Operator algebras and subproduct systems arising from stochastic matrices

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• 作者：Adam Dor-On ^{adamd@math.bgu.ac.il" class="auth_mail} ; Daniel Markiewicz ; ^{danielm@math.bgu.ac.il" class="auth_mail}
• 关键词：primary ; 47L30 ; 46L55 ; 46L57 ; secondary ; 46L08 ; 60J10
• 刊名：Journal of Functional Analysis
• 出版年：15 August 2014
• 年：2014
• 卷：267
• 期：4
• 页码：1057-1120
• 全文大小：685 K

文摘

We study subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselter's generalization of the Cuntz–Pimsner C*-algebra to the context of subproduct systems. Suppose that X   and Y   are Arveson–Stinespring subproduct systems associated to two stochastic matrices over a countable set 惟  , and let T₊(X)${\mathcal{T}}_{+}(X)$ and T₊(Y)${\mathcal{T}}_{+}(Y)$ be their tensor algebras. We show that every algebraic isomorphism from T₊(X)${\mathcal{T}}_{+}(X)$ onto T₊(Y)${\mathcal{T}}_{+}(Y)$ is automatically bounded. Furthermore, T₊(X)${\mathcal{T}}_{+}(X)$ and T₊(Y)${\mathcal{T}}_{+}(Y)$ are isometrically isomorphic if and only if X   and Y   are unitarily isomorphic up to a *-automorphism of 鈩?sup>∞(惟)${\mathrm{鈩?/mi>}}^{\infty }(惟)$. When 惟   is finite, we prove that T₊(X)${\mathcal{T}}_{+}(X)$ and T₊(Y)${\mathcal{T}}_{+}(Y)$ are algebraically isomorphic if and only if there exists a similarity between X   and Y   up to a *-automorphism of 鈩?sup>∞(惟)${\mathrm{鈩?/mi>}}^{\infty }(惟)$. Moreover, we provide an explicit description of the Cuntz–Pimsner algebra O(X)$\mathcal{O}(X)$ in the case where 惟 is finite and the stochastic matrix is essential.