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Register a new userReverse Cholesky factorization and tensor products of nest algebras
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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.
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We show that the tensor product of two cyclic $A_infty$-algebras is, in general, not a cyclic $A_infty$-algebra, but an $A_infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an $A_infty$-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $A_infty$-algebra can be thought of as an $A_infty$-algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic $A_infty$-algebras are not necessarily trivial.
Given two complex Banach spaces $X_1$ and $X_2$, a tensor product of $X_1$ and $X_2$, $X_1tilde{otimes}X_2$, in the sense of J. Eschmeier ([5]), and two finite tuples of commuting operators, $S=(S_1,ldots ,S_n)$ and $T=(T_1,ldots ,T_m)$, defined on $X_1$ and $X_2$ respectively, we consider the $(n+m)$-tuple of operators defined on $X_1tilde{otimes}X_2$, $(Sotimes I,Iotimes T)= (S_1otimes I,ldots ,S_notimes I,Iotimes T_1,ldots ,I otimes T_m)$, and we give a description of the semi-Browder joint spectra introduced by V. Kordula, V. Muller and V. Rako$check{c}$evi$acute{ c}$ in [7] and of the split semi-Browder joint spectra (see section 3), of the $(n+m)$-tuple $(Sotimes I ,Iotimes T)$, in terms of the corresponding joint spectra of $S$ and $T$. This result is in some sense a generalization of a formula obtained for other various Browder spectra in Hilbert spaces and for tensor products of operators and for tuples of the form $(Sotimes I ,Iotimes T)$. In addition, we also describe all the mentioned joint spectra for a tuple of left and right multiplications defined on an operator ideal between Banach spaces in the sense of [5].
We study ring-theoretic (in)finiteness properties -- such as emph{Dedekind-finiteness} and emph{proper infiniteness} -- of ultraproducts (and more generally, reduced products) of Banach algebras. Whilst we characterise when an ultraproduct has these ring-theoretic properties in terms of its underlying sequence of algebras, we find that, contrary to the $C^*$-algebraic setting, it is not true in general that an ultraproduct has a ring-theoretic finiteness property if and only if ultrafilter many of the underlying sequence of algebras have the same property. This might appear to violate the continuous model theoretic counterpart of {L}oss Theorem; the reason it does not is that for a general Banach algebra, the ring theoretic properties we consider cannot be verified by considering a bounded subset of the algebra of emph{fixed} bound. For Banach algebras, we construct counter-examples to show, for example, that each component Banach algebra can fail to be Dedekind-finite while the ultraproduct is Dedekind-finite, and we explain why such a counter-example is not possible for $C^*$-algebras. Finally the related notion of having textit{stable rank one} is also studied for ultraproducts.
The aim of this article is to characterize unitary increment process by a quantum stochastic integral representation on symmetric Fock space. Under certain assumptions we have proved its unitary equivalence to a Hudson-Parthasarathy flow.
Given two correspondences X and Y, we show that (under mild hypotheses) the Cuntz-Pimsner algebra of the tensor product of X and Y embeds as a certain subalgebra of the tensor product of the Cuntz-Pimsner algebra of X and the Cuntz=Pimsner algebra of Y. Furthermore, this subalgebra can be described in a natural way in terms of the gauge actions on the Cuntz-Pimsner algebras. We explore implications for graph algebras, crossed products by the integers, and crossed products by completely positive maps. We also give a new proof of a result of Kaliszewski and Quigg related to coactions on correspondences.
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