonging to a class of groups that includes $SL(2,R)$ and all almost connected nilpotent locally compact groups., Comment: 13 pages

We determine when contractive idempotents in the measure algebra of a locally compact group commute. We consider a dynamical version of the same result. We also look at some properties of groups of measures whose identity is a contactive idempotent., Comment: 13 pages

We show that for a connected Lie group $G$, its Fourier algebra $A(G)$ is weakly amenable only if $G$ is abelian. Our main new idea is to show that weak amenability of $A(G)$ implies that the anti-diagonal, $\check{\Delta}_G=\{(g,g^{-1}):g\in G\}$, is a set of local synthesis for $A(G\times G)$. We then show that this cannot happen if $G$ is non-abelian. We conclude for a locally compact group $G$, that $A(G)$ can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group $G$, $A(G)$ is weakly amenable if and only if its connected component of the identity $G_e$ is abelian., Comment: The final version to appear in Adv. Math.;Proposition 3.4 and related comments from the previous version have been deleted; 25 pages

Let $G$ be a compact group. For $1\leq p\leq\infty$ 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 $p\not=2$ 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 weighted versions, which we call $p$-Beurling-Fourier algebras. We study various amenability and operator amenability properties, Arens regularity and representability as operator algebras. For a connected Lie $G$ and $p>1$, our techniques of estimation of when certain $p$-Beurling-Fourier algebras are operator algebras rely more on the fine structure of $G$, than in the case $p=1$. We also study restrictions to subgroups. In the case that $G=SU(2)$, restrict to a torus and obtain some exotic algebras of Laurent series. We study amenability properties of these new algebras, as well., Comment: Introduction has been rewritten, 41 pages

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 let the central Fourier algebra, ZA(G), be the subalgebra of functions u in the Fourier algebra A(G) of a compact group, for which u(xyx^{-1})=u(y) for all x,y in G. We show that this algebra admits bounded point derivations whenever G contains a non-abelian closed connected subgroup. Conversely when G is virtually abelian, then ZA(G) is amenable. Furthermore, for virtually abelian G, we establish which closed ideals admit bounded approximate identities. We also show that if ZA(G) is weakly amenable, even hyper-Tauberian, exactly when G admits no non-abelian connected subgroup. We also study the amenability constant of ZA(G) for finite G and exhibit totally disconnected groups G for which ZA(G) is non-amenable., Comment: 21 pages, mildy expository improvements made