Norming Algebras and Automatic Complete Boundedness of Isomorphisms of Operator Algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if A_1 and A_2 are operator algebras, then any bounded epimorphism of A_1 onto A_2 is completely bounded provided that A_2 contains a norming C*-subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a C*-algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a C*-algebra is similar to a *-representation precisely when the image operator algebra \lambda-norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras A_i of C*-diagonals (C_i,D_i) (i=1,2) satisfying D_i \subseteq A_i \subseteq C_i extends uniquely to a *-isomorphism of the C*-algebras generated by A_1 and A_2; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.Comment: 9 page

STRUCTURE FOR REGULAR INCLUSIONS

We study pairs (C,D) of unital C∗-algebras where D is an abelian C∗-subalgebra of C which is regular in C in the sense that the span of {v 2 C : vDv∗ [ v∗Dv D} is dense in C. When D is a MASA in C, we prove the existence and uniqueness of a completely positive unital map E of C into the injective envelope I(D) of D whose restriction to D is the identity on D. We show that the left kernel of E, L(C,D), is the unique closed two-sided ideal of C maximal with respect to having trivial intersection with D. When L(C,D) = 0, we show the MASA D norms C in the sense of Pop-Sinclair-Smith. We apply these results to significantly extend existing results in the literature on isometric isomorphisms of norm-closed subalgebras which lie between D and C. The map E can be used as a substitute for a conditional expectation in the construction of coordinates for C relative to D. We show that coordinate constructions of Kumjian and Renault which relied upon the existence of a faithful conditional expectation may partially be extended to settings where no conditional expectation exists. As an example, we consider the situation in which C is the reduced crossed product of a unital abelian C∗-algebra D by an arbitrary discrete group acting as automorphisms of D. We charac- terize when the relative commutant Dc of D in C is abelian in terms of the dynamics of the action of and show that when Dc is abelian, L(C,Dc) = (0). This setting produces examples where no conditional expectation of C onto Dc exists. In general, pure states of D do not extend uniquely to states on C. However, when C is separable, and D is a regular MASA in C, we show the set of pure states on D with unique state extensions to C is dense in D. We introduce a new class of well behaved state extensions, the compatible states; we identify compatible states when D is a MASA in C in terms of groups constructed from local dynamics near an element 2 ˆD. A particularly nice class of regular inclusions is the class of C∗-diagonals; each pair in this class has the extension property, and Kumjian has shown that coordinate systems for C∗-diagonals are particularly well behaved. We show that the pair (C,D) regularly embeds into a C∗-diagonal precisely when the intersection of the left kernels of the compatible states is trivial

Characterizing groupoid C*-algebras of non-Hausdorff \'etale groupoids

Given a non-necessarily Hausdorff, topologically free, twisted etale groupoid $(G, L)$, we consider its "essential groupoid C*-algebra", denoted $C^*_{ess}(G, L)$, obtained by completing $C_c(G, L)$ with the smallest among all C*-seminorms coinciding with the uniform norm on $C_c(G^0)$. The inclusion of C*-algebras $(C_0(G^0), C^*_{ess}(G, L))$ is then proven to satisfy a list of properties characterizing it as what we call a "weak Cartan inclusion". We then prove that every weak Cartan inclusion $(A, B)$, with $B$ separable, is modeled by a topologically free, twisted etale groupoid, as above. In another main result we give a necessary and sufficient condition for an inclusion of C*-algebras $(A, B)$ to be modeled by a twisted etale groupoid based on the notion of "canonical states". A simplicity criterion for $C^*_{ess}(G, L)$ is proven and many examples are provided.Comment: New references and a new main result characterizing arbitrary twisted etale groupoid C*-algebras were added. The title was changed to account for the inclusion of the new main result. Still a preliminary versio

TPGITF user's manual

The primary purpose of TPG interface (TPGITF) software is to create test patterns, according to a mathematical model, for a real device. In order for the device actually to be tested or to be accurately simulated, a means is required to transform the TPGITF generated test into forms acceptable as input to logic test equipment or to a software logic simulator. The MACRODATA-200 Logic Test System and the LOGSIM logic simulator are the particular systems supported by the TPGITF software. A description is presented of the TPGITF software processor. It is assumed that the reader is familiar with the TPG, LOGSIM, and TOIL systems. Flow charts are given

INVARIANT SUBSPACES AND HYPER-REFLEXIVITY FOR FREE SEMIGROUP ALGEBRAS

In this paper, we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras. This enables us to prove that they are reflexive, and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces. Such algebras are called hyper-reflexive. This property is very strong, but it has been established in only a very few cases. Moreover the prototypes of this class of algebras are the natural candidate for a non-commutative analytic Toeplitz algebra on n variables. The case we make for this analogy is very compelling. In particular, in this paper, the key to the invariant subspace analysis is a good analogue of the Beurling theorem for invariant subspaces of the unilateral shift. This leads to a notion of inner-outer factorization in these algebras. In a sequel to this paper [13], we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in Cn

Unique Pseudo-Expectations for C∗C^*-Inclusions

Given an inclusion D $\subseteq$ C of unital C*-algebras, a unital completely positive linear map $\Phi$ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. The set PsExp(C,D) of all pseudo-expectations is a convex set, and for abelian D, we prove a Krein-Milman type theorem showing that PsExp(C,D) can be recovered from its extreme points. When C is abelian, the extreme pseudo-expectations coincide with the homomorphisms of C into I(D) which extend the inclusion of D into I(D), and these are in bijective correspondence with the ideals of C which are maximal with respect to having trivial intersection with D. Natural classes of inclusions have a unique pseudo-expectation (e.g., when D is a regular MASA in C). Uniqueness of the pseudo-expectation implies interesting structural properties for the inclusion. For example, when D $\subseteq$ C $\subseteq$ B(H) are W*-algebras, uniqueness of the pseudo-expectation implies that D' $\cap$ C is the center of D; moreover, when H is separable and D is abelian, we characterize which W*-inclusions have the unique pseudo-expectation property. For general inclusions of C*-algebras with D abelian, we characterize the unique pseudo-expectation property in terms of order structure; and when C is abelian, we are able to give a topological description of the unique pseudo-expectation property. Applications include: a) if an inclusion D $\subseteq$ C has a unique pseudo-expectation $\Phi$ which is also faithful, then the C*-envelope of any operator space X with D $\subseteq$ X $\subseteq$ C is the C*-subalgebra of C generated by X; b) for many interesting classes of C*-inclusions, having a faithful unique pseudo-expectation implies that D norms C. We give examples to illustrate the theory, and conclude with several unresolved questions.Comment: 26 page

Isomorphisms of lattices of Bures-closed bimodules over Cartan MASAs

For i = 1; 2, let (Mi;Di) be pairs consisting of a Cartan MASA Di in a von Neumann algebra Mi, let atom(Di) be the set of atoms of Di, and let Si be the lattice of Bures-closed Di bimodules in Mi. We show that when Mi have separable preduals, there is a lattice isomorphism between S1 and S2 if and only if the sets f(Q1;Q2) 2 atom(Di) atom(Di) : Q1MiQ2 6= (0)g have the same cardinality. In particular, when Di is nonatomic, Si is isomorphic to the lattice of projections in L1([0; 1];m) where m is Lebesgue measure, regardless of the isomorphism classes of M1 and M2

Absolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra $A_n$. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a $*$-extendible representation $\sigma$. A $*$-extendible representation of $A_n$ is ``regular'' if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given $*$-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of $\sigma(A_n)$ is weak-$*$ dense in the unit ball of the associated free semigroup algebra if and only if $\sigma$ is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts.Comment: 26 pages, prepared with LATeX2e, submitted to Journal of Functional Analysi

Implementation of Large Scale Integrated (LSI) circuit design software

Portions of the Computer Aided Design and Test system, a collection of Large Scale Integrated (LSI) circuit design programs were modified and upgraded. Major modifications were made to the Mask Analysis Program in the form of additional operating commands and file processing options. Modifications were also made to the Artwork Interactive Design System to correct some deficiencies in the original program as well as to add several new command features related to improving the response of AIDS when dealing with large files. The remaining work was concerned with updating various programs within CADAT to incorporate the silicon on sapphire silicon gate technology