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Register a new userOn Intrinsic Characterization of Real Locally C*- and Locally JB-Algebras
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In the present paper we obtain an intrinsic characterization of real locally C*-algebras (projective limits of projective families of real C*-algebras) among complete real lmc *-algebras, and of locally JB-algebras (projective limits of projective families of JB-algebras) among complete fine Jordan locally multiplicatively-convex topological algebras.
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It has been established by Inoue that a complex locally C*-algebra with a dense ideal posesses a bounded approximate identity which belonges to that ideal. It has been shown by Fritzsche that if a unital complex locally C*-algebra has an unbounded element then it also has a dense one-sided ideal. In the present paper we obtain analogues of the aforementioned results of Inoue and Fritzsche for real locally C*-algebras (projective limits of projective families of real C*-algebras), and for locally JB-algebras (projective limits of projective families of JB-algebras).
We study the structure of certain classes of homologically trivial locally C*-algebras. These include algebras with projective irreducible Hermitian A-modules, biprojective algebras, and superbiprojective algebras. We prove that, if A is a locally C*-algebra, then all irreducible Hermitian A-modules are projective if and only if A is a direct topological sum of elementary C*-algebras. This is also equivalent to A being an annihilator (dual, complemented, left quasi-complemented, or topologically modular annihilator) topological algebra. We characterize all annihilator $sigma$-C*-algebras and describe the structure of biprojective locally C*-algebras. Also, we present an example of a biprojective locally C*-algebra that is not topologically isomorphic to a Cartesian product of biprojective C*-algebras. Finally, we show that every superbiprojective locally C*-algebra is topologically *-isomorphic to a Cartesian product of full matrix algebras.
We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group $mathbb{G}$ and certain left invariant C*-subalgebras of $C_0(mathbb{G})$. We also prove that every compact quantum subgroup of a co-amenable quantum group is co-amenable. Moreover, there is a one-to-one correspondence between open subgroups of an amenable locally compact group $G$ and non-zero, invariant C*-subalgebras of the group C*-algebra $C^*(G)$.
This is a short survey on idempotent states on locally compact groups and locally compact quantum groups. The central topic is the relationship between idempotent states, subgroups and invariant C*-subalgebras. We concentrate on recent results on locally compact quantum groups, but begin with the classical notion of idempotent probability measure. We also consider the `intermediate case of idempotent states in the Fourier--Stieltjes algebra: this is the dual case of idempotent probability measures and so an instance of idempotent states on a locally compact quantum group.
Let $A$ be a $C^*$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $theta : Eto F$ is a linear map preserving orthogonality, i.e., $<theta(x), theta(y) > = 0$ whenever $<x, y > = 0$. We show in this article that if, in addition, $A$ has real rank zero, and $theta$ is an $A$-module map (not assumed to be bounded), then there exists a central positive multiplier $uin M(A)$ such that $<theta(x), theta(y) > = u < x, y>$ ($x,yin E$). In the case when $A$ is a standard $C^*$-algebra, or when $A$ is a $W^*$-algebra containing no finite type II direct summand, we also obtain the same conclusion with the assumption of $theta$ being an $A$-module map weakened to being a local map.
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