In this paper we address some basic questions of the Banach space structure of the nest algebras in the trace class; in particular, we study whether any two of them are isomorphic to each other, and show that the nest algebras in the trace class have bases. We construct three non-isomorphic examples of nest algebras in $c_1$; present a new proof of the primarity of $c_1$ (Arazy, [Ar1], [Ar2]), and prove that $K(H)$, and the nest algebras in $B(H)$ are primary.
Let N_1 (resp.N_2) be a nest A (resp. B) be the corresponding nest algebra, A_0 (resp. B_0) be the subalgebra of compact operators. We prove that the nests N_1, N_2 are isomorphic if and only if the algebras A, B are weakly-* Morita equivalent if and only if the algebras A_0, B_0 are strongly Morita equivalent. We characterize the nest isomorphisms which implement stable isomorphism between the corresponding nest algebras.
We prove that every positive semidefinite matrix over the natural numbers that is eventually 0 in each row and column can be factored as the product of an upper triangular matrix times a lower triangular matrix. We also extend some known results about factorization with respect to tensor products of nest algebras. Our proofs use the theory of reproducing kernel Hilbert spaces.
Using the notion of passport of a normed Boolean algebra, necessary and sufficient conditions for a $\ast$-isomorphism of $\ast$-algebras of log-integrable measurable functions are found.
Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a~Calkin algebra has been found so far. This naturally led to the question of possible limitations of such assignments. In the present note we provide examples of unital Banach algebras meeting the necessary density condition for being the Calkin algebra of a separable Banach space that are not isomorphic to Calkin algebras of such spaces, nonetheless. The examples may …
The standard view is that PDEs are much more complex than ODEs, but, as will be shown below, for finite derivatives this is not true. We consider the $C^*$-algebras ${\mathscr H}_{N,M}$ consisting of $N$-dimensional finite differential operators with $M\times M$-matrix-valued bounded periodic coefficients. We show that any ${\mathscr H}_{N,M}$ is $*$-isomorphic to the universal uniformly hyperfinite algebra (UHF algebra) $ \bigotimes_{n=1}^{\infty}\mathbb{C}^{n\times n}. $ This is a complete characterization of the differential algebras. In particular, for different $N,M\in\mathbb{N}$ the alge…
We extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra $\mathcal A$, we charaterise the maximal and the minimal $\mathcal A$-bimodules having a given essential support function or support function pair. These characterisations are complete except for the minimal $\mathcal A$-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the wea…
We establish a general result about extending a right invertible row over a Banach algebra to an invertible matrix. This is applied to the computation of right topological stable rank of a split exact sequence. We also introduce a quantitative measure of stable rank. These results are applied to compute the right (left) topological stable rank for all nest algebras. This value is either 2 or infinity, and rtsr(T(N)) = 2 occurs only when N is of ordinal type less than omega^2 and the dimensions of the atoms grows sufficiently quickly. We introduce general results on `partial matrix algebras' ov…
This paper is a comprehensive study of the nest representations for the free semigroupoid algebra $\flgee$ of countable directed graph $G$ as well as its norm-closed counterpart, the tensor algebra $\T^{+}(G)$. We prove that the finite dimensional nest representations separate the points in $\flgee$, and a fortiori, in $\T^{+}(G)$. The irreducible finite dimensional representations separate the points in $\flgee$ if and only if $G$ is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every verte…
We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the invertible group $A^{-1}$ of a unital Banach algebra $A$ onto an open subgroup of the invertible group $B^{-1}$ of a unital Banach algebra $B$, then $T$ is extended to a real-linear isometry up to translation between these Banach algebras. We consider multiplicativity or unti-multiplicativity of the isometry. Note that a unital linear isometry between unital semisimple commutative Banach algebra need be multiplicative. On the other hand, we show that if $A$ is commutative and $A$ or $B$ are semisimple, then $(T(…
We shall consider a locally compact groupoid endowed with a Haar system and having proper orbit space. We shall construct a groupoid C*-algebra which is independent of the Haar system (up to a *-isomorphism).
Let ${\mathcal N}$ be a nest on a complex Banach space $X$ and let $\mbox{ Alg}{\mathcal N}$ be the associated nest algebra. We say that an operator $Z\in \mbox{ Alg}{\mathcal N}$ is an all-derivable point of $\mbox{ Alg}{\mathcal N}$ if every linear map $δ$ from $\mbox{ Alg}{\mathcal N}$ into itself derivable at $Z$ (i.e. $δ$ satisfies $δ(A)B+Aδ(B)=δ(Z)$ for any $A,B \in \mbox{ Alg}{\mathcal N}$ with $AB=Z$) is a derivation. In this paper, it is shown that every injective operator and every operator with dense range in $\mbox{Alg}{\mathcal N}$ are all-derivable points of $\mbox{Alg}{\mathcal …
We prove that two unital dual operator algebras A, B are stably isomorphic if and only if they are Delta-equivalent, if and only if they have completely isometric normal representations a, b on Hilbert spaces H, K respectively and there exists a ternary ring of operators M \subset B(H,K) such that a(A)=[M* b(B) M]^{-w^*} and b(B)=[M a(A) M*]^{-w^*}.
In this paper we prove that several operator algebras are completely isomorphic to each other; e.g., the $C^*_λ(F_k)$, $k\geq 2$, the $C^*$-algebras generated by the regular left representation $λ:F_k\to B(\ell_2(F_k))$, are completely isomorphic to each other. We also study the ``non-commutative'' analytic spaces introduced by G. Popescu [Po], and give applications to Popescu's version of Von Neumann's inequality.
Let ${\mathcal N}$ and ${\mathcal M}$ be nests on Banach spaces $X$ and $Y$ over the (real or complex) field $\mathbb F$ and let $\mbox{\rm Alg}{\mathcal N}$ and $\mbox{\rm Alg}{\mathcal M}$ be the associated nest algebras, respectively. It is shown that a map $Φ:{\rm Alg}{\mathcal N}\rightarrow{\rm Alg}{\mathcal M}$ is a Lie ring isomorphism (i.e., $Φ$ is additive, Lie multiplicative and bijective) if and only if $Φ$ has the form $Φ(A) = TAT^{-1} + h(A)I$ for all $A\in \mbox{\rm Alg}{\mathcal N}$ or $Φ(A)=-TA^*T^{-1}+h(A)I$ for all $A\in \mbox{\rm Alg}{\mathcal N}$, where $h$ is an additive f…
Suppose that $\mathscr{A}$ is an operator algebra on a Hilbert space $H$. An element $V$ in $\mathscr{A}$ is called an all-derivable point of $\mathscr{A}$ for the strong operator topology if every strong operator topology continuous derivable mapping $ϕ$ at $V$ is a derivation. Let $\mathscr{N}$ be a complete nest on a complex and separable Hilbert space $H$. Suppose that $M$ belongs to $\mathscr{N}$ with $\{0\}\neq M\neq\ H$ and write $\hat{M}$ for $M$ or $M^{\bot}$. Our main result is: for any $Ω\in alg\mathscr{N}$ with $Ω=P(\hat{M})ΩP(\hat{M})$, if $Ω|_{\hat{M}}$ is invertible in $alg\math…
We study the *-double functor between the categories of associative and involutive algebras. It is proved that an associative algebra is isomorphic to a subalgebra of a $C\sp*$-algebra if and only if its *-double is *-isomorphic to a *-subalgebra of a $C\sp*$-algebra. Some applications in the theory of operator algebras are presented. In particular each operator algebra is shown to be completely boundedly isomorphic to an operator algebra $B$ with the greatest $C\sp*$-subalgebra consisting of the multiples of the unit and such that each element in $B$ is determined by its module up to a scalar…
The generalized state space $ S_{\mathcal{H}}(\mathcal{\mathcal{A}})$ of all unital completely positive (UCP) maps on a unital $C^*$-algebra $\mathcal{A}$ taking values in the algebra $\mathcal{B}(\mathcal{H})$ of all bounded operators on a Hilbert space $\mathcal{H}$, is a $C^\ast$-convex set. In this paper, we establish a connection between $C^\ast$-extreme points of $S_{\mathcal{H}}(\mathcal{A})$ and a factorization property of certain algebras associated to the UCP map. In particular, this factorization property of some nest algebras is used to give a complete characterization of those $C^…
A subalgebra $\mathcal{A}$ of a $C^*$-algebra $\mathcal{M}$ is logmodular (resp. has factorization) if the set $\{a^*a; a\text{ is invertible with }a,a^{-1}\in\mathcal{A}\}$ is dense in (resp. equal to) the set of all positive and invertible elements of $\mathcal{M}$. There are large classes of well studied algebras, both in commutative and non-commutative settings, which are known to be logmodular. In this paper, we show that the lattice of projections in a von Neumann algebra $\mathcal{M}$ whose ranges are invariant under a logmodular algebra in $\mathcal{M}$, is a commutative subspace latti…
Let $\mathcal{X}$ be a Banach space over the complex field $\mathbb{C}$ and $\mathcal{B(X)}$ be the algebra of all bounded linear operators on $\mathcal{X}$. Let $\mathcal{N}$ be a non-trivial nest on $\mathcal{X}$, ${\rm Alg}\mathcal{N}$ be the nest algebra associated with $\mathcal{N}$, and $L\colon {\rm Alg}\mathcal{N}\longrightarrow \mathcal{B(X)}$ be a linear mapping. Suppose that $p_n(x_1,x_2,\cdots,x_n)$ is an $(n-1)$-th commutator defined by $n$ indeterminates $x_1, x_2, \cdots, x_n$. It is shown that $L$ satisfies the rule $$ L(p_n(A_1, A_2, \cdots, A_n))=\sum_{k=1}^{n}p_n(A_1, \cdots…
In this paper we study the structure of the $C^*$-algebra, generated by the representation of the paths semigroup on a partially ordered set (poset) and get the net of isomorphic $C^*$-algebras over this poset. We construct the extensions of this algebra, such that the algebra is an ideal in that extensions and quotient algebras are isomorphic to the Cuntz algebra.
