Mathematics

We unite elements of category theory, K-theory, and geometric group theory, by defining a class of groups called k -cube groups, which act freely and transitively on the product of k trees, for arbitrary k . The quotient of this action on the product of trees defines a k -dimensional cube complex, which induces a higher-rank graph. We make deductions about the K-theory of the corresponding k -rank graph C*-algebras, and give explicit examples of k -cube groups and their K-theory. We give explicit computations of K-theory for an infinite family of k -rank graphs for k≥3 , which is not a direct consequence of the Künneth Theorem for tensor products.

Some well known results by Haagerup, Jolissaint and de la Harpe may be extended to the setting of a reduced crossed product of a C*-algebra A by a discrete group G. We show that for many discrete groups, which include Gromov's hyperbolic groups and finitely generated discrete groups of polynomial growth, an inequality of the form ?�X?�≤C ??g?�G (1+|g| ) 4 ??X g ??2 ????????????????????????????????holds for any finitely supported operator X in the reduced crossed product.

A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify to the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.

A semicrossed product is a non-selfadjoint operator algebra encoding the action of a semigroup on an operator or C*-algebra. We prove that, when the positive cone of a discrete lattice ordered abelian group acts on a C*-algebra, the C*-envelope of the associated semicrossed product is a full corner of a crossed product by the whole group. By constructing a C*-cover that itself is a full corner of a crossed product, and computing the Shilov ideal, we obtain an explicit description of the C*-envelope. This generalizes a result of Davidson, Fuller, and Kakariadis from Z n + to the class of all discrete lattice ordered abelian groups.

Recently it is proved in arXiv:1906.05493v1 [math.OA] that CCR flows over convex cones are cocycle conjugate if and only if the associated isometric representations are conjugate. We provide a very short, simple and direct proof of that. Using the same idea we prove the analogous statement for CAR flows as well. Further we show that CCR flows are not cocycle conjugate to the CAR flows when the (multi-parameter) isometric representation is `proper', a condition which is satisfied by all known examples.

These notes are the output of a decade of research on how the results about dilations of one-parameter CP-semigroups with the help of product systems, can be put forward to d-parameter semigroups - and beyond. While preliminary work on the two- and d-parameter case is based on the approach via the Arveson-Stinespring correspondence of a CP-map by Muhly and Solel (and limited to von Neumann algebras), here we explore consequently the approach via Paschke's GNS-correspondence of a CP-map by Bhat and Skeide. (A comparison is postponed to Appendix A(iv).) The generalizations are multi-fold, the difficulties often enormous. In fact, our only true if-and-only-if theorem, is the following: A Markov semigroup over (the opposite of) an Ore monoid admits a full (strict or normal) dilation if and only if its GNS-subproduct system embeds into a product system. Already earlier, it has been observed that the GNS- (respectively, the Arveson-Stinespring) correspondences form a subproduct system, and that the main difficulty is to embed that into a product system. Here we add, that every dilation comes along with a superproduct system (a product system if the dilation is full). The latter may or may not contain the GNS-subproduct system; it does, if the dilation is strong - but not only. Apart from the many positive results pushing forward the theory to large extent, we provide plenty of counter examples for almost every desirable statement we could not prove. Still, a small number of open problems remains. The most prominent: Does there exist a CP-semigroup that admits a dilation, but no strong dilation? Another one: Does there exist a Markov semigroup that admits a (necessarily strong) dilation, but no full dilation?

We introduce certain C ∗ -algebras and k -graphs associated to k finite dimensional unitary representations ρ 1 ,..., ρ k of a compact group G . We define a higher rank Doplicher-Roberts algebra O ρ 1 ,..., ρ k , constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this C ∗ -algebra is isomorphic to a corner in the C ∗ -algebra of a row finite rank k graph Λ with no sources. For G finite and ρ i faithful of dimension at least 2 , this graph is irreducible, it has vertices G ^ and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate with some examples when O ρ 1 ,..., ρ k is simple and purely infinite, and with some K -theory computations.

Let M and N be two unital JB ∗ -algebras and let U(M) and U(N) denote the sets of all unitaries in M and N , respectively. We prove that the following statements are equivalent: (a) M and N are isometrically isomorphic as (complex) Banach spaces; (b) M and N are isometrically isomorphic as real Banach spaces; (c) There exists a surjective isometry Δ:U(M)→U(N). We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry Δ:U(M)→U(N) we can find a surjective real linear isometry Ψ:M→N which coincides with Δ on the subset e i M sa . If we assume that M and N are JBW ∗ -algebras, then every surjective isometry Δ:U(M)→U(N) admits a (unique) extension to a surjective real linear isometry from M onto N . This is an extension of the Hatori--Moln{á}r theorem to the setting of JB ∗ -algebras.

Renault proved in 2008 that if G is a topologically principal groupoid, then C 0 ( G (0) ) is a Cartan subalgebra in C ∗ r (G,Σ) for any twist Σ over G . However, there are many groupoids which are not topologically principal, yet their (twisted) C ∗ -algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle c on G and a subgroupoid S⊆G under which C ∗ r (S,c) is Cartan in C ∗ r (G,c) . When G is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.

Suppose A is a unital subhomogeneous C*-algebra. We show that every central sequence in A is hypercentral if and only if every pointwise limit of a sequence of irreducible representations is multiplicity free. We also show that every central sequence in A is trivial if and only if every pointwise limit of irreducible representations is irreducible. We also give a nice repesentation of the latter algebras.