I. G. Todorov
Generalizing a result of Arveson on finite width CSL algebras, we prove that finite width masa-bimodules satisfy spectral synthesis. Introducing a new class of masa-bimodules, we show that there exists a non-synthetic masa-bimodule, such that the maximal algebras over which it is a bimodule, are synthetic.
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Lattices of strongly reflexive masa-bimodules
通过支撑刻画masa-双模中正秩一子空间的稠密性，证明强自反masa-双模在自然运算下构成布尔格，并研究其作为算子系统时的格论性质。
We generalize the notion of `diagonal' from the class of CSL algebras to masa bimodules. We prove that a reflexive masa bimodule decomposes as a sum of two bimodules, the diagonal and a module generalizing the w*-closure of the Jacobson radical of a CSL algebra. The latter module turns out to be reflexive, a result which is new even for CSL algebras. We show that the projection onto the direct summand contained in the diagonal is contractive and preserves compactness and reduces rank of operators. Stronger results are obtained when the module is the reflexive hull of its rank-one subspace. Key…
We define an equivalence relation between bimodules over maximal abelian selfadjoint algebras (masa bimodules) which we call spatial Morita equivalence. We prove that two reflexive masa bimodules are spatially Morita equivalent iff their (essential) bilattices are isomorphic. We also prove that if S^1, S^2 are bilattices which correspond to reflexive masa bimodules U_1, U_2 and f: S^1\rightarrow S^2 is an onto bilattice homomorphism, then: (i) If U_1 is synthetic, then U_2 is synthetic. (ii) If U_2 contains a nonzero compact (or a finite or a rank 1) operator, then U_1 also contains a nonzero …
The measure-multiplicity-invariant for masas in $\rm{II}_{1}$ factors was introduced in \cite{MR2261688} to distinguish masas that have the same Pukánszky invariant. In this paper we study the measure class in the measure-multiplicity-invariant. This is equivalent to studying the standard Hilbert space as an associated bimodule. We characterize the type of any masa depending on the left-right-measure using Baire category methods (selection principle of Jankov and von Neumann). We present a second proof of Chifan's result on normalisers and a measure theoretic proof of the equivalence of weak a…
We develop a general framework for reflexivity in dual Banach spaces, motivated by the question of when the weak* closed linear span of two reflexive masa-bimodules is automatically reflexive. We establish an affirmative answer to this question in a number of cases by examining two new classes of masa-bimodules, defined in terms of ranges of masa-bimodule projections. We give a number of corollaries of our results concerning operator and spectral synthesis, and show that the classes of masa-bimodules we study are operator synthetic if and only if they are strong operator Ditkin.
For i=1,2, let (M_i,D_i) be pairs consisting of a Cartan MASA D_i in a von Neumann algebra M_i, let atom(D_i) be the set of atoms of D_i, and let S_i be the lattice of Bures-closed D_i bimodules in M_i. We show that when M_i have separable preduals, there is a lattice isomorphism between S_1 and S_2 if and only if the sets {(Q_1, Q_2) \in atom(D_i) x atom(D_i): Q_1 M_i Q_2 \neq (0)} have the same cardinality. In particular, when D_i is non-atomic, S_i is isomorphic to the lattice of projections in L^\infty([0,1],m) where m is Lebesgue measure, regardless of the isomorphism classes of M_1 and M…
In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism between Cartan bimodule algebras A_1 and A_2 extends uniquely to a normal *-isomorphism of the von Neumann algebras generated by A_1 and A_2 [13, Corollary 4.3]. Mercer's argument relied upon the Spectral Theorem for Bimodules of Muhly, Saito and Solel [15, Theorem 2.5]. Unfortunately, the arguments in the literature supporting [15, Theorem 2.5] contain gaps, and hence Mercer's proof is incomplete. In this paper, we use the outline in [16, Remark 2.17] to give a proof of Mercer's Theorem under the additional hypothesis that…
We show that Kraus' property $S_σ$ is preserved under taking weak* closed sums with masa-bimodules of finite width, and establish an intersection formula for weak* closed spans of tensor products, one of whose terms is a masa-bimodule of finite width. We initiate the study of the question of when operator synthesis is preserved under the formation of products and prove that the union of finitely many sets of the form $κ\times λ$, where $κ$ is a set of finite width, while $λ$ is operator synthetic, is, under a necessary restriction on the sets $λ$, again operator synthetic. We show that propert…
We construct a contractive, idempotent, MASA bimodule map on B(H), whose range is not a ternary subalgebra of B(H). Our method uses a crossed-product to reduce the existence of such an idempotent map to an analogous problem about the ranges of idempotent maps that are equivariant with respect to a group action and Hamana's theory of G-injective envelopes.
In this note we extend the spectral theorem for bimodules to the higher rank graph C*-algebra context. Under the assumption that the graph is row finite and has no sources, we show that a bimodule over a natural abelian subalgebra is determined by its spectrum iff it is generated by the Cuntz-Krieger partial isometries which it contains iff the bimodule is invariant under the gauge automorphisms. We also show that the natural abelian subalgebra is a masa iff the higher rank graph satisfies an aperiodicity condition.
We study Borel homomorphisms $θ: G\rightarrow H$ for arbitrary locally compact second countable groups $G$ and $H$ for which the measure $$θ_*(μ)(α)=μ(θ^{-1}(α))\quad \text{for } \quad α\subseteq H $$ is absolutely continuous with respect to $ν,$ where $μ$ (resp. $ν$) is a Haar measure for $G,$ (resp. $H$). We define a natural mapping $\mathcal G$ from the class of maximal abelian selfadjoint algebra bimodules (masa bimodules) in $B(L^2(H))$ into the class of masa bimodules in $B(L^2(G))$ and we use it to prove that if $k\subseteq G\times G$ is a set of operator synthesis, then $(θ\times θ)^{-…
This paper is concerned with weak* closed masa-bimodules generated by A(G)-invariant subspaces of VN(G). An annihilator formula is established, which is used to characterise the weak* closed subspaces of B(L^2(G)) which are invariant under both Schur multipliers and a canonical action of M(G) on B(L^2(G)) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.
We apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph $C^*$-algebras. First we give a new proof of non-inner conjugacy of the diagonal MASA ${\mathcal D}_n$ to its non-trivial image under a quasi-free automorphism, where $E$ is a finite transitive graph. Changing graphs representing the algebras, this result applies to some non quasi-free automorphisms as well. Then we exhibit a large class of MASAs in the Cuntz algebra ${\mathcal O}_n$ that are not inner conjugate to the diagonal ${\mathcal D}_n$.
We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G)\otimes_{\rm h} A(G)$ of two copies of the Fourier algebra $A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let $E^{\sharp} = \{(s,t) : st\in E\}$ and show that if $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$ then $E$ is a set of local spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$. …
Maximal amenability of the radial subalgebra in free quantum group factors
本文证明了在N足够大时，自由量子群代数L(FO_N)的径向MASA具有极大可amenability性，通过构造量子Rădulescu基并研究双模，还得到了Pukánszky不变量的值。
For a Banach D-bimodule M over an abelian unital C*-algebra D, we define E(M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual of M. Equip E(M) with the weak-* topology. We develop general properties of E(M). It is properly viewed as a coordinate system for M when M is a subset of C, where C is a unital C*-algebra containing D as a regular MASA with the extension property; moreover, E(C) coincides with Kumjian's twist in the context of C*-diagonals. We identify the C*-envelope of a subalgebra A of a C*-diagonal which contains D. For triangular subalg…
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