arXiv subjects
Publications and source records attributed to Ulrich Haag.
Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected (but different from) group homology. It also gives a version of algebraic $K$-theory for rings by the simple functorial mapping assigning to a ring $R$ the (perfect>) commutator subgroup $E ( R )$ of the infinitedimensional general linear group over $R$.
The article gives the second part of the treatise on Regular Algebraic $K$-theory (Sections V & VI) of the author. Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected to (but different from) ordinary group homology. It also gives a version of algebraic $K$-theory for rings by the simple functorial mapping assigning to the ring $R$ the (perfect) commutator subgroup $E ( R )$ of the infinitedimensional general linear group over $R$.
The article associates two fundamental lattice constructions with each regular unital real ordered Banach space (function system). These are used to establish certain results in the theory of operator algebras, specifically relating the injective envelope of a separable C*-algebra with its enveloping von Neumann algebra in a given faithful separable representation. The last section investigates on lattices of projections arising in injective C*-algebras and von Neumann algebras and certain nonlinear maps sending projections to projections which are essentially determined by their values on positive projections.
The main result of the paper is an extension of the Dirichlet problem from (closures of) bounded open domains U to arbitrary compact subsets X of the complex plane, i.e. the closure of the corresponding space of functions which are harmonic in a neighbourhood of X and equipped with the supremums norm on X is shown to be isometric with the space of continuous functions C (/delta X) on its Shilov boundary (a given compact subset of X). This is used to define an extension of holomorphic function calculus with respect to certain (weakly normal) elements x of a unital operator algebra A to a completely isometric harmonic function calculus into the enveloping operator system of A. It is also shown that in case of a super C*-algebra A (operator algebra with involution) any weakly normal superpositive element x has a square root in A.
The paper gives a short account of the contents of "Regular Algebraic K-Theory For Groups" by the author and its connections with other homology and K-theories.
The article exhibits certain relations between the injective envelope I(A) of a C*-algebra A and the von Neumann algebra generated by a representation lambda of A provided it is injective. More specifically we show that there exist positive retractions sigma : /lambda (A)'' ---> I(A) which are close to being *-homomorphisms in the sense that they are Jordan homomorphisms of the underlying Jordan algebras, and the kernel of /sigma is given by a twosided ideal.
A notion of super operator system is defined which generalizes the usual notion of operator systems to include certain unital involutive operator spaces which cannot be represented completely isometric as a concrete operator system on some Hilbert space. They can nevertheless be represented by bounded operators on a standard Z_2-graded Hilbert space equipped with a superinvolution. We apply this theory to investigate on the relation between certain tensor products defined for operator spaces and C^*-algebras, such as the projective tensor product, the Haagerup tensor product and the maximal C^*-tensor product.
We consider *-linear maps into a commutative C*-algebra C (X) of continuous functions on a locally compact Hausdorff space X with certain specified properties and prove two results: (1) an extension result for a class of *-linear maps Y --> C (X) which may be called of locally compact type (locally finite) with respect to an inclusion Y < X of normed vector spaces, and (2) a minimal decomposition for certain *-linear maps into C (X) (absolutely continuous) as a difference of two positive maps.
The author presents a new proof of injectivity of the composition of the inverse of the rational Chern Character in homology applied to the classifying space BG of a (countable) discrete group G, restricted to dimensions less or equal than two, with the rationalized Assembly map of Kasparov into the (operator) K-Theory of the full group C^*-algebra C^*(G) (tensored with the rational numbers).