Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph ${\rm\Lambda}$ is a natural generalization of a $k$-graph. A pullback of ${\rm\Lambda}$ is constructed by pulling it back over a given monoid morphism to $P$, while a pushout of ${\rm\Lambda}$ is obtained by modding out its periodicity, which is deduced from a natural equivalence relation on ${\rm\Lambda}$. One of our main results in this paper shows that, for some $k$-graphs ${\rm\Lambda}$, ${\rm\Lambda}$ is isomorphic to the pullback of its pushout via a natural quotient map, and that its graph $\text{C}^{\ast }$-algebra can be embedded into the tensor product of the graph $\text{C}^{\ast }$-algebra of its pushout and $\text{C}^{\ast }(\text{Per}\,{\rm\Lambda})$. As a consequence, in this case, the cycline algebra generated by the standard generators corresponding to equivalent pairs is a maximal abelian subalgebra, and there is a faithful conditional expectation from the graph $\text{C}^{\ast }$-algebra onto it.

Let G be a finitely generated group with polynomial growth, and let ω be a weight, i.e. a sub-multiplicative function on G with positive values. We study when the weighted group algebra ℓ1 (G, ω) is isomorphic to an operator algebra. We show that ℓ1 (G, ω) is isomorphic to an operator algebra if ω is a polynomial weight with large enough degree or an exponential weight of order 0 < α < 1. We demonstrate that the order of growth of G plays an important role in this problem. Moreover, the algebraic centre of ℓ1 (G, ω) is isomorphic to a Q-algebra, and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when G consists of the d-dimensional integers ℤd or the three-dimensional discrete Heisenberg group ℍ3(ℤ). The case of the free group with two generators is considered as a counter-example of groups with exponential growth.

We introduce an approximation property ( ${\mathcal{K}}_{\mathit{up}}$ -AP, $1\leq p<\infty$ ), which is weaker than the classical approximation property, and discover the duality relationship between the ${\mathcal{K}}_{\mathit{up}}$ -AP and the ${\mathcal{K}}_{p}$ -AP. More precisely, we prove that for every $1<p<\infty$ , if the dual space $X^{\ast }$ of a Banach space $X$ has the ${\mathcal{K}}_{\mathit{up}}$ -AP, then $X$ has the ${\mathcal{K}}_{p}$ -AP, and if $X^{\ast }$ has the ${\mathcal{K}}_{p}$ -AP, then $X$ has the ${\mathcal{K}}_{\mathit{up}}$ -AP. As a consequence, it follows that every Banach space has the ${\mathcal{K}}_{u2}$ -AP and that for every $1<p<\infty$ , $p\neq 2$ , there exists a separable reflexive Banach space failing to have the ${\mathcal{K}}_{\mathit{up}}$ -AP.

It has been a long-standing question whether every amenable operator algebra is isomorphic to a (necessarily nuclear) $\mathrm{C}^*$ -algebra. In this note, we give a nonseparable counterexample. Finding out whether a separable counterexample exists remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in $\mathrm{C}^*$ -algebras and show that our method cannot produce a separable counterexample.

We study the reflexivity and transitivity of a double triangle lattice of subspaces in a Hilbert space. We show that the double triangle lattice is neither reflexive nor transitive when some invertibility condition is satisfied (by the restriction of a projection under another). In this case, we show that the reflexive lattice determined by the double triangle lattice contains infinitely many projections, which partially answers a problem of Halmos on small lattices of subspaces in Hilbert spaces.

We study reflexivity and structural properties of operator algebras generated by representations of the discrete Heisenberg semigroup. We show that the left regular representation of this semigroup gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation that gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of .

Let A and B be C*-algebras. We prove the slice map conjecture for ideals in the operator space projective tensor product . As an application, a characterization of the prime ideals in the Banach *-algebra is obtained. In addition, we study the primitive ideals, modular ideals and the maximal modular ideals of . We also show that the Banach *-algebra possesses the Wiener property and that, for a subhomogeneous C*-algebra A, the Banach * -algebra is symmetric.

There is an unfortunate error in Theorem 4.1 of our paper. However, the statement of the theorem remains true with a correct construction of adding a tail to enlarge the dynamical system.

We compare two influential ways of defining a generalized notion of space. The first, inspired by Gelfand duality, states that the category of ‘noncommutative spaces’ is the opposite of the category of C*-algebras. The second, loosely generalizing Stone duality, maintains that the category of ‘point-free spaces’ is the opposite of the category of frames (that is, complete lattices in which the meet distributes over arbitrary joins). Earlier work by the first three authors shows how a noncommutative C*-algebra gives rise to a commutative one internal to a certain sheaf topos. The latter, then, has a constructive Gelfand spectrum, also internal to the topos in question. After a brief review of this work, we compute the so-called external description of this internal spectrum, which in principle is a fibred point-free space in the familiar topos of sets and functions. However, we obtain the external spectrum as a fibred topological space in the usual sense. This leads to an explicit Gelfand transform, as well as to a topological reinterpretation of the Kochen–Specker theorem of quantum mechanics.

Recently, Daws introduced a notion of co-representation of abelian Hopf–von Neumann algebras on general reflexive Banach spaces. In this note, we show that this notion cannot be extended beyond subhomogeneous Hopf–von Neumann algebras. The key is our observation that, for a von Neumann algebra 𝔐 and a reflexive operator space E, the normal spatial tensor product is a Banach algebra if and only if 𝔐 is subhomogeneous or E is completely isomorphic to column Hilbert space.

We give a new very concrete description of the C*-envelope of the tensor algebra associated to a multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism and as a groupoid C*-algebra. In the non-surjective case, it is a full corner of such an algebra. We also show that when the space is compact the C*-envelope is simple if and only if the system is minimal.

We analyze the automorphism group for the norm closed quiver algebras 𝒯+(Q). We begin by focusing on two normal subgroups of the automorphism group which are characterized by their actions on the maximal ideal space of 𝒯+(Q). To further discuss arbitrary automorphisms we factor automorphism through subalgebras for which the automorphism group can be better understood. This allows us to classify a large number of noninner automorphisms. We suggest a candidate for the group of inner automorphisms.

We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of is Arens regular, and give some evidence that this is so if and only if is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebraffi We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of is amenable. We provide an abstract characterization in terms of something like an approximate diagonal, and consider when every ultrapower of a C*-algebra, or a group L1-convolution algebra, is amenable.

We introduce the spaces Vℬp(X) (respectively 𝒱ℬp(X)) of the vector measures ℱ:Σ→X of bounded (p,ℬ)-variation (respectively of bounded (p,ℬ)-semivariation) with respect to a bounded bilinear map ℬ:X×Y →Z and show that the spaces Lℬp(X) consisting of functions which are p-integrable with respect to ℬ, defined in by Blasco and Calabuig [‘Vector-valued functions integrable with respect to bilinear maps’, Taiwanese Math. J. to appear], are isometrically embedded in Vℬp(X). We characterize 𝒱ℬp(X) in terms of bilinear maps from Lp′×Y into Z and Vℬp(X) as a subspace of operators from Lp′(Z*) into Y*. Also we define the notion of cone absolutely summing bilinear maps in order to describe the (p,ℬ)-variation of a measure in terms of the cone-absolutely summing norm of the corresponding bilinear map from Lp′×Y into Z.

Every directed graph defines a Hilbert space and a family of weighted shifts that act on the space. We identify a natural notion of periodicity for such shifts and study their C* -algebras. We prove the algebras generated by all shifts of a fixed period are of Cuntz-Krieger and Toeplitz-Cuntz-Krieger type. The limit C* -algebras determined by an increasing sequence of positive integers, each dividing the next, are proved to be isomorphic to Cuntz-Pimsner algebras and the linking maps are shown to arise as factor maps. We derive a characterization of simplicity and compute the K-groups for these algebras. We prove a classification theorem for the class of algebras generated by simple loop graphs.

This note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly σ-complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space ℓ∞.

We prove that two quiver operator algebras can be isometrically isomorphic only if the quivers (=directed graphs) are isomorphic. We also show how the graph can be recovered from certain representations of the algebra.

We give an example of a Banach space X such that K (X, X) is not an ideal in K (X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions H B(x* ⊗ z*) ⊆ (Z*, Y)* of a functional x* ⊗ Z* ∈ (Z ⊗ X)* is equal to the set H B(x*) ⊗ {z*}. Using this result, we show that if X is an M-ideal in Y and Z is a reflexive Banach space, then K (Z, X) is an M-ideal in K(Z, Y) whenever K (Z, X) is an ideal in K (Z, Y). We also show that K (Z, X) is an ideal (respectively, an M-ideal) in K (Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X * has the compact approximation property with conjugate operators.

A left ideal on any C*-algebra is an example of an operator algebra with a right contractive approximate indentiy (r.c.a.i.). Indeed, left ideal in C*-algebras may be charcterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. shoulod be studied in terms of a certain let ideal of a C*-algebra. We study left ideals from the perspective of ‘Hamana theory’ and using the multiplier algebras of an operator space studied elsewhere by the author. More generally, we develop some general theory for operator algebras which have a 1-sided identity or approzimate indentity, including a Banach-Stone theorem for these algebras, and an analysis of the ‘multiplier operator algebra’.