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Severino T. Melo

Publications and source records attributed to Severino T. Melo.

12 recordsLinked to original sources
arXiv2025

On Rieffel's conjecture characterizing a deformed algebra as Heisenberg smooth operators

Let $\mathscr{A}$ be a unital C$^*$-algebra and $E_n$ be the Hilbert $\mathscr{A}$-module defined as the completion of the $\mathscr{A}$-valued Schwartz function space $\mathcal{S}^\mathscr{A}(\mathbb{R}^n)$ with respect to the norm $\|f\|_2 := \left\| \int_{\mathbb{R}^n} f(x)^*f(x) \, dx \right\|_\mathscr{A}^{1 / 2}$. Also, let $\text{Ad }\mathcal{U}$ be the canonical action of the $(2n + 1)$-dimensional Heisenberg group by conjugation on the algebra of adjointable operators on $E_n$ and let $J$ be a skew-symmetric linear transformation on $\mathbb{R}^n$. We characterize the smooth vectors under $\text{Ad }\mathcal{U}$ which commute with a certain algebra of right multiplication operators $R_h$, with $h \in \mathcal{S}^\mathscr{A}(\mathbb{R}^n)$, where the product is ``twisted'' with respect to $J$ according to a deformation quantization procedure introduced by M.A. Rieffel. More precisely, we establish that they coincide with an algebra of left multiplication operators and show that this solves, in particular, a conjecture posed by Rieffel.

math.OA↗
arXiv2025

Differential Norms and Rieffel Algebras

We develop criteria to guarantee uniqueness of the C$^*$-norm on a *-algebra $\mathcal{B}$. Nontrivial examples are provided by the noncommutative algebras of $\mathcal{C}$-valued functions $\mathcal{S}_J^\mathcal{C}(\mathbb{R}^n)$ and $\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$ defined by M.A. Rieffel via a deformation quantization procedure, where $\mathcal{C}$ is a C$^*$-algebra and $J$ is a skew-symmetric linear transformation on $\mathbb{R}^n$ with respect to which the usual pointwise product is deformed. In the process, we prove that the Fréchet *-algebra topology of $\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$ can be generated by a sequence of submultiplicative *-norms and that, if $\mathcal{C}$ is unital, this algebra is closed under the C$^\infty$-functional calculus of its C$^*$-completion. We also show that the algebras $\mathcal{S}_J^\mathcal{C}(\mathbb{R}^n)$ and $\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$ are spectrally invariant in their respective C$^*$-completions, when $\mathcal{C}$ is unital. As a corollary of our results, we obtain simple proofs of certain estimates in $\mathcal{B}_J^\mathcal{C}(\mathbb{R}^n)$.

math.OA↗
arXiv2019

The principal-symbol index map for an algebra of pseudodifferential operators

A C*algebra A generated by a class of zero-order classical pseudodifferential operator on a cylinder RxB, where B is a compact riemannian manifold, containing operators with periodic symbols, is considered. A description of the K-theory index map associated to the continuous extension to A of the principal-symbol map is given. That index map takes values in K_0 of the commutator ideal E of the algebra, which is isomorphic to Z^2. It maps the K_1-class of an operator invertible modulo E to the Fredholm indices of a pair of elliptic pseudodifferentail operators on SxB, where S denotes the circle.

math.DG↗
arXiv2016

Operators with analytic orbit for the torus action

Let $T^n$ denote the n-dimensional torus. The class of the bounded operators on $L^2(T^n)$ with analytic orbit under the action of $T^n$ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on $T^n$ whose discrete symbol $(a_j)_{j\in Z^n}$ is uniformly analytic, in the sense that there exists $C>1$ such that the derivatives of $a_j$ satisfy $|\partial^αa_j(x)|\leq C^{1+|α|}α!$ for all $x\in T^n$, all $j\in Z^n$ and all $α\in N^n$. This implies that this class of pseudodifferential operators is a spectrally invariant *-subalgebra of the algebra of all bounded operators on $L^2(T^n)$.

math.FA↗
arXiv2013

$K$-Theory of Boutet de Monvel algebras with classical SG-symbols on the half space

We compute the $K$-groups of the $C^{*}$-algebra of bounded operators generated by the Boutet de Monvel operators with classical SG-symbols of order (0,0) and type 0 on $\mathbb{R}_{+}^{n}$, as defined by Schrohe, Kapanadze and Schulze. In order to adapt the techniques used in Melo, Nest, Schick and Schrohe's work on the K-theory of Boutet de Monvel's algebra on compact manifolds, we regard the symbols as functions defined on the radial compactifications of $\mathbb{R}_{+}^{n}\times\mathbb{R}^{n}$ and $\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}$. This allows us to give useful descriptions of the kernel and the image of the continuous extension of the boundary principal symbol map, which defines a $C^{*}$-algebra homomorphism. We are then able to compute the $K$-groups of the algebra using the standard K-theory six-term cyclic exact sequence associated to that homomorphism.

math.OA↗
arXiv2011

The Chern-Connes Character for pseudodifferential operators on the sphere

We compute the Chern-Connes character (a map from the $K$-theory of a C$^*$-algebra under the action of a Lie group to the cohomology of its Lie algebra) for the $L^2$-norm closure of the algebra of all classical zero-order pseudodifferential operators on the sphere under the canonical action of ${\rm SO}(3)$. We show that its image is $\mathbb{R}$ if the trace is the integral of the principal symbol.

math.OA↗
arXiv2008

Cordes characterization for pseudodifferential operators with symbols valued on a noncommutative C*-algebra

Given a separable unital C*-algebra A, let E denote the Banach-space completion of the A-valued Schwartz space on Rn with norm induced by the A-valued inner product $ =\int f(x)^*g(x) dx$. The assignment of the pseudodifferential operator B=b(x,D) with A-valued symbol b(x,ξ) to each smooth function with bounded derivatives b defines an injective mapping O, from the set of all such symbols to the set of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E. It is known that O is surjective if A is commutative. In this paper, we show that, if O is surjective for A, then it is also surjective for the algebra of k-by-k matrices with entries in A.

math.OA↗
arXiv2006

Boutet de Monvel's Calculus and Groupoids I

Can Boutet de Monvel's algebra on a compact manifold with boundary be obtained as the algebra $Ψ^0(G)$ of pseudodifferential operators on some Lie groupoid $G$? If it could, the kernel ${\mathcal G}$ of the principal symbol homomorphism would be isomorphic to the groupoid {$C^*$-algebra} $C^*(G)$. While the answer to the above question remains open, we exhibit in this paper a groupoid $G$ such that $C^*(G)$ possesses an ideal ${\mathcal I}$ isomorphic to ${\mathcal G}$. %ES, the kernel of the principal symbol homomorphism on Boutet de Monvel's algebra. In fact, we prove first that ${\mathcal G}\simeqΨ\otimes{\mathcal K}$ with the $C^*$-algebra $Ψ$ generated by the zero order pseudodifferential operators on the boundary and the algebra $\mathcal K$ of compact operators. As both $Ψ\otimes \mathcal K$ and $\mathcal I$ are extensions of $C(S^*Y)\otimes {\mathcal{K}}$ by ${\mathcal{K}}$ ($S^*Y$ is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.

math.KT↗
arXiv2006

C*-Algebra-valued-symbol pseudodifferential operators: abstract characterizations

Given a separable unital C*algebra $C$, let $E_n$ denote the Hilbert module equal to the completion of the Schwartz space of rapidly decreasing smooth functions from $R^n$ to $C$ equipped with the $C$-valued inner product given by integration. Let $B$ denote the space of all smooth functions with bounded derivatives from $R^n$ to $C$. For each $a$ in $B$, let $O(a)$ denote the pseudodifferential operator of symbol $a$. $O$ maps $B$ to $H$, the set of all adjointable operators on $E_n$ which have smooth orbit under the canonical action of the Heisenberg group. We construct a left inverse for $O$, $S:H\to B$, and prove that $S$ is an inverse for $O$ if $C$ is commutative. The case when $C$ is the complex numbers was proven by Cordes in 1979. As a consequence, we prove, for commutative separable unital C*algebras, a characterization of a certain class of pseudodifferential operators conjectured by Rieffel in 1993.

math.OA↗
arXiv2005

K-Theory of pseudodifferential operators with semi-periodic symbols

Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a\in C[-\infty,+\infty], (ii) all multiplications by 2π-periodic continuous functions and (iii) all Fourier multipliers F^{-1}b(M)F, where F denotes the Fourier transform and b is in C[-\infty,+\infty]. The Fredholm property for operators in A is governed by two symbols, the principal symbol σand an operator-valued symbol γ. We give two proofs of the fact that K0(A) and K1(A) are isomorphic, respectively, to Z and Z\oplus Z: by computing the connecting mappings in the standard K-theory six-term exact sequences associated to σand to γ. For the second computation, we prove that the image of γis isomorphic to the direct sum of two copies of the crossed product of C[-\infty,+\infty] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.

math.OA↗
arXiv2005

A K-Theoretic Proof of Boutet de Monvel's Index Theorem for Boundary Value Problems

We study the C*-closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with non-empty boundary. We find short exact sequences in K-theory 0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.

math.KT↗