An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those automorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G --> H. Unlike the group of inner automorphisms of G itself, the group of such extended systems of automorphisms is always isomorphic to G. A similar characterization holds for inner automorphisms of an associative algebra R over a field K; here the group of functorial systems of automorphisms is isomorphic to the group of units of R modulo units of K. If one substitutes "endomorphism" for "automorphism" in these considerations, then in the group case, the only additional example is the trivial endomorphism; but in the K-algebra case, a construction unfamiliar to ring theorists, but known to functional analysts, also arises. Systems of endomorphisms with the same functoriality property are examined in some other categories; other uses of the phrase "inner endomorphism" in the literature, some overlapping the one introduced here, are noted; the concept of an inner {\em derivation} of an associative or Lie algebra is looked at from the same point of view, and the dual concept of a "co-inner" endomorphism is briefly examined. Several questions are posed.Comment: 20 pages. To appear, Publicacions Mathem\`{a}tiques. The 1-1-ness result in the appendix has been greatly strengthened, an "Overview" has been added at the beginning, and numerous small rewordings have been made throughou

Unknown Quantum States: The Quantum de Finetti Representation

We present an elementary proof of the quantum de Finetti representation theorem, a quantum analogue of de Finetti's classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an ``unknown quantum state'' in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than states of nature. We further demonstrate that the theorem fails for real Hilbert spaces and discuss the significance of this point.Comment: 30 pages, 2 figure

Do anyons solve Heisenberg's Urgleichung in one dimension

We construct solutions to the chiral Thirring model in the framework of algebraic quantum field theory. We find that for all positive temperatures there are fermionic solutions only if the coupling constant is $\lambda = \sqrt{2(2n + 1)\pi}, n \in \bf N$.Comment: 19 pages LaTeX, to appear in Eur. Phys. J.