Narrow Results

We establish several characterizations of tracial functionals $\phi$ on the finite $C^{\ast }$-algebra ${𝕄}_n$ (that is, $\phi =k$ tr for some number $k>0$) via any one of the inequalities $\phi (A^{\alpha }{B}^{1-\alpha })\le \alpha \phi (A)+(1-\alpha )\phi (B)$ and $\phi (|e^A|)\le \phi (e^{\mathrm{\text{Re}}A})$, which are well-known when $\phi =\mathrm{\text{tr}}$.

In addition, we characterize the trace on ${𝕄}_n$ among all positive linear functionals $\phi$ with $\phi (I)=n$ through an inequality for determinant. We also establish that such a functional is equal to the usual trace if and only if $\phi ({(B^{\frac{1}{2}}A{B}^{\frac{1}{2}})}^m)\le \phi {(A)}^m\phi {(B)}^m$ for all positive integers $m$ and all $A,B\in {𝕄}_n^{+}$. Furthermore, we show that there is no state $\phi$ on ${𝕄}_n,n\ge 2$ such that $\phi (B^{1/2}A{B}^{1/2})\le \phi (A)\phi (B)$ for all $A,B\in {𝕄}_n^{+}$.

Finally, we establish that for a positive linear functional $\phi$ on a $C^{\ast }$-algebra ${𝒜}$, the following conditions are equivalent: (i) $\phi$ is tracial; (ii) $\phi (e^{AB}-I)\ge 0$ for all $A,B\in {{𝒜}}^{+}$. A new criterion for the commutativity of $C^{\ast }$-algebras is also provided.

A theory of representation of semiautomata by canonical words and equivalences was developed in [7]. That work was motivated by trace-assertion specifications of software modules, but its focus was entirely on the underlying mathematical model. In the present paper we extend that theory to automata with Moore and Mealy outputs, and show how to apply the extended theory to the specification of modules. In particular, we present a unified view of the trace-assertion methodology, as guided by our theory. We illustrate this approach, and some specific issues, using several nontrivial examples. We include a discussion of finite versus infinite modules, methods of error handling, some awkward features of the trace-assertion method, and a comparison to specifications by automata. While specifications by trace assertions and automata are equivalent in power, there are cases where one approach appears to be more natural than the other. We conclude that, for certain types of system modules, formal specification by automata, as opposed to informal state machines, is not only possible, but practical.

Let τ be a faithful semi-finite normal trace on a semi-finite von Neumann algebra, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality states τ(f(a* xa)) ≤ τ(a* f(x)a) for a contraction a and a self-adjoint operator x. Under certain strict convexity assumption on f(t), we will study when this inequality reduces to the equality.

It is well known that the elements of PSL(2, ℂ) are classified as elliptic, parabolic or loxodromic according to the dynamics and their fixed points; these three types are also distinguished by their trace. If we now look at the elements in PU(2,1), then there are the equivalent notions of elliptic, parabolic or loxodromic elements; Goldman classified these transformations by their trace. In this work we extend the classification of elements of PU(2,1) to all of PSL(3, ℂ); we also extend to this setting the theorem that classifies them according to their trace. We use the notion of limit set introduced by Kulkarni, and calculate the limit set of every cyclic subgroup of PSL(3, ℂ) acting on . Given a classical Kleinian group it is possible to "suspend" this group to a subgroup of PSL(3, ℂ); we also calculate the limit set of this suspended group.

Given a C*-algebra with trace τ, we compute the first and second variation formulas for the p-energy functional F_p of the unitary group of , for p = 2n an even integer, namely:

For finite dimensional abelian subalgebras of a finite von Neumann algebra, we obtain the value of conditional relative entropy defined by Choda. We also consider the modified invariant defined by Pimsner and Popa.

We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K₀ group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the latter result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag–Solitar groups BS(m, n).

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms,

Let be a semi-finite von Neumann algebra equipped with a faithful semi-finite normal trace τ, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality in our previous work states τ(f(a*xa)) ≤ τ(a*f(x)a) (as long as the both sides are well-defined) for a contraction and a semi-bounded τ-measurable operator x. Validity of this inequality for (not necessarily semi-bounded) self-adjoint τ-measurable operators is investigated.

The Kubo–Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras, it is impossible to avoid unbounded operators. In this paper, we try to extend a notion of connections to cover various classes of positive unbounded operators (or unbounded objects such as positive forms and weights) appearing naturally in the setting of von Neumann algebras, and we must keep all the expected properties maintained. This generalization is carried out for the following classes: (i) positive $\tau$-measurable operators (affiliated with a semifinite von Neumann algebra equipped with a trace $\tau$), (ii) positive elements in Haagerup’s $L^p$-spaces and (iii) semifinite normal weights on a von Neumann algebra. Investigation on these generalizations requires some analysis (such as certain upper semi-continuity) on decreasing sequences in various classes. Several results in this direction are proved, which may be of independent interest. Ando studied Lebesgue decomposition for positive bounded operators by making use of parallel sums. Here, such decomposition is obtained in the setting of noncommutative (Hilsum) $L^p$-spaces.

We introduce the $C^{\ast }$-algebra $C^{\ast }(\kappa )$ generated by the Koopman representation $\kappa$ of an étale groupoid $G$ acting on a measure space $(X,\mu )$. We prove that for a level transitive self-similar action $(G,E)$ with $E$ finite and $\left|u{E}^1\right|$ constant, there is an invariant measure $\nu$ on $X=E^{\infty }$ and that $C^{\ast }(\kappa )$ is residually finite dimensional with a normalized self-similar trace. We also discuss $p$-fold similarities of Hilbert spaces in connection to representations of the graph algebra $C^{\ast }(E)$ and self-similar representations of $G$ in connection to the Cuntz–Pimsner algebra $C^{\ast }(G,E)$.

In this paper a Homfly-pt type Laurant polynomial¹ for links in S¹ × D² is defined. This invariant is defined by giving an algebra and a trace, analogous to the Hecke algebra construction of the Homfly-pt polynomial for links in S³.

In this paper, we focus on the characterizations of some properties of fuzzy relations by using the notions of traces. The investigation is two-fold. On one hand, we attempt to reformulate the results obtained by Fodor under weakened conditions. On the other hand, we present some new characterizations of properties, mainly T-asymmetry, T-S-semitransitivity and T-S-Ferrers property, in terms of traces.

We establish a noncommutative Blackwell–Ross inequality for supermartingales under a suitable condition which generalizes Khan’s work to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.

We introduce the notion of acceptable noncommutative random variables and investigate their essential properties. More precisely, we provide several efficient estimation of tail probabilities of sums of noncommutative random variables under some mild conditions. Moreover, we investigate the complete convergence of a sequence of the form $\frac{1}{{\lambda }_n}{\sum }_{i=1}^n{x}_j$ in which $(x_n)$ is a sequence of acceptable noncommutative random variables. Our results extend some classical inequalities as well.

This paper proposes a new approach of multiple and hybrid watermarks using two linked fields of insertion. This approach associates two different series of marks: one is sturdy guaranteeing the supervision of diffused media through the network and testifying to the identity of the owner of the medical image; the other series of marks, which is frail, insuring the integrity, the trace, and the archives of the medical diagnoses. In fact, this approach allows us to be benefited from the advantages of different insertion spaces (spatial and frequency field) and kind of two different fields of watermark. In the spatial field, the diagnosis of the doctor (frail marks) is inserted guaranteeing its records and its trace. In the frequency, the sturdy mark is inserted allowing to ensure the automatic control of the media through the network and testifying to the proprietor of the medical image.

Linear feedback shift registers (LFSRs) are widely used cryptographic primitives for generating pseudorandom sequences. Here, we consider systems which are efficient generalizations of LFSRs and produce pseudorandom vector sequences. We study problems related to the cardinality, existence and construction of these systems and give certain results in this direction.

To each symmetric graded Frobenius superalgebra we associate a W-algebra. We then define a linear isomorphism between the trace of the Frobenius Heisenberg category and a central reduction of this W-algebra. We conjecture that this is an isomorphism of graded superalgebras.

For $n\ge 1$ and $\alpha \in (-1,1)$, let $H^{\alpha ,2}$ be the space of harmonic functions $u$ on the upper half-space ${ℝ}_{+}^{n+1}$ satisfying

We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.