A contractive $n$-tuple $A\,=\,({{A}_{1}},...,{{A}_{n}})$ has a minimal joint isometric dilation $S\,=\,({{S}_{1}},...,{{S}_{n}})$ where the ${{S}_{i}}$’s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$ acts on a finite dimensional space, the wot-closed nonself-adjoint algebra $\mathfrak{S}$ generated by $S$ is completely described in terms of the properties of $A$. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra $\mathfrak{S}$ is always hyper-reflexive. In the last section, we describe similarity invariants. In particular, an $n$-tuple $B$ of $d\,\times \,d$ matrices is similar to an irreducible $n$-tuple $A$ if and only if a certain finite set of polynomials vanish on $B$.

The proof of Lemma 3.4 in $\left[ \text{F} \right]$ relies on the incorrect equality ${{\mu }_{j}}(AB)={{\mu }_{j}}(BA)$ for singular values (for a counterexample, see [S, p. 4]). Thus, Theorem 3.1 as stated has not been proven. However, with minor changes, we can obtain a bound for the counting function in terms of the growth of the Fourier transform of $\left| V \right|$.

We have seen, in the previous works [5], [6], that the argument of a certain product is closely connected to that of the cubic Gauss sum. Here the absolute value of the product will be investigated.

We prove a number of results about the stable and particularly the real ranks of tensor products of ${{C}^{*}}$ -algebras under the assumption that one of the factors is commutative. In particular, we prove the following:

1. (1) If $X$ is any locally compact $\sigma $-compact Hausdorff space and $A$ is any ${{C}^{*}}$-algebra, then $\text{RR(}{{C}_{0}}\text{(}X\text{)}\otimes A\text{)}\le \text{dim(}X\text{)+RR(}A\text{)}$ .

2. (2) If $X$ is any locally compact Hausdorff space and $A$ is any purely infinite simple ${{C}^{*}}$ -algebra, then $\text{RR(}{{C}_{0}}\text{(}X\text{)}\otimes A\text{)}\le 1$ .

3. (3) $\text{RR(}C([0,\,1]\,)\otimes \,A)\,\ge \,1$ for any nonzero ${{C}^{*}}$-algebra $A$, and $\text{sr(}C({{[0,\,1]}^{2}})\,\otimes \,A\text{)}\,\ge \,2$ for any unital ${{C}^{*}}$-algebra $A$.

4. (4) If $A$ is a unital ${{C}^{*}}$-algebra such that $\text{RR(}A\text{)}\,\text{=}\,\text{0,}\,\text{s}r\text{(}A\text{)}\,\text{=}\,\text{1}$, and ${{K}_{1}}(A)=0$ , then $\text{sr(}C([0,\,1])\,\otimes \,A\text{)}\,\text{=}\,1$.

5. (5) There is a simple separable unital nuclear ${{C}^{*}}$-algebra $A$ such that $\text{RR(}A\text{)}\,\text{=}\,\text{1}$ and $\text{sr(}C([0,\,1])\,\otimes \,A\text{)}\,\text{=}\,1$.

In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra $B$ over $\mathbf{Q}$; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on ${{B}^{\times }}$, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of ${{B}^{\times }}$. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to $\text{CM}$ points on these curves, and are thus isogenous to a product $E\,\times \,E$, where $E$ is an elliptic curve with complex multiplication. For these $\text{CM}$ points one can make a relation between the action of the $p$-th Hecke operator and Frobenius at $p$, similar to the well-known congruence relation of Eichler and Shimura.

We show that the Elliott invariant is a classifying invariant for the class of ${{C}^{*}}$-algebras that are simple unital infinite dimensional inductive limits of finite direct sums of building blocks of the form

$$\left\{ f\in C\left( \mathbb{T} \right)\otimes {{M}_{n}}:f\left( {{x}_{i}} \right)\in {{M}_{{{d}_{i}}}},i=1,2,...,N \right\},$$

where ${{x}_{1}},\,{{x}_{2.}},...,{{x}_{N\,}}\in \mathbb{T},{{d}_{1}},{{d}_{2}},.\,.\,.,{{d}_{N}}$ are integers dividing $n$, and ${{M}_{{{d}_{i}}}}$ is embedded unitally into ${{M}_{n}}$. Furthermore we prove existence and uniqueness theorems for $*$-homomorphisms between such algebras and we identify the range of the invariant.

The question of counting minimal factorizations of permutations into transpositions that act transitively on a set has been studied extensively in the geometrical setting of ramified coverings of the sphere and in the algebraic setting of symmetric functions.

It is natural, however, from a combinatorial point of view to ask how such results are affected by counting up to equivalence of factorizations, where two factorizations are equivalent if they differ only by the interchange of adjacent factors that commute. We obtain an explicit and elegant result for the number of such factorizations of permutations with precisely two factors. The approach used is a combinatorial one that rests on two constructions.

We believe that this approach, and the combinatorial primitives that have been developed for the “cut and join” analysis, will also assist with the general case.

Giordano, Putnam and Skau showed that the transformation group ${{C}^{*}}$-algebra arising from a Cantor minimal system is an $AT$-algebra, and classified it by its $K$-theory. For approximately inner automorphisms that preserve $C\left( X \right)$, we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

We prove that the operators $\int{_{\mathbb{R}_{+}^{2}}{{e}^{i{{x}^{a}}\cdot {{y}^{b}}}}\varphi (x,y)f(y)dy}$ map ${{L}^{p}}({{\mathbb{R}}^{2}})$ into itself for $p\,\in \,J\,=\,\,\left[ \frac{{{a}_{1}}+{{b}_{1}}}{{{a}_{1}}+(\frac{{{b}_{1}}r}{2})},\frac{{{a}_{1}}+{{b}_{1}}}{{{a}_{1}}+(1-\frac{r}{2})} \right]$ if ${{a}_{l}},{{b}_{l}}\ge 1$ and $\varphi (x,y)=|x-y{{|}^{-r}},0\le r<2$ , the result is sharp. Generalizations to dimensions $d\,>\,2$ are indicated.

We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite $\text{I}{{\text{I}}_{1}}$ factor $R$ in $[\text{E1}]$—which arise by considering unitary braid representations with certain properties—to multi-sided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples—where the braid representations are obtained in connection with the representation theory of Lie algebras of types $A,\,B,\,C,\,D$. We also compute the (first) relative commutants.

A theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact Kähler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces.

We show that the Seiberg-Witten invariants of a lens space determine and are determined by its Casson-Walker invariant and its Reidemeister-Turaev torsion.

The concept of $R$-diagonal element was introduced in [5], and was subsequently found to have applications to several problems in free probability. In this paper we describe a new approach to $R$-diagonality, which relies on freeness with amalgamation. The class of $R$-diagonal elements is enlarged to contain examples living in non-tracial $*$-probability spaces, such as the generalized circular elements of [7].

We prove comparison theorems for the probability of life in a Lipschitz domain between Brownian motion and random walks.

We study when the spaces of Lorentz multipliers from ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ are distinct. Our main interest is the case when $s\,<\,t$, the Lorentz-improving multipliers. We prove, for example, that the space of multipliers which map ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ is different from those mapping ${{L}^{r,v}}\,\to \,{{L}^{r,u}}$ if either $r\,=\,p$ or ${p}'$ and $1/s\,-\,1/t\,\ne \,1/u\,-\,1/v$, or $r\,\ne \,p$ or ${p}'$. These results are obtained by making careful estimates of the Lorentz multiplier norms of certain linear combinations of Fejer or Dirichlet kernels. For the case when the first indices are different the linear combination we analyze is in the spirit of a Rudin-Shapiro polynomial.

Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$, with anisotropic centre. Given a rational action of $G$ on a finite-dimensional vector space $V$, we analyze the truncated integral of the theta series corresponding to a Schwartz-Bruhat function on $V\left( \mathbb{A} \right)$. The Poisson summation formula then yields an identity of distributions on $V\left( \mathbb{A} \right)$. The truncation used is due to Arthur.

We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of $\sigma $-unital simple ${{C}^{*}}$-algebras $A$ with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra $\mathcal{M}(A)$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mathcal{M}(A)$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mathcal{M}(A)$ is reflected in the fact that $\mathcal{M}(A)$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

If $\mathfrak{A}$ is a finite alphabet, $\mathcal{U}\,\subset \,{{\mathbb{Z}}^{D}}$, and ${{\mu }_{\mathcal{U}}}$ is a probability measure on ${{\mathfrak{A}}^{\mathcal{U}}}$ that “looks like” the marginal projection of a stationary stochastic process on ${{\mathfrak{A}}^{{{\mathbb{Z}}^{D}}}}$, then can we “extend” ${{\mu }_{\mathcal{U}}}$ to such a process? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when $D\,=\,1$, we provide some sufficient conditions and some necessary conditions for ${{\mu }_{\mathcal{U}}}$ to be extendible for $D\,>\,1$, and show that, in general, the problem is not formally decidable.

We prove that pre-classifiable (see 3.1) simple nuclear tracially $\text{AF}\,\,{{C}^{*}}$-algebras $\left( \text{TAF} \right)$ are classified by their $K$-theory. As a consequence all simple, locally $\text{AH}$ and $\text{TAF}\,\,\,{{C}^{*}}$-algebras are in fact $\text{AH}$ algebras (it is known that there are locally $\text{AH}$ algebras that are not $\text{AH}$). We also prove the following Rationalization Theorem. Let $A$ and $B$ be two unital separable nuclear simple $\text{TAF}\,\,\,{{C}^{*}}$-algebras with unique normalized traces satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the same (ordered and scaled) $K$-theory and ${{K}_{0}}{{\left( A \right)}_{+}}$ is locally finitely generated, then $A\,\otimes \,Q\,\cong \,B\,\otimes \,Q$, where $Q$ is the $\text{UHF}$-algebra with the rational ${{K}_{0}}$. Classification results (with restriction on ${{K}_{0}}$ - theory) for the above ${{C}^{*}}$-algebras are also obtained. For example, we show that, if $A$ and $B$ are unital nuclear separable simple $\text{TAF}\,\,\,{{C}^{*}}$-algebras with the unique normalized trace satisfying the $\text{UCT}$ and with ${{K}_{1}}\left( A \right)\,=\,{{K}_{1}}\left( B \right)$, and $A$ and $B$ have the same rational (scaled ordered) ${{K}_{0}}$, then $A\,\cong \,B$. Similar results are also obtained for some cases in which ${{K}_{0}}$ is non-divisible such as ${{K}_{0}}\left( A \right)\,=\,\mathbf{Z}\left[ 1/2 \right]$.