Alexander A. Katz, Oleg Friedman
A theorem is presented on existence and uniqueness up to the topological *-isomorphism of universal locally C*-algebra for an arbitrary locally JB-algebra.
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In the present note we show that the involution in locally C*-algebras is uniquely determined.
Let $\mathfrak{g}$ be a compact simple Lie algebra. We modify the quantized enveloping $^*$-algebra associated to $\mathfrak{g}$ by a real-valued character on the positive part of the root lattice. We study the ensuing Verma module theory, and the associated quotients of these modified quantized enveloping $^*$-algebras. Restricting to the locally finite part by means of a natural adjoint action, we obtain in particular examples of quantum homogeneous spaces in the operator algebraic setting.
We introduce a notion of I-factorial quantum torsor, which consists of an integrable ergodic action of a locally compact quantum group on a type I-factor such that also the crossed product is a type I-factor. We show that any such I-factorial quantum torsor is at the same time a I-factorial quantum torsor for the dual locally compact quantum group, in such a way that the construction is involutive. As a motivating example, we show that quantized compact semisimple Lie groups, when amplified via a crossed product construction with the function algebra on the associated weight lattice, admit I-f…
In the present article, it is proved that every nuclear operator in a locally convex space E has a well-defined trace if E possesses the approximation property. However, even if a space possesses the approximation property this still does not guarantee a positive solution of A. Grothendieck's uniqueness problem for this space. Below, we present an example of a quasi-complete space with the approximation property in which it is not possible to define the trace for all Fredholm operators (in the sense of A. Grothendieck). We prove that the uniqueness problem has a positive solution if E possesse…
We introduce and study continuous fields of JB-algebras (which are real non-associate analogues of C*-algebras). In particular, we show that for the universal enveloping C*-algebra C*sub-u(B) for the JB-algebra B defined by a continuous field of JB-algebras A-sub-t, t belongs to T, on a locally compact space T, there exists a decomposition of C*-sub-u(B) into a continuous field of C*-algebras C*u(A-sub-t), t belongs to T, on the same space T, composed entirely of the universal enveloping C*-algebras of the corresponding JB-algebras from the aforementioned decomposition of the algebra B.
In this paper we associate to every reduced C*-algebraic quantum group A a universal C*-algebraic quantum group. We fine tune a proof of Kirchberg to show that every *-representation of a modified L1-space is generated by a unitary corepresentation. By taking the universal enveloping C*-algebra of a dense sub *-algebra of A we arrive at the uinversal C*-algebra. We show that this universal C*-algebra carries a quantum group structure which is as rich as its reduced companion.
For a second-countable locally compact Hausdorff étale groupoid $\mathcal{G}$ with a continuous $2$-cocycle $σ$ we find conditions that guarantee that $\ell^1 (\mathcal{G},σ)$ has a unique $C^*$-norm.
Motivated by the notion of a set of uniqueness in a locally compact group $G$, we introduce and study ideals of uniqueness in the Fourier algebra $A(G)$ of $G$, and their accompanying operator version, masa-bimodules of uniqueness. We establish a transference between the two notions, and use this result to show that the property of being an ideal of uniqueness is preserved under natural operations.
We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf \cst-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the \cst-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorph…
We prove for an arbitrary complex $^*$-algebra $A$ that every topologically irreducible $^*$-representation of $A$ on a Hilbert space is finite dimensional precisely when the Lebesgue decomposition of representable positive functionals over $A$ is unique. In particular, the uniqueness of the Lebesgue decomposition of positive functionals over the $L^1$-algebras of locally compact groups provides a new characterization of Moore groups.
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