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Register a new userIdeals of the Fourier algebra, supports and harmonic operators
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We examine the common null spaces of families of Herz-Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in J. Funct. Anal. 266 (2014), 6473-6500 can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.
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In a recent paper on exotic crossed products, we included a lemma concerning ideals of the Fourier-Stieltjes algebra. Buss, Echterhoff, and Willett have pointed out to us that our proof of this lemma contains an error. In fact, it remains an open question whether the lemma is true as stated. In this note we indicate how to contain the resulting damage. Our investigation of the above question leads us to define two properties emph{ordered} and emph{weakly ordered} for invariant ideals of Fourier-Stieltjes algebras, and we initiate a study of these properties.
Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand-Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group G of Kac type implies *-regularity of the Fourier algebra A(G), that is every closed ideal of C(G) has a dense intersection with A(G). In particular, A(G) has a unique C*-norm.
The Kubo-Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras it is impossible to avoid unbounded operators. In this article we try to extend a notion of connections to cover various classes of positive unbounded operators (or unbounded objects such as positive forms and weights) appearing naturally in the setting of von Neumann algebras, and we must keep all the expected properties maintained. This generalization is carried out for the following classes: (i) positive $tau$-measurable operators (affiliated with a semi-finite von Neumann algebra equipped with a trace $tau$), (ii) positive elements in Haagerups $L^p$-spaces, (iii) semi-finite normal weights on a von Neumann algebra. Investigation on these generalizations requires some analysis (such as certain upper semi-continuity) on decreasing sequences in various classes. Several results in this direction are proved, which may be of independent interest. Ando studied Lebesgue decomposition for positive bounded operators by making use of parallel sums. Here, such decomposition is obtained in the setting of non-commutative (Hilsum) $L^p$-spaces.
Starting with a left ideal $J$ of $L^1(G)$ we consider its annihilator $J^{perp}$ in $L^{infty}(G)$ and the generated ${rm VN}(G)$-bimodule in $mathcal{B}(L^2(G))$, ${rm Bim}(J^{perp})$. We prove that ${rm Bim}(J^{perp})=({rm Ran} J)^{perp}$ when $G$ is weakly amenable discrete, compact or abelian, where ${rm Ran} J$ is a suitable saturation of $J$ in the trace class. We define jointly harmonic functions and jointly harmonic operators and show that, for these classes of groups, the space of jointly harmonic operators is the ${rm VN}(G)$-bimodule generated by the space of jointly harmonic functions. Using this, we give a proof of the following result of Izumi and Jaworski - Neufang: the non-commutative Poisson boundary is isomorphic to the crossed product of the space of harmonic functions by $G$.
Let ${cal Z}$ be the Jiang-Su algebra and ${cal K}$ the C*-algebra of compact operators on an infinite dimensional separable Hilbert space. We prove that the corona algebra $M({cal Z}otimes {cal K})/{cal Z}otimes {cal K}$ has real rank zero. We actually prove a more general result.
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