arXiv subjects
Publications and source records attributed to Larry B. Schweitzer.
Let $A$ be a dense Fréchet subalgebra of the $C^\star$-algebra of compact operators $\cal K$ on a seprable Hilbert space. Assume that $A$ is spectral invariant in $\cal K$. We show that every algebraically cyclic subrepresentation of a topologically irreducible representation of $A$ is contained in a $\cal K$-module.
We construct dense Banach subalgebras $A$ of the null sequence algebra $c_0$ which are spectral invariant, but do not satisfy the $D_1$-condition $\|ab \|_A \leq K(\|a\|_{\infty} \|b\|_A + \|a \|_A \|b \|_{\infty})$, for all $a, b \in A$. The sequences in $A$ vanish in a skewed manner with respect to an unbounded function $σ\colon {\mathbb N} \rightarrow [1, \infty)$.
We show that a $C^\star$-algebra $B$ contains a dense left or right Fréchet ideal $A$, with $A$ a nuclear locally convex space, if and only if the primitive ideal space Prim$(B)$ of $B$ is discrete and countable, and $B/I$ is finite dimensional for each $I \in $ Prim$(B)$. We show the forward implication holds for a general Banach algebra $B$, if the ideal is assumed two-sided. For $C^\star$-algebras, we construct all two-sided dense nuclear ideals by defining a set of matrix-valued Schwartz functions on the countable discrete space Prim$(B)$.
We construct a sequence of $n-1$ cyclic exact sequences that can be used to compute the $K$-theory of the $C^\star$-algebra crossed product $A \ltimes {\mathbb Z}_n$.
We construct an algebra $A=\ell^{\infty \infty}({\Bbb Z})$ of smooth functions which is dense in the pointwise multiplication algebra $\ell^\infty({\Bbb Z})$ of sup-norm bounded functions on the integers $\Bbb Z$. The algebra $A$ properly contains the sum of the algebra $A_c=\ell_c^\infty({\Bbb Z})$ and the ideal ${\cal S}({\Bbb Z})$, where $A_c$ is the algebra of finite linear combinations of projections in $\ell^\infty({\Bbb Z})$ and ${\cal S}({\Bbb Z})$ is the pointwise multiplication algebra of Schwartz functions. The algebra $A$ is characterized as the set of functions whose "first derivatives" vanish rapidly at each point in the Stone-${\check {\rm C}}$ech compactification of $\Bbb Z$.
We define the notion of strong spectral invariance for a dense Frechet subalgebra A of a Banach algebra B. We show that if A is strongly spectral invariant in a C*-algebra B, and G is a compactly generated polynomial growth Type R Lie group, not necessarily connected, then the smooth crossed product G\rtimes A is spectral invariant in the C*-crossed product G\rtimes B. Examples of such groups are given by finitely generated polynomial growth discrete groups, compact or connected nilpotent Lie groups, the group of Euclidean motions on the plane, the Mautner group, or any closed subgroup of one of these. Our theorem gives the spectral invariance of G\rtimes A if A is the set of C^{\infty}-vectors for the action of G on B, or if B= C_{0}(M), and A is a set of G-differentiable Schwartz functions S(M) on M. This gives many examples of spectral invariant dense subalgebras for the C*-algebras associated with dynamical systems. We also obtain relevant results about exact sequences, subalgebras, tensoring by smooth compact operators, and strong spectral invariance in L_{1}(G, B).
We show that if E is a Frechet G\rtimes S(M)-module, for which the canonical map from the projective completion G\rtimes S(M) {\widehat \otimes} E to E is surjective, then every element of E can be written as a finite sum of elements of the form ae where e\in E and a is an element of the smooth crossed product G\rtimes S(M). We require that the Schwartz functions S(M) vanish rapidly with repsect to a continuous, proper map \s : M ---> [0, \infty).
The author's recent results on spectral invariant dense subalgebras of C*-algebras associated with dynamical systems are summarized. If G is a compactly generated polynomial growth Type R Lie group, and the action of G on S(M) (Schwartz functions on a locally compact G-space M) is tempered in a certain sense, then there is a natural smooth crossed product S(G X M) which is dense and spectral invariant in the C*-crossed product C*(G X M).
Let A be a dense Frechet *-subalgebra of a C*-algebra B. (We do not require Frechet algebras to be m-convex.) Let G be a Lie group, not necessarily con- nected, which acts on both $A$ and B by *-automorphisms, and let \s be a sub- multiplicative function from G to the nonnegative real numbers. If \s and the action of G on A satisfy certain simple properties, we define a dense Frechet *-subalgebra G\rtimes^{\s} A of the crossed product L^{1}(G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in \s. We give conditions on the action of G on A which imply the m-convexity of the dense subalgebra G\rtimes^{\s}A. A locally convex algebra is said to be m-con- vex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Frechet algebra, and is useful in modern operator theory. If G acts as a transformation group on a manifold M, we develop a class of dense subalgebras for the crossed product L^{1}(G, C_{0}(M)), where C_{0}(M) denotes the continuous functions on M vanishing at infinity with the sup norm topology.We define Schwartz functions S(M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense m-convex Frechet *-subalgebra G\rtimes^ {\s} S(M) of rapidly vanishing, G-differentiable functions from G to S(M). If the reciprocal of \s is in L^{p}(G) for some p, we prove that our group algebras S^{\s}(G) are nuclear Frechet spaces, and that G\rtimes^{\s}A is the projective completion S^{\s}(G) \otimes A.
We give a short and very general proof of the fact that the property of a dense Fréchet subalgebra of a Banach algebra being local, or closed under the holomorphic functional calculus in the Banach algebra, is preserved by tensoring with the $n\times n$ matrix algebra of the complex numbers.
We give an example of a dense, simple, unital Banach subalgebra $A$ of the irrational rotation C*-algebra $B$, such that $A$ is not a spectral subalgebra of $B$. This answers a question posed in T.W. Palmer's paper [1].
We show that the Thom isomorphism and the Pimsner-Voiculescu exact sequence both hold for smooth crossed products of Frechet algebras by R and Z respectively. We also obtain the same results for L^{1}-crossed products of Banach algebras by R and Z.