Logic Programming with Graph Automorphism: Integrating naut with Prolog (Tool Description)

This paper presents the plnauty~library, a Prolog interface to the nauty graph-automorphism tool. Adding the capabilities of nauty to Prolog combines the strength of the "generate and prune" approach that is commonly used in logic programming and constraint solving, with the ability to reduce symmetries while reasoning over graph objects. Moreover, it enables the integration of nauty in existing tool-chains, such as SAT-solvers or finite domain constraints compilers which exist for Prolog. The implementation consists of two components: plnauty, an interface connecting \nauty's C library with Prolog, and plgtools, a Prolog framework integrating the software component of nauty, called gtools, with Prolog. The complete tool is available as a SWI-Prolog module. We provide a series of usage examples including two that apply to generate Ramsey graphs. This paper is under consideration for publication in TPLP.Comment: Paper presented at the 32nd International Conference on Logic Programming (ICLP 2016), New York City, USA, 16-21 October 201

Isomorphisms of Hilbert C*-Modules and ∗*-Isomorphisms of Related Operator C*-Algebras

Let $\cal M$ be a Banach C*-module over a C*-algebra $A$ carrying two $A$-valued inner products $_1$, $_2$ which induce equivalent to the given one norms on $\cal M$. Then the appropriate unital C*-algebras of adjointable bounded $A$-linear operators on the Hilbert $A$-modules $\{ {\cal M}, _1 \}$ and $\{ {\cal M}, _2 \}$ are shown to be $*$-isomorphic if and only if there exists a bounded $A$-linear isomorphism $S$ of these two Hilbert $A$-modules satisfying the identity $_2 \equiv < S(.),S(.) >_1$. This result extends other equivalent descriptions due to L.~G.~Brown, H.~Lin and E.~C.~Lance. An example of two non-isomorphic Hilbert C*-modules with $*$-isomorphic C*-algebras of ''compact''/adjointable bounded module operators is indicated.Comment: 5 pages, LaTeX fil

A multiplier approach to the Lance-Blecher theorem

A new approach to the Lance-Blecher theorem is presented resting on the interpretation of elements of Hilbert C*-module theory in terms of multiplier theory of operator C*-algebras: The Hilbert norm on a Hilbert C*-module allows to recover the values of the inducing C*-valued inner product in a unique way, and two Hilbert C*-modules {M_1, _1}, {M_2, _2} are isometrically isomorphic as Banach C*-modules if and only if there exists a bijective C*-linear map S: M_1 --> M_2 such that the identity _1 \equiv _2 is valid. In particular, the values of a C*-valued inner product on a Hilbert C*-module are completely determined by the Hilbert norm induced from it. In addition, we obtain that two C*-valued inner products on a Banach C*-module inducing equivalent norms to the given one give rise to isometrically isomorphic Hilbert C*-modules if and only if the derived C*-algebras of ''compact'' module operators are *-isomorphic. The involution and the C*-norm of the C*-algebra of ''compact'' module operators on a Hilbert C*-module allow to recover its original C*-valued inner product up to the following equivalence relation: _1 \sim _2 if and only if there exists an invertible, positive element $a$ of the center of the multiplier C*-algebra M(A) of A such that the identity _1 \equiv a \cdot _2 holds.Comment: Plain TeX, 9 pages, without figures, submitted to Zeitschrift Anal. Anwendunge

Injective envelopes and local multiplier algebras of C*-algebras

The local multiplier C*-algebra M_{loc}(A) of any C*-algebra A can *-isomorphicly embedded into the injective envelope I(A) of A in such a way that the canonical embeddings of A into both these C*-algebras are identified. If A is commutative then M_{loc}(A) = I(A) . The injective envelopes of A and M_{loc}(A) always coincide, and every higher order local multiplier C*-algebra of A is contained in the regular monotone completion \bar{A} in I(A) of A . In case the set Z(A).A is dense in A the center of the local multiplier C*-algebra of A is the local multiplier C*-algebra of the center of A, and both they are *-isomorphic to the injective envelope of the center of A . A Wittstock type extension theorem for completely bounded bimodule maps on operator bimodules taking values in M_{loc}(A) is proven to hold if and only if M_{loc}(A) = I(A). In general, a solution of the problem for which C*-algebras A the C*-algebras M_{loc}(A) is injective is shown to be equivalent to the solution of I. Kaplansky's 1951 problem whether all AW*-algebras are monotone complete.Comment: 10 pages, LaTeX2e, one statement and its proof correcte

Spectral and polar decomposition in AW*-algebras

We show the possibility and the uniqueness of polar decomposition of elements of arbitrary AW*-algebras inside them. We prove that spectral decomposition of normal elements of normal AW*-algebras is possible and unique inside them. The possibility of spectral decomposition of normal elements does not depend on the normality of the AW*-algebra under consideration.Comment: 7 pages, LATEX, preprint NTZ-25/91 (Universitaet Leipzig, Naturwissenschaftlich-Theoretisches Zentrum, 1991), submitted to Zeitschr. Anal. An

The Standard Model - the Commutative Case: Spinors, Dirac Operator and de Rham Algebra

The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac operators, exterior algebra bundles and Connes' differential algebras in the commutative case, among other elements. We avoid the introduction of principal bundles and put the emphasis on a module-based approach using Serre-Swan's theorem, Hermitian structures and module frames. A new proof (due to Harald Upmeier) of the differential algebra isomorphism between the set of smooth sections of the exterior algebra bundle and Connes' differential algebra is presented.Comment: 19 pages, LaTeX2

Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules

B. Magajna and J. Schweizer showed in 1997 and 1999, respectively, that C*-algebras of compact operators can be characterized by the property that every norm-closed (and coinciding with its biorthogonal complement, resp.) submodule of every Hilbert C*-module over them is automatically an orthogonal summand. We find out further generic properties of the category of Hilbert C*-modules over C*-algebras which characterize precisely the C*-algebras of compact operators.Comment: 9 pages, part of a collection dedicated to the memory of Yu. P. Solovyov (Moscow State University). to appear in K-Theor

Direct integrals and Hilbert W*-Modules

Investigating the direct integral decomposition of von Neumann algebras of bounded module operators on self-dual Hilbert W*-moduli an equivalence principle is obtained which connects the theory of direct disintegration of von Neumann algebras on separable Hilbert spaces and the theory of von Neumann representations on self-dual Hilbert {\bf A}-moduli with countably generated {\bf A}-pre-dual Hilbert {\bf A}-module over commutative separable W*-algebras {\bf A}. Examples show posibilities and bounds to find more general relations between these two theories, (cf. R. Schaflitzel's results). As an application we prove a Weyl--Berg--Murphy type theorem: For each given commutative W*-algebra {\bf A} with a special approximation property (*) every normal bounded {\bf A}-linear operator on a self-dual Hilbert {\bf A}-module with countably generated {\bf A}-pre-dual Hilbert {\bf A}-module is decomposable into the sum of a diagonalizable normal and of a ''compact'' bounded {\bf A}-linear operator on that module.Comment: 20 pages, LATEX file, preprint 23/91, NTZ, Univ. Leipzig, German

Hilbertian versus Hilbert W*-modules, and applications to L2L^2- and other invariants

Hilbert(ian) A-modules over finite von Neumann algebras A with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared, and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in $L^2$-invariant theory providing alternative proofs. We indicate new invariants for finitely generated projective B-modules, where B is supposed any unital C*-algebra, (usually the full group C*-algebra $C^*(\pi)$ of the fundamental group $\pi=\pi_1(M)$ of a manifold $M$). The results are of interest to specialists in operator algebras and global analysis.Comment: 12 pages, Latex2

On Hahn-Banach type theorems for Hilbert C*-modules

We show three Hahn-Banach type extension criteria for (sets of) bounded C*-linear maps of Hilbert C*-modules to the underlying C*-algebras of coefficients. One criterion establishes an alternative description of the property of (AW*-) C*-algebras to be monotone complete or additively complete.Comment: 16 pages, LaTeX2e, major changes and additions, updated literature lis