arXiv subjects
Publications and source records attributed to Ebrahim Samei.
We provide necessary and sufficient characterizations of the existence of an amenable trace on a C$^*$-algebra in terms of the joint free numerical radius of tuples of unitaries, isometries, and partial isometries in the algebra. We apply these results to obtain new obstructions to various lifting properties.
For a locally compact group G we introduce and study the reduced Beurling-Fourier-Stieltjes algebra, a weighted analogue of the reduced Fourier-Stieltjes algebra, together with the algebra of completely bounded multipliers of the associated weighted Fourier algebra. We show, in particular, that these two algebras coincide when G is amenable. For a general locally compact group G, we identify them as subspaces of the reduced Fourier-Stieltjes algebra and of the space of functions that locally belong to the Fourier algebra, respectively. Furthermore, we establish sufficient conditions on the group and the weight under which the algebra of completely bounded multipliers of the weighted Fourier algebra embeds into its unweighted counterpart.
We show that the $\ell^p$-pseudofunctions, which were recently shown to lead to exotic completions of group $C^*$-algebras by Wiersma and the second named author, can be used to construct well-behaved crossed product functors in the sense of Buss, Echterhoff and Willett. The construction proceeds via introducing certain Banach algebras, related to operators acting on Hilbert valued $\ell^p$-spaces, which a priori depend on the choice of a Hilbert space representation of the underlying C*-algebra. We prove that, in fact, the resulting algebras are isomorphic (with the isomorphism constant depending only on $p$), and hence their C*-envelopes are isometrically isomorphic. This, in particular, means that the construction genuinely generalises the one studied earlier in the group case. The tools we develop allow us to show that for certain non-amenable actions, the resulting crossed product completions must indeed be exotic.
Let $G$ be a countable group and $\mu$ a probability measure on $G$. We build a new framework to compute asymptotic quantities associated with the $\mu$-random walk on $G$, using methods from harmonic analysis on groups and Banach space theory, most notably complex interpolation. It is shown that under mild conditions, the Lyapunov exponent of the $\mu$-random walk with respect to a weight $\omega$ on $G$ can be computed in terms of the asymptotic behavior of the spectral radius of $\mu$ in an ascending class of weighted group algebras, and we prove that for natural choices of $\omega$ and $\mu$, the Lyapunov exponent vanishes. Also, we show that the Avez entropy of the $\mu$-random walk can be realized as the Lyapunov exponent of $\mu$ with respect to a suitable weight. We apply our results to stationary dynamical systems consisting of an action of a group with the property of rapid decay on a probability space. We prove that whenever the associated Koopman representation is weakly contained in the left-regular representation of the group, then the Avez entropy coincides with the Furstenberg entropy of the stationary space. This gives a characterization of (Zimmer) amenability for actions of rapid decay groups on stationary spaces. Next, by considering the spectral radius in the algebras of $p$-pseudofunctions on $G$, we introduce a new asymptotic quantity, which we call convolution entropy. We show that for groups with the property of rapid decay, the convolution entropy coincides with the Avez entropy.
We construct several new classes of bifunctors $(A,B)\mapsto A\otimes_{\alpha} B$, where $A\otimes_\alpha B$ is a cross norm completion of $A\odot B$ for each pair of C*-algebras $A$ and $B$. For the first class of bifunctors considered $(A,B)\mapsto A\otimes_p B$ ($1\leq p\leq\infty$), $A\otimes_p B$ is a Banach algebra cross-norm completion of $A\odot B$ constructed in a fashion similar to $p$-pseudofunctions of a locally compact group. We also consider $\otimes_{p,q}$ for H\"older conjugate $p,q\in [1,\infty]$ -- a Banach $*$-algebra analogue of the tensor product $\otimes_p$. By taking enveloping C*-algebras of $A\otimes_{p,q} B$, we arrive at a third bifunctor $(A,B)\mapsto A\otimes_{\mathrm C^*_{p,q}} B$ where the resulting algebra $A\otimes_{\mathrm C^*_{p,q}} B$ is a C*-algebra. For groups belonging to a large class of non-amenable discrete groups possessing both the rapid decay and Haagerup property, we show that the tensor products $\mathrm C^*_{\mathrm r}(G_1)\otimes_{\mathrm C^*_{p,q}}\mathrm C^*_{\mathrm r}(G_2)$ coincide with a Brown-Guentner type C*-completion of $\mathrm \ell^1(G_1\times G_2)$ and conclude that if $2\leq p'<p\leq\infty$, then the canonical quotient map $\mathrm C^*_{\mathrm r}(G)\otimes_{\mathrm C^*_{p,q}}\mathrm C^*_{\mathrm r}(G)\to \mathrm C^*_{\mathrm r}(G)\otimes_{\mathrm C^*_{p',q'}}\mathrm C^*_{\mathrm r}(G)$ is not injective. A Banach $*$-algebra $A$ is \emph{rigidly symmetric} if $A\otimes_{\gamma} B$ is symmetric for every C*-algebra $B$. A theorem of Kugler asserts that every type I C*-algebra is rigidly symmetric. Leveraging our new constructions, we establish the converse of Kugler's theorem by showing for C*-algebras $A$ and $B$ that $A\otimes_{\gamma}B$ is symmetric if and only if $A$ or $B$ is type I.
We consider a new class of potentially exotic group C*-algebras $C^*_{PF_p^*}(G)$ for a locally compact group $G$, and its connection with the class of potentially exotic group C*-algebras $C^*_{L^p}(G)$ introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show $C^*_{L^p}(G)=C^*_{PF_p^*}(G)$ for $p\in (2,\infty)$, and the C*-algebras $C^*_{L^p}(G)$ are pairwise distinct for $p\in (2,\infty)$ when $G$ belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of $C^*_{L^p}(G)$ and $C^*_{PF_p^*}(G)$. This greatly generalizes earlier results of Okayasu and the second author on the pairwise distinctness of $C^*_{L^p}(G)$ for $2 0$ (recall $A_\pi\subseteq B_\pi$).
Let $G$ be a discrete group, let $p\ge1$, and let $ω$ be a weight on $G$. Using the approach from [9], we provide sufficient conditions on a weight $ω$ for $\ell^p(G,ω)$ to be a Banach algebra admitting a norm-controlled inversion in the reduced C$^*$-algebra of $G$, namely $C^*_r(G)$. We show that our results can be applied to various cases including locally finite groups as well as finitely generated groups of polynomial or intermediate growth and a natural class of weights on them. These weights are of the form of polynomial or certain subexponential functions. We also consider the non-discrete case and study the existence of norm-controlled inversion in $B(L^2(G))$ for some related convolution algebras.
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.
A locally compact group $G$ is called Hermitian if the spectrum $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in L^1(G)$ satisfying $f=f^*$, and called quasi-Hermitian if $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in C_c(G)$ satisfying $f=f^*$. We show that every quasi-Hermitian locally compact group is amenable. This, in particular, confirms the long-standing conjecture that every Hermitian locally compact group is amenable, a problem that has remained open since the 1960s. Our approach involves introducing the theory of "spectral interpolation of triple Banach $*$-algebras" and applying it to a family ${\rm PF}_p^*(G)$ ($1\leq p\leq \infty$) of Banach $*$-algebras related to convolution operators that lie between $L^1(G)$ and $C^*_r(G)$, the reduced group C$^*$-algebra of $G$. We show that if $G$ is quasi-Hermitian, then ${\rm PF}_p^*(G)$ and $C^*_r(G)$ have the same spectral radius on Hermitian elements in $C_c(G)$ for $p\in (1,\infty)$, and then deduce that $G$ must be amenable. We also give an alternative proof to Jenkins' result that a discrete group containing a free sub-semigroup on two generators is not quasi-Hermitian. This, in particular, provides a dichotomy on discrete elementary amenable groups: either they are non quasi-Hermitian or they have subexponential growth. Finally, for a non-amenable group $G$ with either rapid decay or Kunze-Stein property, we prove the stronger statement that ${\rm PF}_p^*(G)$ is not "quasi-Hermitian relative to $C_c(G)$" unless $p=2$.
Let G be a locally compact group, let $Ω:G\times G\to \mathbb{C}$ be a 2-cocycle, and let ($Φ$,$Ψ$) be a complementary pair of strictly increasing continuous Young functions. It is shown in \cite{OS2} that $(L^Φ(G),\circledast)$ becomes an Arens regular dual Banach algebra if \begin{align}\label{Eq:2-cocycle bdd sum-abstract} |Ω(s,t)|\leq u(s)+v(t) \ \ \ (s,t\in G) \end{align} for some $u,v\in \mathcal{S}^Ψ(G)$. We prove if $L^Φ(G)\subseteq L^2(G)$ and $u,v$ can be chosen to belong to $L^2(G)$, then $(L^Φ(G),\circledast)$ with the maximal operator space structure is completely isomorphic to an operator algebra. We also present further classes of 2-cocycles for which one could obtain such algebras generalizing in part the results of \cite{OS1}. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.
We introduced the concept of strong property $(\mathbb{B})$ with a constant for Banach algebras and, by applying certain analysis on the Fourier algebra of a unit circle, we show that all C$^*$-algebras and group algebras have the strong property $(\mathbb{B})$ with a constant given by $288π(1+\sqrt{2})$. We then use this result to find a concrete upper bound for the hyperreflexivity constant of $\mathcal{C}^n(A,X)$, the space of bounded $n$-cocycles from $A$ into $X$, where $A$ is a C$^*$-algebra or the group algebra of a group with an open subgroup of polynomial growth and $X$ is a Banach $A$-bimodule for which $\mathcal{H}{n+1}(A,X)$ is a Banach space. As another application, we show that for a locally compact amenable group $G$ and $1<p<\infty$, the space $CV_P(G)$ of convolution operators on $L^p(G)$ are hyperreflexive with a constant given by $288π(1+\sqrt{2})$. This is the generalization of a well-known result of E. Christiensen for $p=2$.
Let G be a locally compact abelian group, $ω:G\to (0,\infty)$ be a weight, and ($Φ$,$Ψ$) be a complementary pair of strictly increasing continuous Young functions. We show that for the weighted Orlicz algebra $L^Φ_ω(G)$, the weak amenability is obtained under conditions similar to the one considered by Y. Zhang for weighted group algebras. Our methods can be applied to various families of weighted Orlicz algebras, including weighted $L^p$-spaces.
Let G be a locally compact group, let $Ω:G\times G\to \mathbb{C}^*$ be a 2-cocycle, and let $Φ$ be a Young function. In this paper, we consider the Orlicz space $L^Φ(G)$ and investigate its algebraic property under the twisted convolution $\circledast$ coming from $Ω$. We find sufficient conditions under which $(L^Φ(G),\circledast)$ becomes a Banach algebra or a Banach $*$-algebra; we call it a {\it twisted Orlicz algebra}. Furthermore, we study its harmonic analysis properties, such as symmetry, existence of functional calculus, regularity, and having Wiener property, mostly for the case when $G$ is a compactly generated group of polynomial growth. We apply our methods to several important classes of polynomial as well as subexponential weights and demonstrate that our results could be applied to variety of cases.
Let G be a locally compact group, let $Ω:G\times G\to \mathbb{C}^*$ be a 2-cocycle, and let ($Φ$,$Ψ$) be a complementary pair of strictly increasing continuous Young functions. We continue our investigation of the algebraic properties of the Orlicz space $L^Φ(G)$ with respect to the twisted convolution $\circledast$ coming from $Ω$. We show that the twisted Orlicz algebra $(L^Φ(G),\circledast)$ posses a bounded approximate identity if and only if it is unital if and only if $G$ is discrete. On the other hand, under suitable condition on $Ω$, $(L^Φ(G),\circledast)$ becomes an Arens regular, dual Banach algebra. We also look into certain cohomological properties of $(L^Φ(G),\circledast)$, namely amenability and Connes-amenability, and show that they rarely happen. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.
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 $π: A(G)\to B(H)$ admits an invertible $S$ in $B(H)$ for which $\|S\|\|S^{-1}\|\leq ||π||_{cb}^2$ and $S^{-1}π(\cdot)S$ 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 (Münster 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.
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, $\checkΔ_G=\{(g,g^{-1}):g\in G\}$, 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.
Weighted group algebras have been studied extensively in Abstract Harmonic Analysis where complete characterizations have been found for some important properties of weighted group algebras, namely amenability and Arens regularity. One of the generalizations of weighted group algebras is weighted hypergroup algebras. Defining weighted hypergroups, analogous to weighted groups, we study Arens regularity and isomorphism to operator algebras for them. We also examine our results on three classes of discrete weighted hypergroups constructed by conjugacy classes of FC groups, the dual space of compact groups, and hypergroup structure defined by orthogonal polynomials. We observe some unexpected examples regarding Arens regularity and operator isomorphisms of weighted hypergroup algebras.
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.