We show that there exists a separable, nuclear C*-algebra with real rank zero and trivial K-theory such that its multiplier and corona algebra have real rank one. This disproves two conjectures of Brown and Pedersen. We also compute the real rank of the stable multiplier algebra and the stable corona algebra of countably decomposable type $\mathrm{I}_\infty$ and type $\mathrm{II}_\infty$ factors. Together with results of Zhang this completes the computation of the real rank for stable multiplier and corona algebras of countably decomposable factors.
Main result: If a C*-algebra is simple, $σ$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier also has strict comparison of positive elements by traces. The same results holds if "finitely many extremal traces" is replaced by "quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $σ$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If the algebra is a simple separable stable $σ$-unital with real rank z…
We answer a question of Takesaki by showing that the following can be derived from the thesis of N-T Shen: If A and B are sigma-unital hereditary C*-subalgebras of C such that ||p - q|| < 1, where p and q are the corresponding open projections, then A and B are isomorphic. We give some further elaborations and counterexamples with regard to the sigma-unitality hypothesis. We produce a natural one-to-one correspondence between complete order isomorphisms of C*-algebras and invertible left multipliers of imprimitivity bimodules. A corollary of the above two results is that any complete order iso…
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 answer the title question for sigma-unital C*-algebras. The answer is that the algebra must be the direct sum of a dual C*-algebra and a C*-algebra satisfying a certain local unitality condition. We also discuss similar problems in the context of Hilbert C*-bimodules and imprimitivity bimodules and in the context of centralizers of Pedersen's ideal.
Using representation theory techniques we prove that various spaces of derivations or one-sided multipliers over certain operator algebras are reflexive. A sample result: any bounded local derivation (local left multiplier) on an automorphic semicrossed product is a derivation (resp. left multiplier). In the process we obtain various results of independent interest. In particular, the finite dimensional nest representations of the tensor algebra of a topological graph separate points.
We study from the perspective of Borel complexity theory the classification problem for multiplier algebras associated with operator algebraic varieties. These algebras are precisely the multiplier algebras of irreducible complete Nevanlinna-Pick spaces. We prove that these algebras are not classifiable up to algebraic isomorphism using countable structures as invariants. In order to prove such a result, we develop the theory of turbulence for Polish groupoids, which generalizes Hjorth's turbulence theory for Polish group actions. We also prove that the classification problem for multiplier al…
We introduce the Jordan-strict topology on the multipliers algebra of a JB$^*$-algebra, a notion which was missing despite the fourty years passed after the first studies on Jordan multipliers. In case that a C$^*$-algebra $A$ is regarded as a JB$^*$-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C$^*$-strict topology. We prove that every JB$^*$-algebra $\mathfrak{A}$ is J-strict dense in its multipliers algebra $M(\mathfrak{A})$, and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonalit…
Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix…
In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1<p\leq 2 \leq q<\infty$. Our approach is based on the operator algebras techniques. The result depends on a version of the Hausdorff-Young-Paley inequality that we establish on general locally compact separable unimodular groups. In particular, the obtained result implies the corresponding Hörmander's Fourier multiplier theorem on $\mathbb{R}^{n}$ and the corresponding known results for Fourier multipliers on compact Lie groups.
We develop a duality theory for multiplier Banach-Hopf algebras over a non-Archimedean field K. As examples, we consider algebras corresponding to discrete groups and zero-dimensional locally compact groups with K-valued Haar measure, as well as algebras of operators generated by regular representations of discrete groups.
We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study …
Using ultrapowers of $C^{\ast}$-algebras we provide a new construction of the multiplier algebra of a $C^{\ast}$-algebra. This extends the work of Avsec and Goldbring [Houston J. Math., to appear, arXiv:1610.09276.] to the setting of noncommutative and nonseparable $C^{\ast}$-algebras. We also extend their work to give a new proof of the fact that groups that act transitively on locally finite trees with boundary amenable stabilizers are boundary amenable.
We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type inequalities in group von Neumann algebras, prove $L_p$ estimates for noncommutative Riesz transforms and characterize $L_\infty \to \mathrm{BMO}$ boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigrou…
In this paper, we discuss spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold. First, we give a survey of results on generalized smooth distributions on manifolds, Riemannian structures and associated Laplacians. Then, under the assumption that the singular foliation generated by the distribution is regular, we prove that the Laplacian associated with the distribution defines an unbounded multiplier on the foliation $C^*$-algebra. To this end, we give the construction of a parametrix.
In this paper, we introduce the notion of multiplier of a Hilbert algebra. The space of bounded multipliers is a semifinite von Neumann algebra isomorphic to the left von Neumann algebra of the Hilbert algebra, as expected. However, in the unbounded setting, the space of multipliers has the structure of a *-algebra with nice properties concerning commutant and affiliation: it is a pre-GW*-algebra. And this correspondence between Hilbert algebras and its multipliers is functorial. Then, we can endow the Hilbert algebra with a nice topology constructed from unbounded multipliers. As we can see f…
We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak-$*$ closure, much in the spirit of the corresponding classical facts for the disc algebra. Zero sets for multipliers are also considered and are deeply intertwined with the structure of ideals. Our approach is primarily based on duality arguments.
Let $G$ be a locally compact group and $p,q\in \mathbb{R}$ with $p>1$ $p\not=2$ and $q$ between $2$ and $p$ (if $p<2$ then $p<q<2$, if $p>2$ then $2<q<p.$) The main result of the paper is that $A_q(G)$ multiplies $A_p(G)$, more precisely we show that the Banach algebra $A_p(G)$is a Banach module on $A_q(G).$
In this paper we introduce the notion of multiplier of a Hilbert pro-$C^{\ast }$-bimodule and we investigate the structure of the multiplier bimodule of a Hilbert pro-$C^{\ast}$-bimodule. We also investigate the relationship between the crossed product $A\times _{X}\mathbb{Z}$ of a pro-$% C^{\ast }$-algebra $A$ by a Hilbert pro-$C^{\ast }$-bimodule $X$ over $A$, the crossed product $M(A)\times _{M(X)}\mathbb{Z}$ of the multiplier algebra $M(A)$ of $A$ by the multiplier bimodule $M(X)$ of $X$ and the multiplier algebra $M(A\times _{X}\mathbb{Z})$ of $A\times _{X}\mathbb{Z}$.
Consider two non-degenerate algebras B and C over the complex numbers. We study a certain class of idempotent elements E in the multiplier algebra of the tensor product of B with C, called separability idempotents. The conditions include the existence of non-degenerate anti-homomorphisms from B to M(C) and from C to M(B), the multiplier algebras of C and B respectively. They are called the antipodal maps. There also exist what we call distinguished linear functionals. They are unique and faithful in the regular case. The notion is more restrictive than what is generally considered in the case …
We define the local multiplier module of a Hilbert module in analogy to the local multiplier algebra for $C^*$-algebras. We use properties of the local multiplier module to lift non-closed actions on $C^*$-algebras by Hilbert bimodules to closed actions on local multiplier algebras, and descend known results on such closed actions down to their unclosed counterparts. We define aperiodic dynamical inclusions and characterise them as crossed products by inverse semigroup actions. We describe the slice structure for such inclusions and show that all slices arise as linear combinations of slices a…
